Document [original]

An investigation of

the electromagnetic coupling problem

by means of a rational framework and

selected experiments

vorgelegt von

Wilhelm Rickert, M.Sc.

an der Fakultät V – Verkehrs- und Maschinensysteme

der Technischen Universität Berlin

zur Erlangung des akademischen Grades

Doktor der Ingenieurwissenschaften

– Dr.-Ing. –

genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. Dr.-Ing. U. von Wagner

Gutachter: Prof. Dr. rer. nat. W. H. Müller

Prof. Dr. hab. V. A. Eremeyev

Tag der wissenschaftlichen Aussprache: 27. Februar 2023

Berlin 2023

On the electromagnetic coupling problem III

Abstract

To analyze electromechanical interactions, electromagnetic coupling models are required. Identi-

fying a suitable electromagnetic coupling model for a given application is a challenging task

since there exists a multitude of different models in the literature and not all of these models

are equivalent. This thesis proposes a theoretical framework for the evaluation and comparison

of electromagnetic coupling models. This includes the investigation of the interaction between

coupling models and constitutive theory. It is shown that the electromagnetic force model

influences the entropy principle, but is not affected by it for simple material responses. However,

the electromagnetic energy model is subject to restrictions by the second law of thermodynamics

and therefore influenced by the force-model choice. Hence, the electromagnetic force model

must be investigated first. In order to determine the limitations of the most popular force

models, their theoretical predictions are compared to experimental results. To this end, three

experiments are evaluated. The first two experiments encompass the force measurement of

magnets that are immersed in ferrofluid and the measurement of moments between non-coaxial

permanent magnets. Both experiments were designed and conducted in this work. For the corre-

sponding theoretical investigations, an efficient finite element method in curvilinear coordinates

is developed. The method is also used to evaluate the third experiment, which was conducted in

the literature. In the experiment, the local deformation of an oil drop subjected to an electric

field is investigated. The theoretical predictions of the considered force models are compared

to the experimental data. The evaluation shows that the generalized

Lorentz

model is the

only electromagnetic force model to correctly predict the results of all three aforementioned

experiments. All other considered electromagnetic force models show significant differences

between their prediction and the experimental results in at least one of the three experiments.

On the electromagnetic coupling problem V

Zusammenfassung

Elektromagnetische Kopplungsmodelle sind notwendig für die Beschreibung der Interaktion

zwischen Elektrodynamik und Mechanik. Aufgrund der Vielzahl der verschiedenen Modelle in

der Literatur, die nicht alle äquivalent, stellt das Identifizieren eines geeigneten Modells für eine

spezifische Anwendung eine Herausforderung dar. In dieser Arbeit wird ein theoretisches Konzept

zur Auswertung und zum Vergleich verschiedener Kopplungsmodelle entwickelt. Dazu wird auch

das Zusammenspiel zwischen elektromagnetischer Kopplungsmodelle und thermodynamischer

Materialtheorie untersucht. Es wird gezeigt, dass elektromagnetische Kraftmodelle zwar das

Entropieprinzip beeinflussen, diese jedoch selbst nicht davon beeinflusst werden, wenn einfaches

Materialverhalten angenommen wird. Im Gegensatz dazu unterliegt das elektromagnetische Ener-

giemodell den Einschränkungen resultierend aus dem zweiten Hauptsatz der Thermodynamik und

ist damit auch durch die Wahl des Kraftmodells beeinflusst. Daher muss die Kraftmodellanalyse

zuerst erfolgen. Zur Bestimmung der Anwendungsgrenzen der populärsten Kraftmodelle, werden

deren theoretische Vorhersagen mit experimentellen Ergebnissen verglichen. Hierzu werden drei

Experimente untersucht. Die ersten beiden Experimente werden entwickelt und durchgeführt.

Sie umfassen die Messung der Kräfte zwischen in Ferrofluid eingelassenen Magneten sowie

die Messung der Momente zwischen nicht koaxialen Permanentmagneten. Für die zugehörige

theoretische Analyse wird eine effiziente Finite-Elemente-Methode in krummlinigen Koordinaten

entwickelt. Diese wird auch zur Auswertung des dritten Experiments aus der Literatur verwendet.

In diesem wird die Deformation eines Öltröpfchens aufgrund eines angelegten elektrischen Feldes

bestimmt. Die Untersuchungen ergeben, dass nur das verallgemeinerte

Lorentz

Kraftmodell alle

analysierten Experimente korrekt vorhersagen kann. Alle anderen untersuchten Modelle führen

zu signifikanten Unterschieden zwischen den theoretischen Vorhersagen und den experimentellen

Daten für mindestens eines der Experimente.

On the electromagnetic coupling problem VII

Acknowledgements

The research in this thesis was conducted during my tenure as a research assistant in the

research group “Kontinuumsmechanik und Materialtheorie” at Technische Universität Berlin. I

am grateful for being given the chance to pursue my own scientific ambitions. I worked in a

supportive environment in which the scientific exchange was always encouraged and some of the

best ideas were developed together in coffee breaks. Well-received and bidirectional scientific

criticism helped me immensely during the preparation of this thesis.

I would like to thank Professor W.

Müller

for his guidance when it was needed and for his

continuous stimulation of new ideas. All of our scientific discussions were interesting and after

leaving his office I was always a little bit wiser than before I entered. It is safe to say that he is a

science adviser who is unsatisfied with not-understanding or with half-truths, which encouraged

my critical thinking. In that, he enabled my scientific progress in several ways, some of which

only became apparent to me later. I can wholeheartedly say that he is one of the best lecturers

I experienced myself, both as a student and as an assistant, and I tried to absorb as much as

possible from his remarkable lecturing skills. He allowed me to contribute to his book on basic

mechanics, from which I learned to appreciate mechanics on another level.

I would like to extend my gratitude to Professor V.

Eremeyev

for kindly accepting the

refereeship of my thesis as well as his useful comments and suggestions regarding the ferrofluid

experiment. Additionally, I like to thank Professor U. von

Wagner

for accepting the chair-

manship of this dissertation. In my first three years at the institute of mechanics, I was also

assisting Professor A.

Bertram

in the lectures on continuum physics and I am grateful for

being taught the “Berlin school” of tensor calculus.

The work atmosphere in our group was always encouraging. I like to thank my colleagues

Dr. Sebastian Glane, Dr. Anton Köllner, Dr. Aleksandr Morozov and Dr. Aleksey Sokolov, for

their continuous support. Further, I like to express my gratitude to my colleagues Dr. Gregor

Ganzosch and Margarita Dementeva. Not only was their help crucial for the preparation of the

experiments, but staying late in our shared office to work while singing songs from the 90s left

nothing but happy memories. Similarly, I have to thank Miriam Ziert, Arion Juritza, Peter

Sahlmann and Grit Lamprecht for always assisting with little things that are seldom mentioned.

Most of all, I like to thank Dr. Felix Reich, who was my mentor before and who continued to

support me. He had a huge impact on my life and I am grateful for that.

Further, I like to thank my family for their support and understanding. In particular, my

friends Benjamin Häusler and Tom Chojne must be mentioned, for they read the thesis and

endured me until the end of my thesis and hopefully beyond that. I am grateful to my parents

for providing me with the best start I could imagine. Last but not least, I like to thank my

loving wife for all her support and encouragement.

On the electromagnetic coupling problem IX

Contents

List of Symbols XI

Nomenclature XV

1 Introduction 1

1.1 Fundamentals of electromagnetic coupling . . . . . . . . . . . . . . . . . . . . . 2

1.2 Researchgoal..................................... 3

1.3 Outlineofthisthesis................................. 3

2 Classical continuum electromechanics 5

2.1 Kinematics in spatial description . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Continuum electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Continuum thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Localization and summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Electromagnetic coupling models 21

3.1 Stateofresearch ................................... 21

3.2 Reference model and decomposition . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3 Coupling model construction and modifications . . . . . . . . . . . . . . . . . . 25

3.4 Frequently used coupling models . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.4.1 ABRAHAM and MINKOWSKI ......................... 27

3.4.2 EINSTEIN and LAUB ............................. 29

3.4.3 ERINGEN and MAUGIN ............................ 30

3.4.4 KOVETZ ................................... 31

3.4.5 PAO and HUTTER .............................. 33

4 Consequences of the coupling problem and force model invalidation 35

4.1 Thermodynamic consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2 Global forces and global moments on bodies . . . . . . . . . . . . . . . . . . . 38

4.3 Static electromagnetic pressure in fluids . . . . . . . . . . . . . . . . . . . . . . 41

4.4 Local motion and force model equivalence . . . . . . . . . . . . . . . . . . . . . 42

4.5 Invalidation of electromagnetic coupling models . . . . . . . . . . . . . . . . . 44

5 Numerical field calculation with a reduced finite element method 47

5.1 Finite element method in curvilinear coordinates . . . . . . . . . . . . . . . . . 47

5.1.1 Conversion procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

5.1.2 Spherical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.1.3 Cylindrical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.2 Static magnetic field computation . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.2.1 Governing equations and the weak formulation . . . . . . . . . . . . . . 52

5.2.2 Analytical solution and convergence analysis . . . . . . . . . . . . . . . 54

5.3 Numerical magnetic force computation . . . . . . . . . . . . . . . . . . . . . . 55

5.4 Two-phaseflow.................................... 59

5.4.1 Problem formulation and governing equations . . . . . . . . . . . . . . . 59

5.4.2 Weak form and adaptive time stepping . . . . . . . . . . . . . . . . . . . 61

Contents

5.4.3 Interface tracking method and curvature computation . . . . . . . . . . 63

5.4.4 Consistency analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6 Forces and moments between permanent magnets 67

6.1 Global forces on magnets immersed in ferrofluid . . . . . . . . . . . . . . . . . . 67

6.1.1 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.1.2 Theoretical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.1.3 Model dependent electromagnetic force calculation . . . . . . . . . . . . 72

6.1.4 Experimental results and conclusion . . . . . . . . . . . . . . . . . . . . 73

6.2 Moments acting on rigid permanent magnets and electrets . . . . . . . . . . . 74

6.2.1 Uniform external fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6.2.2 A gedankenexperiment ........................... 77

6.3 Moments between cylindrical permanent magnets . . . . . . . . . . . . . . . . 79

6.3.1 Theoretical electromagnetic moment computation . . . . . . . . . . . . 80

6.3.2 Force to moment conversion . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.3.3 Measurement results and comparison to the theoretical prediction . . . 84

7 The oil drop experiment 85

7.1 Governing equations and boundary conditions . . . . . . . . . . . . . . . . . . 85

7.2 Electromagnetic force distribution . . . . . . . . . . . . . . . . . . . . . . . . . 90

7.3 Steady-state deformation and its semi-analytical solution . . . . . . . . . . . . 91

7.3.1 The electric field of oblate and prolate spheroids . . . . . . . . . . . . . 92

7.3.2 Surface force equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 97

7.4 Comparison to experimental data and discussion . . . . . . . . . . . . . . . . . 99

8 Summary and conclusions 101

8.1 Contribution of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

8.2 Outlook ........................................ 103

List of Figures XV

List of Tables XVII

Bibliography XIX

A Magnetic field measurements i

B Observer transformations iii

C Mathematical identities and coordinate transformations ix

C.1 Integral theorems and general balance laws . . . . . . . . . . . . . . . . . . . . ix

C.1.1 Regular integral theorems . . . . . . . . . . . . . . . . . . . . . . . . . . x

C.1.2 Integral theorems with discontinuities . . . . . . . . . . . . . . . . . . . xi

C.1.3 Localization of global balance laws . . . . . . . . . . . . . . . . . . . . . xiii

C.2 An extension of CAUCHY’s tetrahedron argument to surfaces . . . . . . . . . . . xv

C.3 Cylindrical coordinate transformations . . . . . . . . . . . . . . . . . . . . . . xvi

Contents

On the electromagnetic coupling problem XI

List of Symbols

General quantities and relations

𝛺0,𝛺(𝑡)material domain in the reference and current placement

𝑊0,𝑊(𝑡)non-material domain in the reference and current placement

𝑥m,𝑥sposition vector to a point of the material and spatial volume m

𝑋m,𝑋sposition vectors to reference points m

𝑣m,𝑣smaterial and spatial particle velocities m/s

𝑣𝐼particle velocity on a singular surface m/s

𝑤,𝑤𝐼mapping velocity for regular and singular points m/s

𝐹spseudo-deformation gradient 1

d𝐴, d𝐴scalar and directed surface element m2

𝜈,𝜏binormal vector and tangent vector of the boundary of a surface 1

𝑛,𝑒normal vectors to a volume and a surface, respectively 1

𝑒𝐼,𝜏𝐼normal vector and tangent vector to a singular line 1

d𝑊sspatial volume element m3

∇s,∇𝐼nabla operator in spatial description and its surface projection 1/m

𝐼,ℓ𝐼singular surface and singular line

𝑊±,𝛤±regular subdomains of the volume 𝑊and its boundary 𝜕𝑊

𝑆±,ℓ±regular subdomains of the surface 𝑆and its boundary 𝜕𝑆

𝜓±left and right limit of a regular density onto a singular surface

(·)𝐼quantity defined on a singular surface (usually surface density)

(·)indication of a functional dependence

(·)dimensionless quantity

J·Kjump operator

⟨3⟩

𝜖Levi–Civita tensor 1

1unit tensor 1

Electrodynamics

𝜖0vacuum permittivity A2s2/(N m2)

𝜇0vacuum permeability N/A2

𝑐0speed of light in vacuum m/s

𝑞volumetric charge density As/m3

𝑗,𝐽conductive and total electric current A/m2

(·)f,(·)bfree and bound version of a quantity

𝐸,ℰelectric field and electromotive intensity V/m

𝐷,Dtotal and free electric charge potentials As/m2

𝑃bound charge potential or polarization vector As/m2

𝐵magnetic flux density T

𝐻,Htotal and free electric current potential A/m

𝑀bound electric current potential A/m

𝑓(em)

felectromagnetic force acting upon free charges and free currents N/m3

List of Symbols

XII

2 Continuum thermodynamics

(·)∙=D(·)

/D𝑡substantial time derivative 1/s

ℰkin specific kinetic energy m2/s2

𝑝specific mechanical linear momentum m/s

𝑠specific mechanical spin m2/s

𝐽ssymmetric micro-inertia tensor m2

𝜌sspatial mass density kg/m3

𝐹global force quantity N

(·)(mech) mechanical quantity

(·)(em) quantity of electromagnetic origin

(·)(conv) convective quantity

𝑓(mech) specific mechanical force density N/kg

𝑓(em) electromagnetic force density N/m3

𝑡,𝜎traction vector and stress tensor N/m2

𝑃𝛿pillbox domain of height 𝛿

𝑇total moment quantity N m

𝜏specific moment density N m/kg

𝜏(em) electromagnetic moment density N/m2

𝑚(mech),𝜇surface couple and couple stress tensor N/m

𝑈total energy J

𝑢specific internal energy J/kg

𝑊power quantity J/s

𝑞non-convective energy flux J/(m2s)

𝑟specific radiation J/(kg s)

ℎ(em) electromagnetic power J/(m3s)

𝜂specific entropy J/(K kg)

𝜑entropy flux J/(Km2s)

𝜎,𝑧entropy production and entropy supply J/(Km3s)

3 Electromagnetic coupling models

𝑔(em) electromagnetic momentum Ns/m3

𝑢(em) electromagnetic energy J/m3

𝜎(em) electromagnetic stress tensor N/m2

𝑞(em) Poynting vector J/(m2s)

𝐶(em) electromagnetic body couple N/m2

𝜎(em)

aux auxiliary stress tensor N/m2

𝑞(em)

aux auxiliary energy tensor J/(m2s)

(·)name quantity from the model by the author indicated by “name”

Δ(·)difference to the generalized Lorentz model

𝜒eelectric susceptibility 1

𝜒mmagnetic susceptibility 1

𝜖rrelative permittivity 1

𝜇rrelative permeability 1

𝐾𝑖,𝑆𝑖electromagnetic stress tensor coefficients

𝒮modified electromagnetic energy flux J/(m2s)

ℳ,ℋLorentz magnetization and magnetomotive intensity A/m

4 Consequences of the electromagnetic coupling problem and force model invalidation

𝐿velocity gradient 1/s

ℎ(axial) objective relative angular velocity 1/s

𝑊skew-symmetric tensor generated from ℎ1/s

List of Symbols

On the electromagnetic coupling problem XIII

𝜓(em) electromagnetic free energy J/m3

𝑘modified energy-entropy flux J/(m2s)

𝑝thermodynamic pressure N/m2

𝑓(em)

𝐼,eff effective electromagnetic surface force density N/m2

𝛿𝑣virtual velocity 1

5 Finite element method in curvilinear coordinates

𝑉mmagnetic potential A

(·)⋆quantity expressed in a pseudo-Cartesian space

(·)∘quantity in the spherical coordinate space

(·)

∪

∘

quantity in the cylindrical coordinate space

(·)sym indication of a dimensional reduction due to symmetry

𝑠𝑉,𝑠𝐴,𝑠ℓscaling functions for volume, surface and line elements

𝑠𝛿,𝑠∇scaling functions for the test function and the nabla operator

{𝑟, 𝜗, 𝜙}spherical coordinates

{𝜌, 𝜙, 𝑧}cylindrical coordinates

R3,Sphysical space and curvilinear coordinate space

𝒫𝑘continuous piece-wise polynomial function space of order 𝑘

𝒟𝑘discontinuous piece-wise polynomial function space of order 𝑘

𝑒rel,𝑒abs relative and absolute error

Re Reynolds number 1

We Weber number 1

Fr Froude number 1

6 Forces and moments between permanent magnets

𝐹exp

𝑧measured force value in 𝑧-direction N

𝑀ssaturation magnetization of a ferrofluid A/m

𝛼mmagnetic energy ratio 1

𝑘Boltzmann’s constant J/K

𝑇absolute temperature K

𝜌Fmass density of ferrofluid kg/m3

𝜌

∪

∘

radial cylindrical coordinate m

(·)′quantity in the rotated system

𝛿𝑢 virtual displacement 1

𝛼relative rotation angle between two magnets 1

𝜓turntable rotation angle 1

7 Forces and moments between permanent magnets

(·)ref reference value for a parameter

𝑉dimensionless electric potential 1

˜𝜇dimensionless shear viscosity 1

˜𝜌dimensionless mass density 1

Weel electric Weber number 1

˜𝑎,˜

𝑏dimensionless half-axes 1

{𝜇o, 𝜈o, 𝜙o}oblate spheroidal coordinates

{𝜁, 𝜉, 𝜙}modified oblate spheroidal coordinates

{𝜇p, 𝜈p, 𝜙p}prolate spheroidal coordinates

{𝜎, 𝜏, 𝜙}modified prolate spheroidal coordinates

𝐹o,𝐹poblate and prolate coordinate parameters

𝐷deformation measure for ellipses 1

List of Symbols

On the electromagnetic coupling problem XV

Nomenclature

The dyadic product is a non-commutative product, which constitutes a linear mapping from

one vector space into another. For any three vectors

𝑎

𝑏

𝑐∈ 𝒱

the dyadic product is defined

via its application to a vector

dyad(𝑎,𝑏)(𝑐)=[𝑎⊗𝑏](𝑐):= (𝑏·𝑐)𝑎and write [𝑎⊗𝑏](𝑐) = 𝑎⊗𝑏·𝑐.

Consider two vector spaces

𝒰

and

𝒲

that are are constructed from an ordinary three dimensional

vector space 𝒱such that

𝒰=𝒱 ×. . . ×𝒱

⏟ ⏞

𝑚times

,𝒲=𝒱 ×. . . ×𝒱

⏟ ⏞

𝑛times

The most general multilinear map between such two vector spaces is called tensor of rank

𝑘=𝑚+𝑛⟨𝑘⟩

𝑇:𝒰 → 𝒲 .

Upon defining multiple contractions via

(𝑎⊗𝑏)··(𝑐⊗𝑑)=(𝑎·𝑐)(𝑏·𝑑),

(𝑎⊗𝑏⊗𝑐)···(𝑑⊗𝑒⊗𝑓)=(𝑎·𝑑)(𝑏·𝑒)(𝑐·𝑓),

and so on, the component matrix

𝑇

of a tensor

⟨𝑘⟩

𝑇

with respect to the Cartesian orthonormal

basis 𝑒𝑖is given by the component-wise evaluation of

𝑇𝑖...𝑗 =⟨𝑘⟩

𝑇(𝑘)

·𝑒𝑖⊗. . . ⊗𝑒𝑗.

Therein, the Cartesian basis satisfies

𝑒𝑖·𝑒𝑗

𝛿𝑖𝑗

and

𝑒𝑖×𝑒𝑗

𝜀𝑖𝑗𝑘𝑒𝑘

, where

𝛿

is the

Kronecker

delta and

𝜀

is the

Levi

–

Civita

symbol, see [Flügge (1972)]. They form the components of the

unit tensor and the Levi–Civita tensor, respectively,

1=𝛿𝑖𝑗𝑒𝑖⊗𝑒𝑗=𝑒𝑖⊗𝑒𝑗,⟨3⟩

𝜖=𝜀𝑖𝑗𝑘𝑒𝑖⊗𝑒𝑗⊗𝑒𝑘.

Note that the

Einstein

sum convention is tacitly assumed for all Latin indices. The nabla

operator and the versions of the gradient of a vector are defined with respect to the position

vector 𝑥=𝑥𝑖𝑒𝑖via:

∇=𝜕

𝜕𝑥,(𝑎⊗∇) = 𝜕𝑎

𝜕𝑥=𝜕𝑎𝑖

𝜕𝑥𝑗

𝑒𝑖⊗𝑒𝑗,(∇⊗𝑎) = 𝜕𝑎𝑗

𝜕𝑥𝑖

𝑒𝑖⊗𝑒𝑗.

In this thesis, the algebraic position of the nabla operator in a product has no influence on its

action. Rather, it acts upon all quantities in a product, unless its range of influence is indicated

with brackets, e.g.,

𝑎·(∇⊗𝑏)=(𝑏⊗∇)·𝑎=𝜕𝑏𝑖

𝜕𝑥𝑗

𝑎𝑗𝑒𝑖,∇·(𝑎⊗𝑏)=(∇·𝑎)𝑏+𝑎·(∇⊗𝑏).

Nomenclature

On the electromagnetic coupling problem 1

1 Introduction

On a macroscopic scale, electromagnetic fields interact with matter via the exchange of linear

momentum, angular momentum and energy. The corresponding interaction mechanisms are

utilized in the design of a wide range of technical applications, such as the analysis of electric

field disturbances for the development of touch displays or the optimization of electric motors in

the automotive industry. In order to describe these macroscopic phenomena, continuum theory

is employed. More precisely, continuum mechanics is connected to continuum electrodynamics

by means of an electromagnetic coupling model (EMCM), which quantifies the interaction

between the electromagnetic fields and matter in a material-dependent manner. It is customary

to express an EMCM in terms of electromagnetic forces, moments and powers. Unfortunately,

the correct expressions of these quantities are unknown for certain material classes. Thus, the

task of finding suitable coupling models is referred to as electromagnetic coupling problem

(EMCP). In particular, electromechanical applications involving materials that react significantly

to electromagnetic fields, such as magnets or electrets, are affected by the EMCP.

Upon analyzing existing EMCMs that are verified for special circumstances, it becomes

apparent that they all have the same mathematical structure. This gives rise to the belief that

there exists an EMCM that is valid for a wider range of material classes if not all materials.

In the latter case, the coupling model must include the material response functions given

by the polarization and the magnetization generically. Its structure is then not altered by

constitutive theory and the coupling itself could be referred to as material independent. Such a

coupling model would be beneficial because it would reduce the amount of required experimental

verification for the material-dependent modeling. However, if a material-independent EMCM

exists is still an open question. Furthermore, a given EMCM can only be termed generally

applicable until circumstances are found in which it fails to predict experimental evidence.

That is to say, a coupling model can only be verified for material classes as well as classes of

initial-boundary value problems.

Regardless of whether or not the EMCP has a general solution, the limitations of existing

coupling models must be identified. This is particularly important because more advanced

continuum theories build upon their classical foundations, e.g., higher gradient continuum

theories or otherwise generalized theories such as micropolar theory or the theory of liquid

crystals. Subsequently, the employed constitutive theory may be affected by the limitations of a

given EMCM. In an attempt to find a generally applicable EMCM, researchers have developed a

multitude of EMCMs. Naturally, not all of these models are equivalent and thus lead to different

theoretical predictions. In addition to the variety of coupling models, they are embedded in

different frameworks of continuum thermodynamics, which renders their comparison difficult.

Therefore, a physically sound theory for the coupling of classical continuum mechanics and

electrodynamics, which can incorporate different types of coupling models, is required.

This work addresses such necessity by developing a rational framework in which different

coupling models can be embedded. Subsequently, several coupling models are analyzed and

evaluated regarding their predictions on local and global effects. By means of the unitized

investigation, experiments are devised in which coupling model differences can be observed.

1.1 Fundamentals of electromagnetic coupling

The theory of electrodynamics is inherently coupled to mechanics, which can be seen from the

definition of the electric field and the magnetic flux density as force fields acting upon electric

charges and currents, respectively. The EMCP arises only in the presence of magnetizable

or polarizable matter. The awareness for this problem was raised after

Einstein

and

Laub

published an electromagnetic force model in [Einstein and Laub (1908)] that differed from

the one proposed by

Abraham

, [Abraham (1909)]. Later,

Minkowski

provided yet another

expression for the electromagnetic force density accompanied by an electromagnetic momentum

in [Minkowski (1910)]. This sparked a debate in academia that is known as “

Abraham

–

Minkowski

controversy” (AMC). Originally, researchers were mostly concerned with the

different electromagnetic momentum formulations proposed by

Abraham

and

Minkowski

respectively. Over time, several theoretical arguments have been provided to argue in favor of

either

Abraham

Minkowski

. Subsequently, experiments have been designed to decide which

coupling model is correct. For example, the electromagnetic force exerted on thin metal sheets by

means of light beams was analyzed, [Jones (1953); Shockley and James (1967)]. Unfortunately,

the differences in the prediction of the force models were not detectable experimentally. The

same applies to the investigations of electromagnetic forces on dielectrics in [Walker and Walker

(1977)]. There is an abundance of publications on the AMC alone as can be seen from the

list of over 300 references provided in [McDonald (2017)]. As of today, none of the contained

publications resulted in a definite resolution of the debate. Furthermore, it was recognized that

in addition to the electromagnetic force model, the coupling with respect to angular momentum

and energy must also be considered, see [Truesdell and Toupin (1960)]. Subsequently, many

more coupling models were developed. Most notable is the model by

Chu

et al. in [Chu, Haus,

and Penfield (1966)] as well as the more recent ones in [Pao and Hutter (1975); Eringen and

Maugin (1990); Kovetz (2000)].

Due to the multitude of different coupling models and frameworks they are embedded in,

there seems to be no comprehensive coupling theory between continuum electromagnetism and

continuum mechanics, that was thoroughly tested and verified against a sufficiently wide range

of experiments. The differences between theories may be of minor significance, e.g., differences in

notation, different unit systems, and also different definitions of basic electromagnetic quantities

such as electric charges and currents. However, there are also significant deviations between

theories, which result in different predictions of measurable quantities. These differences become

apparent when the complete system of equations for the electromagnetic and mechanical coupling

is taken into account. The electromagnetic equations are given by

Maxwell

’s equations and

the thermomechanical equations on a continuum level are given by the balances of mass, linear

momentum, angular momentum and energy. In order to specify an EMCM, it is not sufficient to

provide the electromagnetic force, moment and power expressions alone. This becomes evident

from, e.g., the electromagnetic theory by

Kovetz

. In [Kovetz (2000)], electromagnetic forces

are assumed to be contact forces only, while most other theories consider both volumetric and

surface forces, see [Müller (1985)]. In particular, in some of the modern continuum physics

textbooks the employed EMCM is presented without investigating its limitations e.g., [Eringen

and Maugin (1990)]. The problem arising from the electromagnetic coupling is denied altogether

in publications like [Pfeifer et al. (2007)], while others firmly promote a particular model to be

correct, e.g., [Obukhov and Hehl (2003)] or [Mansuripur (2010)]. As far as the literature on

the electromagnetic coupling problem is concerned, general confusion and unawareness can be

observed. Finally, in recent investigations like [Datsyuk and Pavlyniuk (2015)] or [Reich (2017)],

progress has been made in terms of the analysis of modern coupling models as well as fruitful

comparisons of theoretical predictions and experimental evidence, which this thesis builds upon.

In particular, the rational procedure provided by

Reich

forms the starting point of the analyses

in this thesis.

Chapter 1. Introduction

On the electromagnetic coupling problem 3

1.2 Research goal

The main goal of this thesis is to determine the limitations of the most popular electromagnetic

force models in order to aid the process of choosing the correct one in a given application.

Therefore, suitable experiments are identified. To this end, the following objectives are defined:

A rational framework is developed, in which the different EMCMs can be incorporated in

a way that their impact becomes apparent.

An easy-to-use classification is devised. Therefore, technically relevant and most commonly

analyzed initial boundary-value problems are assessed as to whether they are affected by

the electromagnetic coupling problem and if so, to what extent.

By means of this classification, suitable experiments for the identification of generally

applicable EMCMs are developed.

Finally, the experiments are conducted and investigated theoretically. Subsequently, the

theoretical predictions are compared to the experimental observations, which (hopefully)

enables the identification of a single class of EMCMs that cannot be discarded utilizing

the experiments in this thesis.

1.3 Outline of this thesis

The first part of this thesis is concerned with the theoretical framework and numerical solution

methods. It is comprised of the Chapters 1–6. In the second part, several experiments are

discussed, conducted and investigated theoretically. The chapters are outlined below:

Chap. 2:

In this chapter, the basic equations for a micropolar continuum in a spatial description

are presented. Therefore, the kinematic relations are introduced and the governing equations

of continuum electromagnetism and continuum thermodynamics are derived for open systems.

This allows for a precise mathematical definition of the EMCP.

Chap. 3:

After the EMCP is defined, a rational approach to modeling the electromagnetic

coupling is analyzed. This encompasses some historical remarks and the resulting universal

constraints that a coupling model is subjected to. Furthermore, a mathematical framework for

the comparison of different coupling models is devised. Subsequently, popular coupling models

are presented and reformulated to fit the developed framework.

Chap. 4:

Using the obtained framework, several initial boundary-value problems that are

relevant in technical applications are classified with respect to the EMCP. Building upon this

classification, appropriate experiments for the investigations in this thesis are defined.

Chap. 5:

In order to analyze the devised experiments theoretically, numerical methods for the

solution of the governing equations must be used. To this end, a new finite element method in

curvilinear coordinates is developed, explained, and subsequently tested for its application to the

problems relevant to this thesis. In particular, the numerical electromagnetic force calculation

is discussed and various approaches are compared. Furthermore, an interface tracking method

for the computation of two-phase fluid flow is derived.

Section 1.2. Research goal

Chap. 6:

The first set of experiments is conducted and evaluated. In particular, the inter-

actions between permanent magnets are analyzed in two different scenarios. First, the total

electromagnetic force between a cylindrical magnet and another one immersed in a ferrofluid is

measured and computed numerically. Second, two cylindrical permanent magnets are rotated

against each other while the resulting electromagnetic moment, which tries to align them, is

measured. Subsequently, the obtained total electromagnetic moment is compared to various

predictions of the frequently used EMCMs.

Chap. 7:

The oil drop experiment as performed by [Torza, Cox, and Mason (1971)] is inves-

tigated theoretically. The developed interface tracking method for two-phase flow is used to

calculate the deformation process as well as the steady-state deformation of an oil droplet. In

the experiment, a droplet of silicone oil is submerged in castor oil, which in turn is confined in

a container subjected to an electric field from a connected plate condenser. For steady states,

deformation shapes are computed for the frequently used EMCM and subsequently compared

to the experimental observations by Torza et al.

Chap. 8:

Conclusions of the thesis are drawn and the scientific contributions are outlined.

Comments on possible future investigations are provided in the outlook.

Chapter 1. Introduction

On the electromagnetic coupling problem 5

2 Classical continuum electromechanics

Classical electromagnetic theory is an important part of physics and is treated in textbooks

like [Stratton (1941)], [Jackson (1999)] or [Hehl and Obukhov (2003)]. In particular, continuum

electromagnetism combined with continuum mechanics is covered extensively in, e.g., [Brown

(1966); Eringen and Maugin (1990); Kovetz (2000); Hutter, Ven, and Ursescu (2006)]. However,

despite the abundance of pertinent literature, it is necessary to introduce all of the kinematic

and physical concepts involved in an electromechanical theory. The reasons for this are threefold:

The descriptions of the kinematics of continuum bodies in the above references are

restricted to material bodies. However, electromagnetic fields are not directly tied to

matter and thus a spatial description of the electromagnetic theory should be constructed.

As soon as electromagnetism is involved, even the basic definitions of quantities such as

the linear momentum are controversial and not uniquely defined, cf. [Eringen and Maugin

(2012)] and [Kovetz (2000)].

There are several ways the coupling between electromagnetism and mechanics can be

introduced. Thus, simply stating an electromagnetic force or moment density (say) on its

own is insufficient. Rather, a complete system of balance laws must be provided.

In particular, physical assertions and mathematical consequences must be carefully distinguished,

see the discussion in [Synge (1974)]. Moreover, this is elaborated in context with the

Maxwell

stress tensor concept in [Rinaldi and Brenner (2002)]. Therefore, an electromagnetic coupling

model must be given in combination with a complete framework of continuum electromechanics.

Otherwise, any statement regarding the electromagnetic coupling problem becomes ambiguous.

In the following, a discussion of the kinematic description is provided. Subsequently, a

framework for a coupled theory of continuum mechanics and electrodynamics is developed in

form of the balance equations

of continuum physics. These presentations closely follow [Rickert

and Müller (2020)].

2.1 Kinematics in spatial description

In this section, all measurements are performed by one inertial observer

, see App.B for

the relevant definitions. Consider a material body that occupies the (material) volume

𝛺

(

𝑡

which is in part or completely contained in the non-material control volume

𝑊

(

𝑡

). The current

positions of material points,

𝑃m

, and spatial points,

𝑃s

, are identified by the corresponding

position vectors

𝑥m

and

𝑥s

, respectively. The considered scenario is depicted in Fig. 2.1. Both

position vectors can be obtained from bijective functions of motion, which satisfy:

𝑥m=𝜒m(𝑋m, 𝑡),𝑥s=𝜒s(𝑋s, 𝑡).(2.1)

Therein, the position vectors

𝑋m

and

𝑋s

refer to points in a time-independent reference

configuration of the material and spatial volume,

𝛺0

and

𝑊0

, respectively. From each motion,

Note that if a proper spatial description is employed, the global balance laws have to be adjusted for open

volumes. In particular, the global balance of the magnetic flux is thereby affected.

𝑂

𝛺0

𝑊0

s𝑃m

𝑋m

𝑋s

𝑃m

𝑥m

𝑃s𝑃s

𝑊s(𝑡)

𝛺(𝑡)

𝑥s

Fig. 2.1:

Schematic depiction of different examples for both the material and spatial description of

kinematics. For simplicity, only two-dimensional representations of the three-dimensional volumes are

sketched.

a velocity can be defined:

𝑣m=^

𝑣m(𝑋m, 𝑡):=𝜕𝜒m

𝜕𝑡 𝑋m

=d𝜒m

d𝑡,𝑤=^

𝑤(𝑋s, 𝑡):=𝜕𝜒s

𝜕𝑡 𝑋s

=d𝜒s

d𝑡.(2.2)

The velocity

𝑣m

is associated with a material point and is subsequently referred to as material

particle velocity. In contrast,

𝑤

is the mapping velocity and corresponds to the non-material

motion of a point

𝑃s

from the control volume

𝑊s

(

𝑡

), where time dependence will be

tacitly assumed in the following. The temporal changes given by the time derivatives in Eq.

(2.2)

are measured by an inertial observer

, which is equipped with the origin

𝑂

as well as the

coordinate system indicated by the blue coordinate lines in Fig. 2.1.

It should be noted that the material particle velocity is naturally a function of the position

vector of a material point,

𝑋m

. Since

𝜒m

is bijective, it is of course possible to trace the position

of a particle backward in time, i.e.,

𝑋m

𝜒−1

(

𝑥m, 𝑡

). Thus, another functional dependence of

the particle velocity can be found,

𝑣m=¯

𝑣m(𝑥m, 𝑡):=^

𝑣m(𝜒−1

m(𝑥m, 𝑡), 𝑡).(2.3)

Since

𝑣m

depends upon

𝑋m

, it is said to be in the

Lagrang

ian description, while

𝑣m

is a

function of

𝑥m

and the description is referred to as

Euler

ean. The former is also termed

“material” and the latter “spatial,” which is not reasonable, because both descriptions are tied to

material points, see [Ivanova, Vilchevskaya, and Müller (2016)]. In a proper spatial description,

another meaningful velocity is introduced, namely the velocity of material particles evaluated at

a fixed spatial position 𝑥sin the current configuration,

𝑣s=^

𝑣s(𝑥s, 𝑡):=^

𝑣m(︀𝜒−1

m(𝑥s, 𝑡), 𝑡)︀.(2.4)

It can be referred to as spatial particle velocity. Note that this quantity represents the velocity of

different material particles passing through the spatial point

𝑥s

. The spatial velocity is usually

not obtained from the definition in Eq.

(2.4)

, as this would require knowledge of the motion

𝜒−1

throughout the whole process history. Rather, the velocity itself is obtained as a solution

of the balance of linear momentum for a given initial boundary-value problem. That is to say,

the velocity 𝑣sis regarded as a primary kinematic quantity.

A proper spatial description incorporates both the pseudo material or virtual motion of the

control domain,

𝜒s

, as well as the spatial particle velocity field

𝑣s

. It is then possible to define

Chapter 2. Classical continuum electromechanics

On the electromagnetic coupling problem 7

a pseudo-deformation gradient via

𝐹s=𝜕𝜒s

𝜕𝑋s

.(2.5)

Similar to a material description, the time derivatives of a spatial volume element d

𝑊s

and a

spatial directed surface element d𝐴s=𝑛sd𝐴sare given by:

d𝑡(d𝑊s)=(∇s·𝑤) d𝑊s,∇s=𝜕

𝜕𝑥s

d𝑡(d𝐴s)=(∇s·𝑤) d𝐴s+ (∇s⊗𝑤)·𝑛d𝐴 ,

(2.6)

Two conceptual differences to a material description must be observed. First, the temporal

changes of the volume and surface element are governed by the non-material mapping velocity

𝑤

instead of a velocity associated with material motion, i.e.,

𝑣s

𝑣m

. Secondly, the nabla

operator,

∇s

, represents a derivative with respect to the spatial position

𝑥s

in space that is not

tied to matter.

If a generalized continuum is considered, e.g., a micropolar medium, additional and indepen-

dent kinematical degrees of freedoms such as the angular velocity field

𝜔m=^

𝜔m(𝑋m, 𝑡)

can

be introduced, see [Eremeyev and Lebedev (2013); Cowin (1974)]. These quantities are always

associated with matter because non-material points themselves do not possess any intrinsic

properties. However, as for the linear velocity

𝑣s

, there exists a meaningful spatial representation

of the angular velocity,

𝜔s=^

𝜔s(𝑥s, 𝑡):=^

𝜔m(𝜒−1

m(𝑥s, 𝑡), 𝑡).(2.7)

In the following, two types of open control domains are considered, see Fig. 2.2. Therein, the

volumetric domain

𝑊

is divided into two regular subdomains

𝑊±

due to the cut by the surface

of discontinuity

𝐼

, also known as singular surface. As a result, the boundary of the domain is

decomposed into regular and singular parts,

𝜕𝑊

𝛤+∪𝛤−∪

(

𝜕𝑊 ∩𝐼

). Similarly, the open

𝜕𝛺

𝛤+

𝛤−

𝛺−𝛺+

𝜕𝐼

𝑒

𝜈

𝑛

𝐼

𝑒

𝜕𝑆

𝜏𝑆+

𝜈

𝑆−

ℓ−

ℓ+

ℓ

𝐼

𝜏𝐼

𝑒

𝐼

Fig. 2.2:

Schematic depictions of a volumetric and a surface control domain from [Rickert and Müller

(2020)], reproduced with permission from Springer Nature. The binormal 𝜈𝐼=𝑒𝐼×𝜏𝐼is not shown.

control surface

𝑆

is divided into regular subdomains

𝑆±

due to a singular line

ℓ𝐼

. Thus, the

boundary line decomposition 𝜕𝑆 =ℓ+∪ℓ−∪(𝜕𝑆 ∩𝜕ℓ𝐼)is employed.

The total content of the physical quantity

𝛹

in a given control volume

𝑊

can be decomposed

using its volumetric density 𝜓=^

𝜓(𝑥s, 𝑡)as well as its singular density 𝜓𝐼,

𝛹=ˆ

𝑊+∪𝑊−

𝜓d𝑊+ˆ

𝐼

𝜓𝐼d𝐴 , 𝐹 =ˆ

𝑆+∪𝑆−

𝑒·𝑓d𝐴 , (2.8)

where the surface flux

𝐹

through

𝑆

is expressed in terms of its flux density

𝑓

. Upon balancing

Section 2.1. Kinematics in spatial description

these quantities, their temporal changes are given by

d𝛹

d𝑡=−ˆ

𝛤+∪𝛤−

(𝛷V+ [𝑛·(𝑣s−𝑤)]𝜓) d𝐴+ˆ

𝑊

(𝑝V+𝑠V) d𝑊+(2.9)

+ˆ

𝐼

(𝑝𝐼+𝑠𝐼) d𝐴−ˆ

𝜕𝑊 ∩𝐼

𝜈·𝜙

𝐼dℓ ,

d𝐹

d𝑡=−ˆ

ℓ+∪ℓ−

(︀𝛷A+ [𝑓×(𝑣s−𝑤)] ·𝜏)︀dℓ+ˆ

𝑆+∪𝑆−

𝑛·(𝑝A+𝑠A) d𝐴+ˆ

ℓ𝐼

𝜈𝐼·𝑠A

𝐼dℓ . (2.10)

Therein,

𝛷V

and

𝛷A

are the non-convective fluxes into the volume and the surface, respectively.

Since the control domains are open, there is also a convective flux due to the relative motion

of the control domain with respect to matter. Of course, the transport terms (

𝑛·𝑣s

)

𝜓

and

(

𝑓×𝑣s

)

·𝜏

are only present if the respective fields are transported with matter. Otherwise,

the corresponding terms must be excluded. For example, a surface flux density that is non-

transported with matter is denoted as

𝑓nc.

. The remaining terms in Eq.

(2.9)

and

(2.10)

consist

of the various supply terms

𝑠

, production terms

𝑝

and the singular flux

𝜙

𝐼

. Note that supply

and production terms are of the same mathematical structure, but their physical interpretation

is different. While supply terms can be controlled externally, production terms are inherent to

the considered system.

Using the tetrahedron argument by

Cauchy

and the generalized surface theorem from

App. C.2, it can be shown that the non-convective fluxes are linear functions of the normal and

tangent vector, respectively,

𝛷V=^

𝛷V(𝑥s, 𝑡;𝑛) = 𝑛·^

𝜙V(𝑥s, 𝑡), 𝛷A=^

𝛷A(𝑥s, 𝑡;𝜏) = 𝜏·^

𝜙A(𝑥s, 𝑡).(2.11)

Thus, the localization techniques shown in App. C.1.3 can be applied. This results in the local

forms of the balance laws in regular points

𝜕𝜓

𝜕𝑡 +∇s·(𝑣s⊗𝜓+𝜙V) = 𝑝V+𝑠V,(2.12a)

𝜕𝑓

𝜕𝑡 + (∇s·𝑓)𝑤+∇s×(𝜙A−𝑣s×𝑓) = 𝑝A+𝑠A,(2.12b)

𝜕𝑓nc.

𝜕𝑡 + (∇s·𝑓nc.)𝑤+∇s×𝜙A=𝑝A+𝑠A,(2.12c)

and in singular points

𝜕𝜓𝐼

𝜕𝑡 +∇𝐼·(𝑣𝐼⊗𝜓𝐼+1𝐼·𝜙

𝐼) = −𝑒·J(𝑣s−𝑤𝐼)⊗𝜓+𝜙K+(2.12d)

+𝑤𝐼·(∇𝐼⊗𝜓𝐼) + 𝑝𝐼+𝑠𝐼,

𝑒𝐼×J𝜙A−(𝑣s−𝑤𝐼)×𝑓K=𝑠A

𝐼,(2.12e)

𝑒𝐼×J𝜙A+𝑤𝐼×𝑓nc.K=𝑠A

𝐼,(2.12f)

where the jump operator is defined via

J𝜓K

𝜓+−𝜓−

, see Eq.

(C.15)

, and the surface nabla

operator results from the projection

∇𝐼

1𝐼·∇s

with the surface projector

1𝐼

= (

1−𝑒⊗𝑒

The singular surface velocity

𝑣𝐼

corresponds to the motion of particles on the singular surface.

Note that the partial time derivative of a surface quantity

𝜓𝐼

is defined as the surface time

derivative in App. C.1.3.

Chapter 2. Classical continuum electromechanics

On the electromagnetic coupling problem 9

2.2 Continuum electromagnetism

In modern continuum electrodynamics,

Maxwell

’s equations are derived from global balances

laws, [Hehl and Obukhov (2003)]. These are given by the conservation of magnetic flux, also

known as

Faraday

’s law of induction as well as the balance of the electric charge. The derivation

Maxwell

’s equations for material domains can be found in, e.g., [Müller (1985); Kovetz

(2000); Hutter, Ven, and Ursescu (2006); Eringen and Maugin (2012); Müller (2014)], to name

just a few. In the following, the corresponding balances are considered for open non-material

control domains. This analysis is advantageous for two reasons. First, the transport term

Faraday

’s law is straightforwardly interpreted and can be introduced naturally without

additional assumptions. Secondly, the electric field and the magnetic flux density, respectively,

𝐸=^

𝐸s(𝑥s, 𝑡),𝐵=^

𝐵s(𝑥s, 𝑡),(2.13)

are not bound to matter, which makes a spatial description appropriate. Note that the final

equations have the same well-known form as in the literature, only their interpretation is slightly

different. In order to simplify the notation, the indication “s” for spatial fields is omitted for

𝐸

and 𝐵.

The axioms for the derivation of rational electrodynamics are given by, cf., [Hehl and Obukhov

(2003)]:

EM1 The conservation of electric charge expressed via the balance of (total) electric charge,

d𝑡ˆ

𝑊+∪𝑊−

𝑞d𝑊+d

d𝑡ˆ

𝐼

𝑞𝐼d𝐴=−ˆ

𝛤+∪𝛤−

𝑛·(𝑗+𝑞[𝑣s−𝑤]) d𝐴−ˆ

𝜕𝑊 ∩𝐼

𝜈·(𝑗𝐼+𝑞𝐼[𝑣s−𝑤𝐼]) dℓ , (2.14)

where the indication for a spatial field is omitted for the charge densities

𝑞

and

𝑞𝐼

The conductive electric current

𝑗

and convective current

𝑞𝑣s

, as well as their singular

counterparts, can be combined into the total currents

𝐽:=𝑗+𝑞𝑣s,𝐽𝐼:=𝑗𝐼+𝑞𝐼𝑣𝐼.(2.15)

EM2

Additive decomposition of electric charges and electric currents into free and bound parts,

𝑞=𝑞f+𝑞b,𝐽=𝐽f+𝐽b,(2.16)

where

𝑞f

and

𝐽f

correspond to charges and currents that move “freely,” while their

counterparts

𝑞b

and

𝐽b

are bound to matter. Furthermore, it is postulated that both the

free charge and the bound charge are conserved separately,

d𝑡ˆ

𝑊

𝑞fd𝑊+d

d𝑡ˆ

𝐼

𝑞f

𝐼d𝐴=−ˆ

𝛤+∪𝛤−

𝑛·(𝐽f−𝑞f𝑤) d𝐴−ˆ

𝜕𝑊 ∩𝐼

𝜈·(𝐽f

𝐼−𝑞f

𝐼𝑤𝐼) dℓ ,

d𝑡ˆ

𝑊

𝑞bd𝑊+d

d𝑡ˆ

𝐼

𝑞b

𝐼d𝐴=−ˆ

𝛤+∪𝛤−

𝑛·(𝐽b−𝑞b𝑤) d𝐴−ˆ

𝜕𝑊 ∩𝐼

𝜈·(𝐽b

𝐼−𝑞b

𝐼𝑤𝐼) dℓ .

(2.17)

EM3 The Lorentz law for the force acting upon free charges and free currents,

𝑓(em)

f=𝑞f𝐸+𝐽f×𝐵=𝑞f(𝐸+𝑣s×𝐵) + 𝑗f×𝐵,(2.18)

which effectively defines the electric field

𝐸

and the magnetic flux density

𝐵

as force

fields.

Section 2.2. Continuum electromagnetism

EM4

The balance of magnetic flux through a non-material surface

𝑆

with a material boundary

𝒞=𝜕𝑆,d

d𝑡ˆ

𝑆

𝐵·𝑒d𝐴=−˛

𝜕𝑆

𝜏·(𝐸+𝑤×𝐵) dℓ . (2.19)

EM5

The space-time relations, also known as

Maxwell

–

Lorentz

aether relations, connect

the force field pair (

𝐸,𝐵

)to the pair of potentials (

𝐷,𝐻

), which will be introduced

below. The corresponding connections are given by:

𝜖0𝐸=𝐷,1

𝜇0𝐵=𝐻.(2.20)

Therein, the vacuum permittivity and the vacuum permeability are connected to the speed

of light in vacuum via 𝑐0=1

/√𝜖0𝜇0and their values are given by, respectively,

𝜖0≈8.854 ×10−12 A2s2/(Nm2), 𝜇0≈4π×10−7N/A2.(2.21)

In order to obtain

Maxwell

’s equations, it turns out to be convenient to start with the

balance of the magnetic flux. However, before this balance of the magnetic flux is localized, its

physical meaning is interpreted. The boundary curve

𝒞

𝜕𝑆

is said to be material because

it can be thought of as an infinitesimally small wire. Therein, the electric field

𝐸

represents

the force per unit charge that drives the motion of charges through the wire, cf., the

Lorentz

force law in Postulate EM3. In turn, the moving charges generate the magnetic flux through the

surface spanned by

𝒞

. Of course, this surface is non-material and thus arbitrary. Furthermore,

neither

𝐸

nor

𝐵

are fields that are transported with matter directly, which is emphasized by

the non-material convective transport

𝑤×𝐵

. Since

𝒞

is material,

𝑤

could be replaced with the

material particle velocity

𝑣s

. However, the interpretation of the “moving charges” through

𝒞

rather secondary to the mathematical statement in Eq.

(2.19)

, which does not even contain the

notion of charges explicitly. Furthermore, upon localizing

Faraday

’s law, the velocity has to

be replaced with

𝑤

again. Despite the aforementioned physical interpretation, it can be noted

that the contained control surface

𝑆

is arbitrary, which allows for the special choice

𝑆

𝜕𝑊

Thus, the balance of magnetic flux reduces to

d𝑡ˆ

𝜕𝑊

𝐵·𝑛d𝐴= 0 ⇒0 = ˆ

𝜕𝑊

𝐵·𝑛d𝐴=ˆ

𝑊−∪𝑊+

∇s·𝐵d𝑉+ˆ

𝐼

𝑒·J𝐵Kd𝐴 . (2.22)

Therein, the constant arising from the time integration vanishes according to the turning-on

argument

. It follows from this expression that both integral kernels must vanish separately.

Thus, the localization of

Faraday

’s law for a general surface

𝑆

is significantly reduced by the

fact that the magnetic flux density is solenoidal,

𝜕𝐵

𝜕𝑡 +∇s×𝐸=0,∇s·𝐵= 0 ,

𝑒×J𝐸K−(𝑒·𝑤𝐼)J𝐵K=0,𝑒·J𝐵K= 0 .

(2.23)

Using the theory of mixtures, it can be shown that the charge balance in Postulate EM1

follows from the mass balance, see [Reich, Stahn, and Müller (2015)]. The local electric charge

2Some authors prefer to state this as an additional axiom.

Chapter 2. Classical continuum electromechanics

On the electromagnetic coupling problem 11

balance can be obtained by means of the localization laws in Eq. (2.12),

𝜕𝑞

𝜕𝑡 +∇s·(𝑞𝑣s+𝑗)=0,

𝜕𝑞𝐼

𝜕𝑡 +∇𝐼·(𝑞𝐼𝑣𝐼+1𝐼·𝑗𝐼) = 𝑤𝐼·(∇𝐼𝑞𝐼)−𝑒·J(𝑣s−𝑤𝐼)𝑞+𝑗K.

(2.24)

However, in order to obtain

Maxwell

’s equations, it is customary to represent the total charge

contained in a control volume via the total electric charge potential 𝐷:

𝜕𝑊

𝐷·𝑛d𝐴:=ˆ

𝑊+∪𝑊−

𝑞d𝑊+ˆ

𝐼

𝑞𝐼d𝐴 . (2.25)

Upon using the generalized

Gauss

theorem from Eq.

(C.16)

as well as the localization theorem,

it becomes obvious why 𝐷is referred to as charge potential,

∇s·𝐷=𝑞 , 𝑒·J𝐷K=𝑞𝐼.(2.26)

Through the introduction of the charge potential, the balance of electric charge effectively

reduces to a surface balance. In particular, the surface 𝑆=𝜕𝑊 is closed and has therefore no

boundary, i.e.,

𝜕𝑆

∅

. Hence, a zero-valued line integral with respect to

ℓ+∪ℓ−

can be added

to the pseudo surface balance, which may be used to gauge the potential

𝐷

that is otherwise

only determined up to a curl-free field. In view of the general surface flux balance in Eq.

(2.10)

such an addition is natural,

d𝑡˛

𝑆

𝐷·𝑛d𝐴=ˆ

ℓ−∪ℓ+

𝑎·𝜏dℓ−ˆ

𝑆+∪𝑆−

𝑛·(︀𝐽−𝑞𝑤)︀d𝐴−ˆ

𝜕𝑊 ∩𝐼

𝜈·(︀𝐽𝐼−𝑞𝐼𝑤𝐼)︀dℓ . (2.27)

Therein, the auxiliary field

𝑎

may be decomposed into a conductive and a convective part, cf.,

the general surface balance in Eq.

(2.12)

. However, at this point it is not clear whether or not

the charge potential should be transported with matter, i.e.,

𝑎→𝐻−𝐷×(𝑣s−𝑤)or 𝑎→𝑎nc. =𝐻+𝐷×𝑤,(2.28)

where

𝐻

is a non-convective charge transport that will be interpreted later. Since

𝐷

is a

mathematical potential, its physical interpretation is not fully determined. On the one hand,

the field

𝐷

is associated with charges, which in turn represent the presence of matter. On

the other hand, there is an argument in favor of the interpretation that

𝐷

is not necessarily

convected with matter. According to PostulateEM5, the charge potential is linearly related to

the electric field

𝐸

, which is certainly not attached to matter and thus not transported with it.

Only in the context of a proper spatial description is the physical interpretation of the choice in

Eq.

(2.28)

obvious. In publications like [Pao and Hutter (1975)], both choices are analyzed, but

they are derived from a purely material description, which obscures their interpretation.

In this thesis, the choice

𝑎

𝑎nc.

𝐻

𝐷×𝑤

is made, which leads to the

Amper

ian

current model of Maxwell’s equations. By means of the identifications

𝑓=𝐷,𝜙A=−𝐻,𝑝A=−𝐽+𝑞𝑤,𝑠A

𝐼=−𝐽𝐼+𝑞𝐼𝑤𝐼,𝑒

𝐼=𝑒,(2.29)

the insertion into Eq. (2.12c) yields Ørsted’s law,

−𝜕𝐷

𝜕𝑡 +∇s×𝐻=𝐽=𝑗+𝑞𝑣s,

(𝑒·𝑤𝐼)J𝐷K+𝑒×J𝐻K=𝐽𝐼=𝑗𝐼+𝑞𝐼𝑣𝐼.

(2.30)

Section 2.2. Continuum electromagnetism

From

Ørsted

’s law, the auxiliary field

𝐻

can be interpreted as total electric current potential.

According to Postulate EM2, the free charge density, as well as its bound counterpart both,

satisfy balance laws of the same form as the total electric charge balance. In view of the

derivation above, it is only natural to introduce charge and current potentials for the free and

bound fields, respectively. Therefore, the free charge potential

and the bound charge potential

𝑃, also known as polarization vector, are defined via:

∇s·D=𝑞f,𝑒·JDK=𝑞f

𝐼,(2.31)

−∇s·𝑃=𝑞b,−𝑒·J𝑃K=𝑞b

𝐼,(2.32)

where the minus sign in front of the polarization vector is conventional. It follows from the

charge decomposition postulate that the charge potentials are connected via

𝐷=D−𝑃.(2.33a)

Similarly, but without the conventional minus sign, the current potential is decomposed into

the potential of free electric currents

and its bound counterpart

𝑀

, which is also referred to

as magnetization,

𝐻=H+𝑀.(2.33b)

The corresponding potentials obey versions of Ørsted’s law:

−𝜕D

𝜕𝑡 +∇s×H=𝐽f=𝑗f+𝑞f𝑣s,

(𝑒·𝑤𝐼)JDK+𝑒×JHK=𝐽f

𝐼=𝑗f

𝐼+𝑞f

𝐼𝑣𝐼,

(2.34)

and 𝜕𝑃

𝜕𝑡 +∇s×𝑀=𝐽b,

−(𝑒·𝑤𝐼)J𝑃K+𝑒×J𝑀K=𝐽b

𝐼.

(2.35)

Note that there are two key differences between Eq.

(2.31)

together with

(2.34)

and Eq.

(2.32)

combined with

(2.35)

. First, the bound currents are usually not decomposed into their convective

and conductive components. Second, the equations differ in their purpose. After constitutive

relations for

𝑞f

and

𝑗f

are provided, Eqs.

(2.31)

and

(2.34)

are used to determine

and

However, the sole purpose of Eqs. (2.32) and (2.35) is to provide alternative representations of

the bound charge density

𝑞b

and the bound current density

𝐽b

. Therefore, the fields

𝑃

and

𝑀

must be regarded as material-response functions, for which constitutive relations are required.

As a result, only the equations for

𝐷

𝐻

and

are independent equations that must be

solved.

In the literature, the symbol “

𝐻

” is commonly used to represent what is termed

in this

thesis. Similarly,

𝐷

is used for what is here referred to as

. Besides these different notations,

there are electromagnetic theories with different unit systems and different definitions of electric

charge and electric current. The unit conversion between the most popular variants can be

found in [Eringen and Maugin (1990), pg. 406] or [Jackson (1999)]. These variants are the

Heaviside

–

Lorentz

units, the

Gauss

units and the SI units. The latter are used in this thesis.

For example, the electric field in the

Heaviside

–

Lorentz

system or in the

Gauss

unit system

are given by:

𝐸[HL] =√𝜖0𝐸[SI] ,𝐸[G] =√4π𝜖0𝐸[SI] .(2.36)

On the other hand, the conversion between the different formulations of

Maxwell

’s equations

requires more effort. The relations in Eqs.

(2.23)

and

(2.34)

represent the

Amper

ian model

of electromagnetism. Other popular choices are the

Minkowski

formulation, the

Lorentz

Chapter 2. Classical continuum electromechanics

On the electromagnetic coupling problem 13

formulation and the

Chu

formulation, see [Hutter, Ven, and Ursescu (2006)]. For example, in

[Pao and Hutter (1975)] the

Chu

model is employed, which is connected to the

Amperian

model via: 𝐵=𝜇0(HCh +𝑀Ch),𝐸+𝑣×𝐵=𝐸Ch +𝜇0𝑣×HCh ,

D=𝜖0𝐸Ch +𝑃Ch ,H−𝑣×D=HCh −𝜖0𝑣×𝐸Ch ,(2.37)

where the index “

” indicates a quantity from the

Chu

model. This system of equations can

be used to obtain the following relations:

𝐴·HCh =𝐴·H+𝑣×(︀1

𝑐2𝑣×𝑀−𝑃)︀,

𝐴·𝐸Ch =𝐴·𝐸+𝜇0𝑣×(𝑀+𝑣×𝑃),

𝐴=1

𝑐2[︀(𝑐2−𝑣·𝑣)1+𝑣⊗𝑣]︀.

(2.38)

Upon invoking the low-velocity approximation,

𝑣·𝑣≪𝑐2

, the tensor

𝐴≈1

can be neglected.

As a result, the approximate transformations are given by:

HCh =H−𝑣×𝑃,𝐸Ch =𝐸+𝜇0𝑣×𝑀,

𝑀Ch =𝑀+𝑣×𝑃,𝑃Ch =𝑃−1

𝑐2𝑣×𝑀.(2.39)

In particular, the following expression occurs frequently:

𝐸Ch +𝜇0𝑣×HCh ≈𝐸+𝑣×𝐵.(2.40)

In summary, the

Chu

formulation only differs from the

Amper

ian formulation for moving

matter.

2.3 Continuum thermodynamics

In this section, a micropolar continuum is considered, because it is a contemporary generalization

of the classical Euler–Cauchy continuum. Furthermore, in the context of electromagnetism,

it is conveniently used for ferrofluids and to demonstrate the influence of the electromagnetic

moment

𝜏(em)

. An introduction to micropolar theory can be found in [Eremeyev and Lebedev

(2013)]. Following, [Müller, Rickert, and Vilchevskaya (2020)], the specific kinetic energy is

introduced as a quadratic form

ℰkin =1

2𝑣s·𝑣s+1

2𝜔s·𝐽s·𝜔s.(2.41)

Subsequently, the specific mechanical linear momentum and the specific mechanical spin are

obtained as

𝑝:=𝜕ℰkin

𝜕𝑣s

=𝑣s,𝑠:=𝜕ℰkin

𝜕𝜔s

=𝐽s·𝜔s.(2.42)

Therein,

𝐽s

is the symmetric micro-inertia tensor. The spin is only one part of the specific angular

momentum, which additionally includes a linear momentum contribution, ℓ=𝑥s×𝑝+𝑠.

Consider a non-material open volumetric control region

𝑊

, that is cut by an interface

𝐼

. The

resulting regular regions

𝑊±

on each side of the interface are combined to

𝑊

𝑊+∪𝑊−

similar to Fig. 2.2. The material subdomain

𝛺

can be partially or fully contained in

𝑊

. Note

that the boundary

𝜕𝛺

always constitutes a part of the interface

𝐼

. For simplicity this interface

shall be without a singular mass density, i.e.,

𝜌𝐼≡

0. This assumption must be dropped if the

dynamics of free surface charges

𝑞f

𝐼

are to be analyzed, see Eq.

(2.24)

, which is beyond the scope

of this thesis.

Section 2.3. Continuum thermodynamics

The balances of inertia

simplify significantly, if the interface

𝐼

is assumed to be free from a

singular mass distribution,

𝜌𝐼≡

0. Therefore, the total mass is given in terms of the spatial

mass density 𝜌sonly and the mass balance reads

d𝑡ˆ

𝑊

𝜌sd𝑊=−ˆ

𝛤+∪𝛤−

𝜌s(𝑣s−𝑤)·𝑛d𝐴 , (2.43)

which reduces in its local form to

𝜕𝜌s

𝜕𝑡 +∇s·(𝜌s𝑣s)=0,𝑒·J(𝑣s−𝑤𝐼)𝜌sK= 0 .(2.44)

In contrast, the equation for the micro-inertia tensor without structural change is not found from

a global balance law. Instead, the arguments provided in [Cowin (1974)] lead to the following

relation 𝜕𝐽s

𝜕𝑡 +𝑣s·(∇s⊗𝐽s) = 𝜔s×𝐽s−𝐽s×𝜔s.(2.45)

As an alternative to the volumetric density

𝜓

in Eq.

(2.9)

, it is customary to consider specific

densities, i.e.,𝜓→𝜌s𝜓. Using the mass balance, the localization in Eq. (2.12) simplifies to

𝜌s

D𝜓

D𝑡+∇s·𝜙V=𝑝V+𝑠V,(2.46)

where the substantial time derivative is introduced by means of

D𝜓

D𝑡:=𝜕𝜓

𝜕𝑡 𝑥s

+𝑣s·(∇s⊗𝜓),(2.47)

see [Ivanova, Vilchevskaya, and Müller (2016)]. Note that this derivative does not represent a

total time derivative, since 𝜓=^

𝜓s(𝑥s, 𝑡)and therefore

d𝜓

d𝑡=𝜕𝜓

𝜕𝑡 𝑥s

+𝑤·(∇s⊗𝜓)⇒D𝜓

D𝑡=d𝜓

d𝑡+ (𝑣s−𝑤)·(∇s⊗𝜓).(2.48)

The spatial description reduces to a material one if the identification

𝜒s≡𝜒m

is made. Subse-

quently, all velocities coincide,

𝑣s

𝑣m

𝑤

, and the functional dependence

𝜓m=^

𝜓m(𝑥m, 𝑡)

results in the material time derivative,

D𝜓

D𝑡→𝜕𝜓

𝜕𝑡 𝑋m

=𝜕𝜓

𝜕𝑡 𝑥m

+𝑣m·(∇m⊗𝜓),∇m=𝜕

𝜕𝑥m

.(2.49)

The balance of linear momentum

is the continuum version of

Newton

’s law of motion, i.e.,

the temporal change of the linear momentum of a body is given by forces. However, in a spatial

description, the control volume does not have to be material and can be open. Therefore, the

balance of linear momentum in the control volume

𝑊

has to incorporate convective momentum

exchange, d

d𝑡ˆ

𝑊

𝜌s𝑝d𝑊=𝐹(mech) +𝐹(em) +𝐹(conv) ,(2.50)

Chapter 2. Classical continuum electromechanics

On the electromagnetic coupling problem 15

where the forces are decomposed into a convective and a non-convective mechanical part as well

as forces of electromagnetic origin:

𝐹(mech) =ˆ

𝑊

𝜌s𝑓(mech) d𝑊+ˆ

𝛤

𝑡(mech) d𝐴+ˆ

𝜕𝑊 ∩𝐼

𝑡(mech)

𝐼dℓ , (2.51a)

𝐹(em) =ˆ

𝑊

𝑓(em) d𝑊+ˆ

𝐼

𝑓(em)

𝐼d𝐴 , (2.51b)

𝐹(conv) =−˛

𝛤+∪𝛤−

𝜌s𝑝s⊗(𝑣s−𝑤)·𝑛d𝐴 . (2.51c)

Therein,

𝑓(mech)

is the specific body force density,

𝑡(mech)

is the mechanical surface force density

usually referred to as traction vector and

𝑡(mech)

𝐼

is the singular traction vector arising from

effects like surface tension. It is connected to the surface stress tensor 𝜎𝐼via

𝑡(mech)

𝐼=𝜈·𝜎𝐼,(2.52)

see [Slattery, Sagis, and Oh (2007)]. The expressions for mechanical and electromagnetic forces

in Eq.

(2.51)

differ due to their different origins. For example, the traction vector represents

mechanical contact forces. There is no such notion for electromagnetic forces, see [Rinaldi and

Brenner (2002)].

The (regular) traction vector is connected to the surface normal by means of

Cauchy

’s

tetrahedron argument such that

𝑡(mech)

𝑛·𝜎

𝜎T·𝑛

. For this argument, a limiting process

is performed in which the volumetric contributions shrink faster than the surface traction. In

the presence of the electromagnetic forces contained in

𝐹(em)

, the volumetric contribution

𝑓(em)

does not alter the tetrahedron argument. However, at the first glance, the surface force density

𝑓(em)

𝐼

may contribute to the stress tensor as the set

𝐼

includes the boundary domain

𝜕𝛺

. In

order to verify that this is not the case, the physical origin of the surface force density has

to be considered. Due to jumps of electromagnetic fields, e.g., the magnetization across the

interface of a magnet with its non-magnetizable surrounding, surface forces arise. Consider the

pill-box domain

𝑃𝛿

, which is a cylinder of the height

𝛿

centered around a patch of

𝐼

. In the

limit

𝛿→

0, the volume of the domain vanishes while its surface approaches the patch of

𝐼

Thus, the surface force density is defined via

𝐼

𝑓(em)

𝐼d𝐴:= lim

𝛿→0ˆ

𝑃𝛿

𝑓(em) d𝑊 , (2.53)

see, e.g., [Toupin (1956), pg. 870]. The balance of linear momentum in Eq.

(2.50)

is in fact not

a conservation law. In a non-relativistic context, the mechanical body force

𝑓(mech)

is usually

given by gravity, which can be interpreted as a volumetric supply term. On the other hand, the

force densities

𝑓(em)

and

𝑓(em)

𝐼

are regarded as production terms, see the discussion in [Reich,

Rickert, and Müller (2017)].

The balance of angular momentum

is obtained from

Eulers

law, which states that the

change of angular momentum of a body is given by moments:

d𝑡ˆ

𝑊

𝜌s(𝑥s×𝑝s+𝑠) d𝑊=𝑇(mech) +𝑇(em) +𝑇(conv) ,(2.54)

Section 2.3. Continuum thermodynamics

where the moments acting on the body are given by

𝑇(mech) =ˆ

𝑊

𝜌s(𝑥s×𝑓(mech) +𝜏(mech)) d𝑊+

+ˆ

𝛤+∪𝛤−

(𝑥s×𝑡(mech) +𝑚(mech)) d𝐴+ˆ

𝜕𝑊 ∩𝐼

𝑥s×𝑡(mech)

𝐼dℓ ,

𝑇(em) =ˆ

𝑊

𝑥s×𝑓(em) d𝑊+ˆ

𝑊

𝜏(em) d𝑊+ˆ

𝐼

𝜏(em)

𝐼d𝐴 ,

𝑇(conv) =−˛

𝛤

𝜌s(𝑥s×𝑝s+𝑠)⊗(𝑣s−𝑤)·𝑛d𝐴 .

(2.55)

Therein,

𝜏(mech)

is the mechanical body moment,

𝑚(mech)

is the mechanical surface moment

density usually referred to as the surface couple. The regular surface couple is connected to

the couple stress tensor via

𝑚

𝑛·𝜇

𝜇T·𝑛

. A mechanical surface moment density similar

𝑡(mech)

𝐼

is usually not introduced. It should be noted that

𝑓(em)

𝐼

is not explicitly contained in

𝑇(em)

. Nevertheless, the surface force density contributes to the total moment via the surface

moment density 𝜏(em)

𝐼, which is defined as

𝐼

𝜏(em)

𝐼d𝐴:= lim

𝛿→0ˆ

𝑃𝛿

(𝑥s×𝑓(em) +𝜏(em)) d𝑊 . (2.56)

It is interesting to note that the electromagnetic forces are required in order to calculate the

electromagnetic moments. This means that the choice for the force model influences the choice

of the moment model.

The energy balance

emphasizes the fact that the linear and angular momentum alone cannot

describe a continuum body completely. With a grain of salt it could be argued that with the

set of equations above, the (absolute) temperature

𝑇s

(

𝑥s, 𝑡

)cannot be computed. Since

temperature is not additive, it cannot be balanced. Thus, the concept of internal energy is

introduced, which could be interpreted as a measure of the “temperature content” inside a body.

However, it is customary to postulate the balance of the total energy as the manifestation of

the first law of thermodynamics [Toupin (1956); Müller (1985)]

d𝑡𝑈=𝑄+𝑊(mech) +𝑊(em) +𝑊(conv) ,(2.57)

where 𝑈is the total energy contained in 𝑊

𝑈(mech) =ˆ

𝑊

𝜌s(ℰkin +𝑢) d𝑊 , (2.58)

Chapter 2. Classical continuum electromechanics

On the electromagnetic coupling problem 17

and the supply, as well as production, are given by, see also [Reich and Müller (2018)]:

𝑄=ˆ

𝑊

𝜌s𝑟d𝑊−ˆ

𝛤+∪𝛤−

𝑛·𝑞d𝐴+ˆ

𝐼

𝑟𝐼d𝐴 ,

𝑊(mech) =ˆ

𝑊

𝜌s(𝑓(mech) ·𝑣s+𝜏(mech) ·𝜔s) d𝑊+

+ˆ

𝛤+∪𝛤−

(𝑡(mech) ·𝑣s+𝑚(mech) ·𝜔s) d𝐴+ˆ

𝜕𝑊 ∩𝐼

𝑡(mech)

𝐼·𝑣sdℓ ,

𝑊(conv) =−ˆ

𝛤+∪𝛤−

𝜌s(ℰkin +𝑢)⊗(𝑣s−𝑤)·𝑛d𝐴 ,

(2.59)

where

𝑞

is the non-convective energy flux,

𝑟

is the specific radiation and

𝑟𝐼

its singular counterpart.

Furthermore, the electromagnetic contribution is given by

𝑊(em) =ˆ

𝑊

ℎ(em) d𝑊+ˆ

𝐼

ℎ(em)

𝐼d𝐴 . (2.60)

Therein, the volumetric energy production due to electromagnetic fields can be denoted as

follows

ℎ(em) =𝑓(em) ·𝑣s+𝜏(em) ·𝜔s+𝑟(em) ,(2.61)

where

𝑟(em)

is an energy supply term that does not require mechanical motion, i.e., it represents

electromagnetic radiation. It is interesting to note that the moments

𝜏(em)

can only contribute

to the electromagnetic work, if, for example, a micropolar continuum is considered. That is

to say,

𝜏(em)

can only contribute if the notion of a rotational kinematic quantity exists. The

resulting singular energy production is defined via

𝐼

ℎ(em)

𝐼d𝐴:= lim

𝛿→0ˆ

𝑃𝛿

(𝑓(em) ·𝑣s+𝜏(em) ·𝜔+𝑟(em)) d𝑊(2.62)

The entropy balance

represents the second law of thermodynamics, for which many different

formulations exist. However, in continuum physics, where a rational theory is desired, most

authors follow the methodology presented in [Truesdell and Toupin (1960)]. First, the existence

of an additive quantity named entropy,

𝑆

, is postulated. Its mass-specific density is given by

𝜂

The entropy balance then reads

d𝑡ˆ

𝑊

𝜌s𝜂d𝑊=−ˆ

𝛤+∪𝛤−

𝜑·𝑛d𝐴+ˆ

𝑊

(𝜎+𝑧) d𝑊−ˆ

𝛤+∪𝛤−

𝜌s𝜂(𝑣s−𝑤)·𝑛d𝐴 . (2.63)

Therein,

𝜑

is referred to as entropy flux and

𝑧

is the entropy supply. Most importantly, the

entropy production 𝜎≥0is assumed to be non-negative.

Section 2.3. Continuum thermodynamics

2.4 Localization and summary

The set of Maxwell’s equations in regular points is given by Eqs. (2.23)–(2.35):

𝜕𝐵

𝜕𝑡 +∇s×𝐸=0,∇s·𝐵= 0 ,(2.64a)

−𝜕𝐷

𝜕𝑡 +∇s×𝐻=𝐽=𝑗+𝑞𝑣s,∇s·𝐷=𝑞 , (2.64b)

−𝜕D

𝜕𝑡 +∇s×H=𝐽f=𝑗f+𝑞f𝑣s,∇s·D=𝑞f,(2.64c)

𝜕𝑃

𝜕𝑡 +∇s×𝑀=𝐽b,−∇s·𝑃=𝑞b.(2.64d)

They must be supplemented by the space-time relations and the field decomposition

𝐻=H+𝑀=1

𝜇0𝐵,𝐷=D−𝑃=𝜖0𝐸,(2.64e)

as well as Maxwell’s equations in singular points:

−𝑤⊥J𝐵K+𝑒×J𝐸K=0,𝑒·J𝐵K= 0 ,(2.64f)

𝑤⊥J𝐷K+𝑒×J𝐻K=𝐽𝐼=𝑗𝐼+𝑞𝐼𝑣𝐼,𝑒·J𝐷K=𝑞𝐼,(2.64g)

𝑤⊥JDK+𝑒×JHK=𝐽f

𝐼=𝑗f

𝐼+𝑞f

𝐼𝑣𝐼,𝑒·JDK=𝑞f

𝐼,(2.64h)

−𝑤⊥J𝑃K+𝑒×J𝑀K=𝐽b

𝐼,−𝑒·J𝑃K=𝑞b

𝐼,(2.64i)

where the normal velocity of the surface is given by 𝑤⊥=𝑒·𝑤𝐼.

The balance equations of continuum thermodynamics in Eqs.

(2.50)

–

(2.63)

are localized to

regular points by means of the relations in Eq.

(2.12)

. Subsequently, the balance of angular

momentum is reduced by the moment of momentum part arising from

𝑥×𝑝

, in order to

obtain the dynamical spin balance. This is achieved by cross multiplying the balance of linear

momentum with

𝑥

and subsequently by subtracting it form the balance of angular momentum.

In a similar fashion, the balance of the total non-electromagnetic energy is reduced in order to

obtain the balance of internal energy. Furthermore, the superscript indicating the mechanical

origin of a quantity will be dropped in the following, e.g.,

𝜎(mech)

𝜎

. Together with the mass

balance, the reduced balances for linear momentum, dynamic spin, internal energy and entropy

are obtained for regular points:

D𝜌s

D𝑡=−𝜌s(∇s·𝑣s),(2.65a)

𝜌s

D𝑣s

D𝑡=∇s·𝜎+𝜌s𝑓+𝑓(em) ,(2.65b)

𝜌s𝐽s·D𝜔s

D𝑡=∇s·𝜇+⟨3⟩

𝜖··𝜎+𝜌s𝜏+𝜏(em) −𝜔s×𝐽s·𝜔s,(2.65c)

𝜌s

D𝑢

D𝑡=−∇s·𝑞+𝜎··(∇s⊗𝑣s−𝜔s·⟨3⟩

𝜖) + 𝜇··(∇s⊗𝜔s) + 𝜌s𝑟+𝑟(em) ,(2.65d)

𝜌s

D𝜂

D𝑡=−∇s·𝜑+𝜎+𝑧 . (2.65e)

Therein, the double contraction is defined via

𝐴··

(

𝑎⊗𝑏

)

𝑎·𝐴·𝑏

and

⟨3⟩

𝜖

is the

Levi

–

Civita

Chapter 2. Classical continuum electromechanics

On the electromagnetic coupling problem 19

tensor. Furthermore, the corresponding equations in singular points are given by

−∇𝐼·(1𝐼·𝜎𝐼) = 𝑒·J𝜎+ (𝑤𝐼−𝑣s)⊗𝜌s𝑣sK+𝑓(em)

𝐼,(2.66a)

⟨3⟩

𝜖··𝜎𝐼+𝑒·𝜎𝐼×𝑒=𝑒·J𝜇+ (𝑤𝐼−𝑣s)⊗𝜌s𝐽s·𝜔sK+𝜏(em)

𝐼−𝑥s×𝑓(em)

𝐼,(2.66b)

−∇𝐼·(1𝐼·𝜎𝐼·𝑣s) = 𝑒·J(𝑤𝐼−𝑣s)⊗𝜌s(ℰkin +𝑢)K+𝑟𝐼+ℎ(em)

𝐼+(2.66c)

+𝑒·J𝜎·𝑣s+𝜇·𝜔s−𝑞K,

0≤𝑒·J(𝑣s−𝑤𝐼)𝜌s𝜂−𝜑K,(2.66d)

For a discussion of the entropy inequality in singular points, see [Eringen and Maugin (1990),

pg. 79]. In the local forms of the balance equations in

(2.65a)

, the electromagnetic coupling is

realized by means of

𝑓(em)

𝜏(em)

and

ℎ(em)

as well as their singular counterparts. These form

the electromagnetic coupling model that is the subject of the next chapter.

Section 2.4. Localization and summary

On the electromagnetic coupling problem 21

3 Electromagnetic coupling models

Mathematically speaking, an electromagnetic coupling model (EMCM) determines the connection

between the balance equations of thermodynamics and

Maxwell

’s equations. This connection

is given in terms of relations for the electromagnetic force density

𝑓(em)

, the electromagnetic

moment density

𝜏(em)

and the electromagnetic energy production

ℎ(em)

. However, in order to

completely specify an EMCM it is not sufficient to state the triple (

𝑓(em),𝜏(em), ℎ(em)

)alone,

because there are different formulations of the thermodynamic balance equations in which they

are inserted. As a result, the electromagnetic coupling quantities must be considered with

respect to the balance equations in local and in global form. The reason for these requirements

lies in the inconsistent terminology in the pertinent literature. For example, in [Eringen (1973)]

the balance of linear momentum differs from Eq.

(2.65b)

by the momentum density.

Eringen

states

𝜌

(

𝑣

𝑔(em)

)as the momentum density, while others only employ the mechanical part

𝜌𝑣

, see [Hutter, Ven, and Ursescu (2006)]. Similarly,

Kovetz

introduces a mixed moment

density in his balance of linear momentum but assumes implicitly that electromagnetic forces

are always short range force, i.e., the balance of linear momentum does not contain volumetric

electromagnetic forces, see [Kovetz (2000), pg. 216]. As a consequence, the definition and the

interpretation of the quantity “force” is altered in

Kovetz

’ theory. This goes to show that even

if two theories employ the same triple (

𝑓(em),𝜏(em), ℎ(em)

), their impact on, e.g., the mechanical

motion, can differ significantly.

In order to render different electromagnetic coupling models comparable, they have to be

adjusted with respect to one common system of balance equations. In Sec.

(3.4)

, frequently used

coupling models are presented and modified such that they fit the framework from this thesis in

Chap. 2. Before the analysis of EMCMs is performed, a short review of the state of research is

provided in the following.

3.1 State of research

In [McDonald (2017)] a list of over 300 references regarding the electromagnetic coupling

problem is presented, which shows that it is a highly discussed topic. Most of the authors

present one of the following viewpoints. One group of authors points out the differences between

electromagnetic coupling models and attempts to analyze the impact of these differences on

observable quantities, e.g., [Mansuripur (2010); Datsyuk and Pavlyniuk (2015); Obukhov and

Hehl (2003)]. The other group claims that the electromagnetic coupling problem is already

solved, [Pfeifer et al. (2007)], or that at least all models are formally equivalent if the second law

of thermodynamics is properly accounted for, [Hutter, Ven, and Ursescu (2006); Penfield and

Haus (1967)]. For the latter argument, it is assumed that any potentially incorrect coupling

model can be corrected by means of the constitutive equations for

1𝜎

and

𝑞

. This point of view

is discussed in Sec. 4.4.

However, there is one special scenario in which the following electromagnetic coupling model is

generally accepted and experimentally verified. In the absence of magnetization and polarization,

𝑀≡0and 𝑃≡0, the “free field coupling model” is known to be, see [Müller (1985)]:

𝑓(em)

f=𝑞f𝐸+𝐽f×𝐵,𝜏(em)

f=0, ℎ(em)

f=𝐽f·𝐸.(3.1)

Most authors do not consider polar media, but as an extension to their argument one could also add

𝜇

to this

list.

In fact, the first expression is even formulated as a postulate in the theory of rational electro-

magnetism and determines the force fields

𝐸

and

𝐵

. Under the free field conditions, every

electromagnetic coupling model must agree with the one given above. The index “f” in Eq.

(3.1)

indicates the vanishing magnetization and polarization similar to the notation in

Maxwell

’s

equations. In this special scenario,

Maxwell

’s equations reduce significantly, because

𝑞

𝑞f

and 𝐽=𝐽f, which results in

𝑀≡0≡𝑃⇒H=1

𝜇0𝐵,D=𝜖0𝐸.(3.2)

By means of these relations, the expressions in Eq.

(3.1)

can be manipulated such that the

so-called pseudo balance laws arise:

𝑓(em)

f=−𝜕𝑔(em)

𝜕𝑡 +∇s·𝜎(em)

f, ℎ(em)

f=−𝜕𝑢(em)

𝜕𝑡 −∇s·𝑞(em)

f,(3.3a)

with the following auxiliary quantities:

𝑔(em)

f=D×𝜇0H,𝜎(em)

f=−1

2(1

𝜇0H·H+D·𝐸)1+D⊗𝐸+1

𝜇0H⊗H,

𝑢(em)

f=1

2D·𝐸+1

2𝜇0H·H,𝑞(em)

f=𝐸×H.

(3.3b)

Since

𝑓(em)

is a volumetric force density,

𝜎(em)

has the same units as a stress tensor. Analogously,

𝑔(em)

could be referred to as an electromagnetic momentum density and

𝑢(em)

could be interpreted

as electromagnetic energy. Similarly,

𝑞(em)

could be termed electromagnetic energy flux. Of

course, this terminology corresponds to a quite far-fetched interpretation, since the expressions

in Eq.

(3.3)

are nothing more than mathematical identities, see [Eringen and Maugin (1990),

Sec. 3.6]. Assigning physical meaning to these identities is the root of the confusion around the

electromagnetic coupling problem that persists in academia, [Kemp (2011)].

The problem remains, the physically correct expressions for

𝑓(em)

𝜏(em)

and

ℎ(em)

are unknown

in the case of

𝑀

𝑃

. In other words, the influence of magnetization and polarization

upon the electromagnetic coupling is not known. Naturally, attempts are made to generalize the

expressions in Eq.

(3.3)

. By means of the manipulation of

Maxwell

’s equations, it is believed

that one can always find electromagnetic identities of the same form as in Eq. (3.3):

𝜕𝑔(em)

𝜕𝑡 −∇·𝜎(em) =−𝑓(em) ,(3.4a)

⟨3⟩

𝜖··𝐶(em) =𝜏(em) ,(3.4b)

𝜕𝑢(em)

𝜕𝑡 +∇·𝑞(em) =−ℎ(em) .(3.4c)

Therein,

𝜎(em)

is referred to as

Maxwell

stress tensor and

𝑔(em)

is the electromagnetic momen-

tum. Similarly,

𝑢(em)

is termed electromagnetic energy and

𝑞(em)

the electromagnetic energy

flux or

Poynting

vector. The heating term

ℎ(em)

is sometimes called

Joule

heating and is

often decomposed into the power due to electromagnetic forces and force-independent radiation

heating via

ℎ(em) =𝑓(em) ·𝑣s+𝑟(em) .(3.4d)

This is the first indicator of the interplay between electromagnetic energy and electromagnetic

momentum. Furthermore, a nonzero electromagnetic moment

𝜏(em)

is introduced, because the

fields

𝑀

and

𝑃

give rise to the notion of distributed point moments acting on dipoles. Whether

or not the introduction of

𝜏(em)

is reasonable, is discussed in Chap. 4. However, from their

alleged origin as moments acting on dipoles, [Maugin (1976)], it is intuitively clear that they are

axial in character, i.e., they are the skew-symmetric part of some second order tensor

𝐶(em)

. An

Chapter 3. Electromagnetic coupling models

On the electromagnetic coupling problem 23

electromagnetic angular momentum density other than

𝑥s×𝑔(em)

is not found in the pertinent

literature.

The Eqs.

(3.4)

are of the form of and are sometimes referred to as balance equations. In

particular, the Eq.

(3.4c)

is also known as

Poynting

’s theorem and the associated flux density

is usually denoted by

𝑞(em)

𝑆

. However, as mentioned before, these are just identities formed

from

Maxwell

’s equations and are interpreted as balance laws simply because they have

the correct unit. In order to stress this fact, the three equations are henceforth referred to

as electromagnetic identities of (electromagnetic) force, moment and energy. Furthermore, it

has to be noted that

𝜎(em)

is not a real stress tensor when compared to the

Cauchy

stress

tensor, which was shown in [Rinaldi and Brenner (2002)]. Similarly, the fact that

Ponyting

’s

theorem is solely the result of mathematical manipulations of

Maxwell

’s equations rather

than a physical law is acknowledged in [Feynman, Leighton, and Sands (1965)]. This criticism

is humorously expressed in a story about freezing chickens in [Synge (1974)].

There are two mathematical advantages of replacing (

𝑓(em),𝜏(em), ℎ(em)

)with the auxiliary

quantities in Eq.

(3.4)

. First, the surface densities can be computed conveniently. This is

achieved by means of substituting the primary electromagnetic coupling quantities in Eqs.

(2.53)

(3.5b) and (3.5c), which results in:

𝑓(em)

𝐼=𝑛·J𝑤𝐼⊗𝑔(em) +𝜎(em)K,(3.5a)

𝜏(em)

𝐼=𝑥s×𝑓(em)

𝐼,(3.5b)

ℎ(em)

𝐼=𝑓(em)

𝐼·𝑣s+𝑟(em)

𝐼, 𝑟(em)

𝐼=𝑛·J𝑢(em)𝑤𝐼−𝑞(em)K,(3.5c)

see [Reich, Rickert, and Müller (2017)]. Therein, the normal vector

𝑒

was replaced with

𝑛

because the distinction between these two was only relevant for the derivations and because

the singular quantities are most often evaluated at the surface of a material body. Since there

exists no notion of an electromagnetic coupling tensor, the electromagnetic moment density

cannot contribute to the electromagnetic heating in singular points. The second advantage of

electromagnetic identities arises from the following modified identity,

𝑟(em) =−𝜌s(𝜌−1

s𝑢(em))∙+ (𝜎(em) +𝑣s⊗𝑔(em))··(∇s⊗𝑣s)−∇s·𝒮(em) +˙

𝑔(em) ·𝑣s,

𝒮(em) =𝑞(em) + (𝜎(em) +𝑣s⊗𝑔(em) −𝑢(em)1)·𝑣s,(3.6)

where the superimposed dot indicates the substantial time derivative from Eq.

(2.47)

. At

this point, it can already be noted that the expression for

𝑟(em)

in Eq.

(3.6)

demonstrates the

interdependence between the electromagnetic momentum and electromagnetic energy quantities.

However, this identity is only meaningful for material domains, since the mass density

𝜌s

well as the particle velocity

𝑣s

are involved. There is another representation for

𝑟(em)

, which will

prove useful for the analysis of the thermodynamic consistency in Sec. 4.1. Using

Maxwell

’s

equations, it follows from Eq. (3.6) that

𝑟(em) =𝑗f·ℰ+˙

𝑃·ℰ−˙

𝐵·ℳ+∇s·𝑞(em)

aux +𝜎(em)

aux ··(∇s⊗𝑣s),(3.7)

where

𝑞(em)

aux

and

𝜎(em)

aux

depend upon the coupling model. The newly introduced quantities

ℰ

and

𝐵are referred to as electromagnetic intensities:

ℰ=𝐸+𝑣s×𝐵,ℳ=𝑀+𝑣s×𝑃,ℋ=H−𝑣s×D,(3.8)

where ℋis used in Sec. 3.4.

Section 3.1. State of research

3.2 Reference model and decomposition

The generalized

Lorentz

model was put forward by [Truesdell and Toupin (1960), Sec. 284]

and is also adopted by [Müller (1985)] as well as [Obukhov and Hehl (2003)]. It is obtained

from the

Lorentz

model in Eq.

(3.1)

by replacing the free charges and the free currents with

their total counterparts:

𝑞f→𝑞=𝑞f+𝑞b,𝐽f→𝐽=𝐽f+𝐽b.(3.9)

This replacement is justified by the assumption that the free and bound electric quantities

behave identically regarding their mechanical response. Therefore, the generalized

Lorentz

force density is given by

𝑓gL = (∇·𝐷)𝐸+𝐽×𝐵.(3.10)

Similarly, the electromagnetic energy in presence of matter is constructed from its vacuum form

by replacements. In total, the generalized Lorentz model is given by

𝑓gL = (𝑞f−∇·𝑃)𝐸+(︂𝑞f𝑣+𝑗f+𝜕𝑃

𝜕𝑡 +∇×𝑀)︂×𝐵,(3.11a)

𝜏gL ≡0, ℎgL =𝐽·𝐸,(3.11b)

with the complementary quantities

𝑔gL =𝐷×𝐵, 𝑢gL =1

2(𝜖0𝐸·𝐸+1

𝜇0𝐵·𝐵),(3.11c)

𝜎gL =−1

2(𝜖0𝐸·𝐸+1

𝜇0𝐵·𝐵)1+𝜖0𝐸⊗𝐸+1

𝜇0𝐵⊗𝐵,(3.11d)

𝐶(em) ≡0,𝑞gL =𝐸×1

𝜇0𝐵.(3.11e)

Furthermore, the surface densities are given by

𝑓gL

𝐼=(︀𝑞f

𝐼−𝑛·J𝑃K)︀⟨𝐸⟩+(︀𝐽f

𝐼−J𝑃K𝑤⊥+𝑛×J𝑀K)︀×⟨𝐵⟩,(3.11f)

𝜏gL

𝐼=𝑥×𝑓gL

𝐼,(3.11g)

𝑟gL

𝐼= (𝐽f

𝐼−𝑤⊥J𝑃K+𝑛×J𝑀K)·⟨𝐸⟩.(3.11h)

The contained quantities have the same form as in Eq. (3.3) and could be obtained from them

by performing the following replacements:

D→𝐷=𝜖0𝐸,H→𝐻=1

𝜇0𝐵.(3.12)

Since the generalized

Lorentz

coupling model naturally reduces to the free field model from

Eq.

(3.1)

, it is convenient to consider it as a reference model. Thus, any electromagnetic coupling

model may be decomposed into a

Lorentz

part as well as a rest part. The decomposition for

primary quantities is denoted as follows:

𝑓(em) =𝑓gL + Δ𝑓,𝜏(em) = Δ𝜏, ℎ(em) =ℎgL + Δℎ , (3.13a)

where

𝜏gL ≡0

was used. For the derived quantities the decomposition is performed analogously:

𝑔(em) =𝑔gL + Δ𝑔,𝜎(em) =𝜎gL + Δ𝜎,𝐶(em) = Δ𝐶,

𝑢(em) =𝑢gL + Δ𝑢 , 𝑞(em) =𝑞gL + Δ𝑞.(3.13b)

All rest parts Δ

(·)

are assumed to be functions of the form

𝑓

(

𝐸,𝐵,𝑃,𝑀

)with the

Chapter 3. Electromagnetic coupling models

On the electromagnetic coupling problem 25

following constraint ^

𝑓(𝐸,𝐵,𝑃=0,𝑀=0) = 0,(3.14)

in order to satisfy the free field conditions in Eq.

(3.1)

. That is to say that the rest parts

contain only material-dependent behavior. The analysis using this decomposition renders the

impact of the differences between forces models immediately visible. Furthermore, the rest parts

themselves fulfill the electromagnetic identities, e.g.,

𝜕Δ𝑔

𝜕𝑡 −∇·(Δ𝜎) = −Δ𝑓.(3.15)

Hence, it is sufficient to state, e.g., the pair

{

𝑔,

𝜎}

as the rest force Δ

𝑓

is then a consequence.

Since the surface force density in Eq.

(3.5a)

is a linear expression of

𝑔(em)

and

𝜎(em)

, the same

decomposition as above applies

𝑓(em)

𝐼=𝑓gL

𝐼+ Δ𝑓𝐼,Δ𝑓𝐼=𝑛·J𝑤⊗Δ𝑔+ Δ𝜎K.(3.16)

3.3 Coupling model construction and modifications

Instead of providing an electromagnetic coupling model directly, the auxiliary quantities

𝑔(em)

and

𝑢(em)

can be used. Subsequently, they are substituted into the pseudo balance laws in

Eq.

(3.4)

. By means of

Maxwell

’s equations, the contained derivatives are manipulated until

the pairs (

𝜎(em),𝑓(em)

)or (

𝑞(em), ℎ(em)

)can be identified, which is demonstrated for the generalized

Lorentz in Sec. 3.2.

Naturally, this procedure raises the question of how

𝑔(em)

and

𝑢(em)

are derived. To this end,

generalizations of the free electromagnetic momentum density

𝑔(em)

and of the free electromag-

netic energy 𝑢(em)

fcould be obtained from one of the following replacements:

(D,H)→(𝐷,𝐻)or (D,H)→(𝐷,H)or (D,H)→(D,𝐻).(3.17)

The first relation was already introduced in Eq.

(3.12)

for the generalized

Lorentz

model. If

these modifications are applied, the following variants for possible generalizations are found:

𝑔gL =𝐷×𝜇0𝐻, 𝑢gL =1

2(𝐷·𝐸+1

𝜇0𝐵·𝐵),(3.18a)

𝑔A=𝐷×𝜇0H, 𝑢 =1

2(𝐷·𝐸+1

𝜇0H·H),(3.18b)

𝑔M=D×𝜇0𝐻, 𝑢 =1

2(D·𝐸+1

𝜇0𝐻·𝐻).(3.18c)

The latter two momentum densities are referred to as

Abraham

momentum density and

Minkowski

momentum density, because they were originally introduced in [Abraham (1909)]

and [Minkowski (1910)], respectively. However, the corresponding energy densities are not

associated with

Abraham

Minkowski

. Unfortunately, not all electromagnetic energy models

are found by this approach, see Sec. 3.4. Therefore, only the electromagnetic force models are

investigated in the remainder of this section. Of course, the employed methods can be applied

to the electromagnetic energy model as well.

Consider the last two momentum densities in Eq.

(3.18)

. It follows from the space-time

relations that their rest parts are given by:

Δ𝑔A=−𝐷×𝜇0𝑀,Δ𝑔M=𝑃×𝐵.(3.19)

Both of these differences to the

Lorentz

model vanish for non-magnetizable and non-polarizable

matter, which is in agreement with the constraint in Eq.

(3.14)

. Furthermore, the rest parts

consist of products of electric and magnetic quantities. Hence, any experiment that aims to

Section 3.3. Coupling model construction and modifications

detect the momentum differences Δ

𝑔A

and Δ

𝑔M

must necessarily incorporate both, electric and

magnetic effects.

After a particular combination of an electromagnetic momentum density and an energy density

is selected, the identifications

𝑔(em) →(𝜎(em),𝑓(em)), 𝑢(em) →(𝑞(em), ℎ(em))(3.20)

are not unique, due to the following two reasons. First, an expression with zero value can be

added to the electromagnetic identities in Eq. (3.4). For example, by considering the identity

0= (∇·𝐵)𝑀=∇·(𝐵⊗𝑀)−𝐵·(∇⊗𝑀),(3.21)

one could add this expression to the electromagnetic momentum balance and conclude the

following addition to any force model:

Δ𝑔=0,Δ𝜎=𝐵⊗𝑀,Δ𝑓=𝐵·(∇⊗𝑀).(3.22)

The second type of ambiguity arises from the rearrangements of already existing expressions in

a given coupling model. In doing so, parts of the electromagnetic force can be transferred into

the electromagnetic stress tensor and vice versa. Consider for example the manipulation of an

expression in a force term like

𝑞𝐸

. Before the product rule is applied, a zero-valued expression

is generated,

0=𝑞𝐸−(∇·𝐷)𝐸=𝑞𝐸−∇·(𝐷⊗𝑃) + 𝐷·(∇⊗𝐸).(3.23)

If this is added to the electromagnetic momentum identity it could be concluded that

Δ𝑔=0,Δ𝜎=−(𝐷⊗𝑃),Δ𝑓=𝐷·(∇⊗𝐸)−𝑞𝐸.(3.24)

To illustrate this further, consider the following identities:

0= (∇×𝑀)×𝐵−∇·(𝐵⊗𝑀)+(∇⊗𝑀)·𝐵(3.25a)

= (∇×𝑀)×𝐵+∇·([𝐵·𝑀]1−𝐵⊗𝑀)−(∇⊗𝐵)·𝑀,(3.25b)

0= (∇×𝐸)×𝑃−∇·(𝑃⊗𝐸)−(∇·𝑃)𝐸−(∇⊗𝐸)·𝑃.(3.25c)

These lead to the following modifications

(3.25a) ⇒Δ𝜎=−𝐵⊗𝑀,Δ𝑓= (∇×𝑀)×𝐵−(∇⊗𝑀)·𝐵,

(3.25b) ⇒Δ𝜎= (𝐵·𝑀)1−𝐵⊗𝑀,Δ𝑓= (∇×𝑀)×𝐵−(∇⊗𝐵)·𝑀,(3.26)

The consequences of this reshuffling between

𝑔(em)

𝜎(em)

and

𝑓(em)

are discussed in Sec. 4.4. All of

the

Maxwell

stress tensor modifications introduced above have the same structure. Therefore,

the following generic form for the electromagnetic stress tensor 𝜎(em) could be considered:

𝜎(em) =𝜅1+

∑︁

𝛼=1

∑︁

𝛽=1

𝜎𝛼𝛽𝑞𝛼⊗𝑞𝛽,𝑞𝛼={𝐵,𝐸,𝑀,𝑃}𝛼.(3.27)

The coefficient

𝜅

may depend upon the fields

𝑞𝛼

arbitrarily, i.e.,

𝜅

^𝜅

(

𝐵,𝐸,𝑀,𝑃

). In contrast,

all

𝜎𝛼𝛽

are assumed to be constants of appropriate units. In particular, the representation in

Eq.

(3.27)

may be reduced significantly, if linear electromagnetic material behavior is considered,

i.e.,

𝑃=𝜖0𝜒e𝐸,𝑀=𝜒mH,(3.28)

where

𝜒e

and

𝜒m

are the electric and magnetic susceptibility, respectively. Thus, it follows that

Chapter 3. Electromagnetic coupling models

On the electromagnetic coupling problem 27

the electric field quantities

{𝐸,D,𝑃}

are proportional to each other and the same holds for

{𝐵,H,𝑀},

D=𝜖0𝜖r𝐸,H=1

𝜇0𝜇r

𝐵,𝑀=𝜇r−1

𝜇0𝜇r

𝐵,(3.29)

where the relative permittivity is given by

𝜖r

= 1 +

𝜒e

and the relative permeability reads

𝜇r= 1 + 𝜒m. This leads to a significant reduction2of the representation in Eq. (3.27),

𝜎(em) =−1

2(𝐾1𝐵·𝐵+𝐾2𝐸·𝐸)1+𝑆1𝐵⊗𝐵+𝑆2𝐸⊗𝐸.(3.30)

This representation will prove useful in analyzing the differences in electromagnetic force

predictions. It is therefore convenient to state the coefficients

𝐾𝑖

and

𝑆𝑖

alongside the presentation

of an electromagnetic coupling model. In particular, for the generalized

Lorentz

model it can

be deduced from Eq. (3.11c) that the contained coefficients are given by:

𝐾gL

1=1

𝜇0

, 𝐾gL

2=𝜖0, 𝑆gL

1=1

𝜇0

, 𝑆gL

2=𝜖0,(3.31)

As the electromagnetic moment identity in Eq.

(3.4b)

is an algebraic equation rather than

a differential equation, no reshuffling between

𝜏(em)

and

𝐶(em)

may occur due to

Maxwell

’s

equations. For the energy model, however, the same situation as for the force model arises. The

corresponding analysis is similar, but the investigation of its impact is beyond the scope of this

thesis.

3.4 Frequently used coupling models

The electromagnetic coupling models proposed by different authors need to be interpreted with

respect to the underlying basic equations, i.e., the balances for linear momentum, angular

momentum and energy in global form. Furthermore, different definitions and units of electro-

magnetic fields need to be accounted for. In this section, some of the prominent coupling models

are presented in their original form and subsequently modified in order to be consistent with

the framework provided in Sec. 2.4. Quantities that are taken directly from another source

are marked with a “

” as subscript or superscript and observable quantities are assumed to

agree, e.g.,

𝜌*

𝜌

𝑣*

𝑣

. Another difference to the framework from Sec. 2.4 arises from

the fact that most authors used a material description instead of a spatial description. By

specializing the spatial description to a material one, a direct comparison becomes possible.

Note that not all quantities introduced in Sec. 3.1 will be provided in the following. For example,

in order to specify the energy model, it is sufficient to state the electromagnetic energy

𝑢(em)

and

the electromagnetic energy flux

𝑞(em)

because the electromagnetic power

ℎ(em)

is subsequently

determined by Eq. (3.4c).

3.4.1 ABRAHAM and MINKOWSKI

In the original paper [Abraham (1909)] by

Abraham

Heaviside

–

Lorentz

units are used

and the right-hand side divergence is employed, such that

𝜎(em)

= (

𝜎(em)

)

. Furthermore,

Maxwell

’s equations and the electromagnetic fields are denoted in a moving system. In the

following, the employed relations are presented with respect to a rest system. The relevant

quantities are given by:

𝑔A=𝐷×𝜇0H, 𝑢A=1

2(𝐸·D+H·𝐵),𝜏A=0,(3.32a)

In principle, combinations like

𝐸⊗𝐵

are possible, but are they are not found in the most commonly used

electromagnetic force models presented in Sec. 3.4 and are neglected.

Section 3.4. Frequently used coupling models

together with the fluxes:

𝜎A=−1

2(D·𝐸+𝐵·H)1+D⊗𝐸+𝐵⊗H,𝑞A=𝐸×H.(3.32b)

From these expressions, the electromagnetic force density can be computed

𝑓A=𝑞f𝐸+𝐽f×𝜇0H+𝑃·(∇⊗𝐸) + 𝑀·(∇⊗𝜇0H) +

+𝜕𝑃

𝜕𝑡 ×𝜇0H−𝜕𝑀

𝜕𝑡 ×𝜇0𝜖0𝐸−1

2∇(𝑃·𝐸+𝑀·𝜇0H).(3.32c)

In the literature, a symmetrized version of the

Abraham

tensor is commonly used [Mansuripur

(2017)]

𝜎A, mod =−1

2(D·𝐸+𝐵·H)1+1

2(D⊗𝐸+𝐸⊗D+𝐵⊗H+H⊗𝐵).(3.32d)

Therefore, the modified force volume density is connected to the original one via

𝑓A, mod =𝑓A+1

2∇·(︀𝐸⊗D−D⊗𝐸+H⊗𝐵−𝐵⊗H)︀

2𝑞f𝐸+𝐽f×𝜇0H+ (𝑃−1

2D)·(∇⊗𝐸)+(𝜇0𝑀−1

2𝐵)·(∇⊗H) +

+𝜕𝑃

𝜕𝑡 ×𝜇0H−𝜕𝑀

𝜕𝑡 ×𝜇0𝜖0𝐸−1

2∇(𝑃·𝐸+𝑀·𝜇0H) + 1

2(∇·𝐸)D+

2𝐸·(∇⊗D) + 1

2H·(∇⊗𝐵)−1

2(∇·𝑀)𝐵

(3.32e)

For the investigations in this work, the following differences to the generalized

Lorentz

model

are most relevant:

Δ𝑔A=−𝐷×𝜇0𝑀,(3.33a)

Δ𝜎A=−1

2(𝑃·𝐸−𝐵·𝑀)1+𝑃⊗𝐸−𝐵⊗𝑀,(3.33b)

Δ𝜎A, mod =−1

2(𝑃·𝐸−𝐵·𝑀)1+1

2(𝑃⊗𝐸+𝐸⊗𝑃−𝐵⊗𝑀−𝑀⊗𝐵).(3.33c)

In particular, the coefficients for the specialization in Eq. (3.30) are given by:

𝐾A

1=1

𝜇0𝜇𝑟

, 𝐾A

2=𝜖0𝜖r, 𝑆A

1=1

𝜇0𝜇r

, 𝑆A

2=𝜖0𝜖r.(3.34)

Note that the symmetrization of the

Abraham

stress tensor has no effect on these coefficients.

Similarly, the model by

Minkowski

employs the same electromagnetic stress tensor,

𝜎M

𝜎A

but the moment density is different which results in another volumetric force density [Minkowski

(1910)]

𝑔M=D×𝐵,(3.35a)

𝑓M=𝑞f𝐸+𝐽f×𝐵+𝑃·(∇⊗𝐸) + 𝑃×(∇×𝐸) + (3.35b)

+𝜇0𝑀·(∇⊗H) + 𝜇0𝑀×(∇×H)−1

2∇(𝑃·𝐸+𝜇0𝑀·H).

As a result of the same stress tensor as the

Abraham

model, the same coefficients for the

specialization in Eq. (3.30) arise:

𝐾M

1=1

𝜇0𝜇𝑟

, 𝐾M

2=𝜖0𝜖r, 𝑆M

1=1

𝜇0𝜇r

, 𝑆M

2=𝜖0𝜖r.(3.35c)

Chapter 3. Electromagnetic coupling models

On the electromagnetic coupling problem 29

3.4.2 EINSTEIN and LAUB

Even though the controversy around the electromagnetic momentum is named after

Abraham

and

Mikowski

, the model by

Einstein

and

Laub

was also a contestant, [Einstein and Laub

(1908)]. To this date, the

Einstein

–

Laub

model is still in use, see for example [Mansuripur,

Zakharian, and Wright (2013)]. In particular, the corresponding formulas are easily retrieved

from the presentations in [Mansuripur (2017)], because almost the same notation is employed in

this thesis, except for the following two replacements:

𝐻*=H,𝑀*=𝜇0𝑀.(3.36)

The Einstein–Laub coupling model consist of the following expressions:

𝑓EL =𝑞f𝐸+(︂𝑞f𝑣+𝑗f+𝜕𝑃

𝜕𝑡 )︂×𝜇0H−𝜇0

𝜕𝑀

𝜕𝑡 ×𝜖0𝐸+(3.37a)

+𝑃·(∇⊗𝐸) + 𝜇0𝑀·(∇⊗H),

𝜏EL =𝑃×𝐸+𝜇0𝑀×H,(3.37b)

ℎEL =𝐽f·𝐸+𝐸·𝜕𝑃

𝜕𝑡 +H·𝜇0

𝜕𝑀

𝜕𝑡 .(3.37c)

By means of

Maxwell

’s equations and the electromagnetic identities from Eq.

(3.4)

the following

associated quantities are obtained:

𝑔EL =𝜖0𝐸×𝜇0H,(3.37d)

𝜎EL =−1

2(𝜖0𝐸·𝐸+𝜇0H·H)1+D⊗𝐸+𝐵⊗H,(3.37e)

𝐶EL =𝑃⊗𝐸+𝑀⊗𝜇0H,(3.37f)

𝑢EL =1

2(𝜖0𝐸·𝐸+𝜇0H·H),(3.37g)

𝑞EL =𝐸×H.(3.37h)

It should be noted that the momentum density is equal to the

Abraham

momentum density,

𝑔EL

𝑔A

. By utilizing the electromagnetic identities in Eq.

(3.4)

one often replaces the primary

quantities

𝑓(em)

𝜏(em)

and

ℎ(em)

by the derived quantities

𝑔(em)

𝜎(em)

𝐶(em)

𝑢(em)

and

𝑞(em)

Therefore, the model decomposition introduced in Sec. 3.2 is presented for the

Einstein

–

Laub

model’s derived quantities only:

Δ𝑔EL =−𝜖0𝐸×𝜇0𝑀,(3.38a)

Δ𝜎EL =𝜇0(H·𝑀+1

2𝑀·𝑀)1+𝑃⊗𝐸−𝐵⊗𝑀,(3.38b)

Δ𝐶EL =𝐶EL =𝑃⊗𝐸+𝑀⊗𝜇0H,(3.38c)

Δ𝑢EL =−𝜇0(H·𝑀+1

2𝑀·𝑀),(3.38d)

Δ𝑞EL =−𝐸×𝑀.(3.38e)

It can be seen that all differences to the generalized

Lorentz

model are material dependent.

Therefore, they vanish in the case of non-magnetizable and non-polarizable matter. In view of

Eq. (3.30), the reduced electromagnetic stress tensor components are given by

𝐾EL

1=1

𝜇0𝜇2

, 𝐾EL

2=𝜖0, 𝑆EL

1=1

𝜇0𝜇r

, 𝑆EL

2=𝜖0𝜖r.(3.39)

Section 3.4. Frequently used coupling models

3.4.3 ERINGEN and MAUGIN

In the textbook by

Eringen

and

Maugin

Heaviside

–

Lorentz

units are used. According to

[Eringen and Maugin (1990), pg. 406], the following replacements must be performed:

𝐷*→1

√𝜖0

D,𝐸*→√𝜖0𝐸,𝑃*→1

√𝜖0

𝑃,(𝑞e)*→1

√𝜖0

𝑞f,

𝐵*→1

√𝜇0

𝐵,H*→√𝜇0H,𝑀*→√𝜇0𝑀,𝒥*→1

√𝜖0

𝑗f.

(3.40)

Besides these differences in notation and units, the employed formulation of

Maxwell

’s

equations as well as the thermomechanical balance laws coincide with the versions presented in

this thesis. The electromagnetic coupling model put forward by

Eringen

and

Maugin

is given

by [Eringen and Maugin (1990), pg. 62],

𝑓Er =𝑞fℰ+(︂𝑗f+𝜕𝑃

𝜕𝑡 +∇·(𝑣⊗𝑃)−𝑃·(∇⊗𝑣))︂×𝐵+(3.41a)

+𝑃·(∇⊗ℰ)+(∇⊗𝐵)·ℳ,

𝜏Er =𝑃×ℰ+ℳ×𝐵,(3.41b)

ℎEr =𝑓Er ·𝑣+ℰ·D𝑃

D𝑡−(∇·𝑣)ℰ·𝑃−ℳ·D𝐵

D𝑡+𝑗f·ℰ.(3.41c)

Furthermore, the derived quantities are given by

𝑔Er =𝜖0𝐸×𝐵,(3.41d)

𝜎Er =−1

2(𝜖0𝐸·𝐸+1

𝜇0𝐵·𝐵−2ℳ·𝐵)1+(3.41e)

+𝑃⊗ℰ−𝐵⊗ℳ+𝜖0𝐸⊗𝐸+1

𝜇0𝐵⊗𝐵,

𝐶Er =𝜎Er ,(3.41f)

𝑢Er =1

2(𝜖0𝐸·𝐸+1

𝜇0𝐵·𝐵),(3.41g)

𝑞Er =𝐸×H−(𝐸·𝑃)𝑣.(3.41h)

It is interesting to note that the magnetization and the polarization enter the electromagnetic

stress tensor differently. In particular, the coefficients in Eq. (3.30) are given by:

𝐾Er

1=2−𝜇r

𝜇0𝜇r

, 𝐾Er

2=𝜖0, 𝑆Er

1=1

𝜇0𝜇r

, 𝑆Er

2=𝜖0𝜖r.(3.42)

Finally, note that in [Eringen and Maugin (1990), Chap. 5] the constitutive relations depend

highly upon the electromagnetic fields. Thus, the stress tensor representations given by the

authors contain parts of the electromagnetic force model. Therefore, the presentations in this

section may not be complete and should be used with caution. However, for an isotropic elastic

solid

Eringen

and

Maugin

denote their stress tensor via, see [Eringen and Maugin (1990),

pg. 170],

𝜎(mech) = (𝛼𝑒−𝛽𝑒Δ𝑇)1+ (𝜆𝑒−𝛼𝑒) tr(𝜀)1+ 2(𝜇𝑒+𝛼𝑒)𝜀,

𝜀=1

2(∇⊗𝑢+𝑢⊗∇),(3.43)

where Δ

𝑇

is the temperature difference with respect to some reference temperature,

𝜀

is the

linear strain tensor,

𝑢

is the displacement vector and the other parameters are constants.

Similarly, the

Cauchy

stress tensor for linear fluids in [Eringen and Maugin (1990), pg. 178] is

Chapter 3. Electromagnetic coupling models

On the electromagnetic coupling problem 31

given by

𝜎(mech) = (𝜆𝑣∇·𝑣−𝑝)1+𝜇𝑣(∇⊗𝑣+𝑣⊗∇), 𝑝 =−𝜕𝜓

𝜕𝜃 ,

𝜓=𝜓0(𝜌, 𝜃)−1

2𝜒eℰ·ℰ−1

2𝜒m𝐵·𝐵,

(3.44)

where

𝜃

is the absolute temperature. Therefore, at least for the simple cases of a

Hooke

ean

solid or a

Newton

ian fluid, the electromagnetic coupling presented in this section is a complete

representation of the theory by Eringen and Maugin.

3.4.4 KOVETZ

In the textbook by

Kovetz

, SI units are used, but the notation

𝐷*

is employed, [Kovetz

(2000)]. Moreover, different global balance laws are used when compared to those presented in

Sec. 2.4. With a slightly altered notation, Kovetz makes use of:

d𝑡ˆ

𝛺

𝑔*d𝑚=˛

𝜕𝛺

𝜎*·𝑛d𝐴+ˆ

𝛺

𝑏*d𝑚 , (3.45a)

d𝑡ˆ

𝛺

(𝑥−𝑥𝑂)×𝑔*d𝑚=˛

𝜕𝛺

(𝑥−𝑥𝑂)×𝜎*·𝑛d𝐴+ˆ

𝛺

(𝑥−𝑥𝑂)×𝑏*d𝑚 , (3.45b)

d𝑡ˆ

𝛺

𝑢*d𝑚=−˛

𝜕𝛺

(𝑞*+ℰ×ℋ)·𝑛d𝐴+(3.45c)

+ˆ

𝛺

𝑟*d𝑚+˛

𝜕𝛺

𝑣·𝜎*·𝑛d𝐴+ˆ

𝛺

𝑣·𝑏*d𝑚 .

There are major differences in these balance laws when compared to the framework presented

in this thesis. First, the specific force density

𝑏*

is supposed to represent a force of non-

electromagnetic origin, i.e., gravity. Therefore, only the stress tensor

𝜎*

can introduce forces

of electromagnetic origin. That is to say, electromagnetic forces are a priori assumed to be a

surface phenomenon. The same holds true for the moments on the right-hand side of Eq.

(3.45b)

At the first glance, it seems that this may be compensated by the specific momentum and

the specific moment of momentum, which both depend upon the electromagnetic fields in the

framework by

Kovetz

. However, the definition of the total force and the total moment acting

on a body, i.e., the right-hand sides of the corresponding balances, have direct consequences for

rigid body mechanics. Following the statements by

Kovetz

, the total force and moment are

given by:

𝐹*=˛

𝜕𝛺

𝜎*·𝑛d𝐴+ˆ

𝛺

𝑏*d𝑚 , (3.46a)

𝑇*=˛

𝜕𝛺

(𝑥−𝑥𝑂)×𝜎*·𝑛d𝐴+ˆ

𝛺

(𝑥−𝑥𝑂)×𝑏*d𝑚 . (3.46b)

According to

Kovetz

𝜎*

itself and subsequently the electromagnetic coupling model must

be acquired from constitutive theory. However, in a footnote he writes “A final assumption is

then made to the effect that

ℰ×ℋ

is itself an energy flux.” [Kovetz (2000), pg. 217]. Through

this choice, the electromagnetic coupling model is defined, as can be seen from every example

for constitutive relations given in [Kovetz (2000)]. Therein, the resulting stress tensor

𝜎*

can

always be divided into some derivative of the free energy

𝜎(mech)

plus an electromagnetic part.

Section 3.4. Frequently used coupling models

The same holds true for the momentum density and the energy flux:

𝜎*=𝜎(mech)

*+𝜎(em)

*+𝜖0𝐸×𝐵⊗𝑣,(3.47a)

𝜎(em)

*=−1

2(𝜖0𝐸·𝐸+1

𝜇0𝐵·𝐵−2ℳ·𝐵)1+𝜖0𝐸⊗𝐸+(3.47b)

𝜇0𝐵⊗𝐵+ℰ⊗𝑃−ℳ⊗𝐵,

𝑔*=𝑣+𝜌−1𝜖0𝐸×𝐵,(3.47c)

𝑢*=𝑢+1

2𝑣·𝑣+𝜌−1(1

2𝜖0𝐸·𝐸+1

2𝜇0𝐵·𝐵+ℰ·𝑃).(3.47d)

Therein,

𝜎(mech)

and

𝑢

may still depend upon the electromagnetic fields, but they do not contain

derivatives of the free energy density with respect to electromagnetic fields. If Eqs.

(3.46)

are

taken seriously, it follows from Eq.(3.47a) that

𝐹(em)

K=˛

𝜕𝛺−(︀𝜎gL

++𝑔gL

+⊗𝑣+ [ℳ+·𝐵+]1+ℰ+⊗𝑃+−ℳ+⊗𝐵+)︀·𝑛d𝐴 , (3.48)

for the (global) electromagnetic force acting on a body

𝛺−

. The index “

” is added in order

to emphasize that, according to

Kovetz

, these are external forces, i.e., the limits are to be

taken from the exterior domain. Therefore, if the body under consideration is embedded in

a non-magnetizable and non-polarizable environment, the integrals containing

ℳ+≡0

and

𝑃+≡0vanish.

Furthermore, the corresponding global moment, where 𝑥𝑂≡0, is given by

𝑇(em)

K=˛

𝜕𝛺−

𝑥×(𝜎gL

++𝑔gL

+⊗𝑣)·𝑛d𝐴+˛

𝜕𝛺−(︀ℳ+·𝐵+)︀𝑥×𝑛d𝐴+

+˛

𝜕𝛺−

𝑥×(︂ℰ+⊗𝑃+−ℳ+⊗𝐵+)︂·𝑛d𝐴 .

(3.49)

The local balance of linear momentum by

Kovetz

in Eq.

(3.45a)

formally agrees with the

material version of Eq. (2.50) in this thesis if:

𝑓(em) =𝜎(em)

*·∇−𝜖0

𝜕

𝜕𝑡(𝐸×𝐵),𝑓(mech) =𝑏*.(3.50)

Note that the resulting expression is at no point explicitly denoted in the book by

Kovetz

. It

follows that

𝑓K=𝑞fℰ+(︂𝑗f+D𝑃

D𝑡+ (∇·𝑣)𝑃−𝑃·(∇⊗𝑣))︂×𝐵+(3.51a)

+𝑃·(∇⊗ℰ) + ℳ·(∇⊗𝐵) + ℳ×(∇×𝐵),

𝜏K=0,(3.51b)

ℎK=𝑓K·𝑣s+𝑗f·ℰ−ℳ·D𝐵

D𝑡−𝑃·Dℰ

D𝑡−∇s·(︀[𝜎K+𝑣s⊗𝑔K−𝑢K1]·𝑣s)︀.(3.51c)

Thus, a choice for the derived quantities consistent with the relations explicitly stated by

Kovetz is given by

𝑔K=𝜖0𝐸×𝐵,(3.51d)

𝜎K=−1

2(𝜖0𝐸·𝐸+1

𝜇0𝐵·𝐵−2ℳ·𝐵)1+(3.51e)

+𝑃⊗ℰ−𝐵⊗ℳ+𝜖0𝐸⊗𝐸+1

𝜇0𝐵⊗𝐵,

Chapter 3. Electromagnetic coupling models

On the electromagnetic coupling problem 33

𝐶K=0,(3.51f)

𝑢K=1

2(𝜖0𝐸·𝐸+1

𝜇0𝐵·𝐵) + ℰ·𝑃,(3.51g)

𝑞K=ℰ×ℋ.(3.51h)

It can be noted that the electromagnetic stress tensor is the as in the model due to

Eringen

and Maugin,i.e.,𝜎K=𝜎Er. Therefore, the coefficients in Eq. (3.30) are also given by,

𝐾K

1=2−𝜇r

𝜇0𝜇r

, 𝐾K

2=𝜖0, 𝑆K

1=1

𝜇0𝜇r

, 𝑆K

2=𝜖0𝜖r.(3.52)

3.4.5 PAO and HUTTER

Another popular formulation of

Maxwell

’s equations is the

Chu

formulation, which is used

in [Pao and Hutter (1975)]. The authors used the same framework of equations as presented

in this thesis. Furthermore, the coupling model proposed by

Pao

and

Hutter

was in part

also considered in [Chu, Haus, and Penfield (1966)]. The primary electromagnetic coupling

quantities are given by:

𝑓P=𝑞f𝐸Ch +𝑗f×𝜇0HCh +𝑃Ch ·(∇⊗𝐸)−𝜇0𝑃Ch ·(∇⊗HCh)×𝑣+

+(︂𝜕𝑃Ch

𝜕𝑡 +∇·[𝑣⊗𝑃Ch])︂×𝜇0HCh −1

𝑐2(︂𝜕𝑀Ch

𝜕𝑡 +∇·(𝑣⊗𝑀))︂×𝐸+(3.53a)

+𝜇0𝑀Ch ·(∇⊗HCh) + 1

𝑐2𝑀Ch ·(∇⊗𝐸Ch)×𝑣,

𝜏P=𝑃Ch ×(𝐸Ch +𝑣×𝜇0HCh) + 𝜇0𝑀Ch ×(HCh −𝑣×𝜖0𝐸Ch),(3.53b)

𝑟P=𝑗f·𝐸Ch +𝐸Ch ·D𝑃Ch

D𝑡+𝑃Ch ·(∇⊗𝐸Ch)·𝑣+(3.53c)

+𝜇0HCh ·D𝑀Ch

D𝑡+𝑀Ch ·(∇⊗𝜇0HCh)·𝑣.

and thus the derived coupling quantities can be denoted as follows:

𝑔P=1

𝑐2𝐸Ch ×HCh ,(3.53d)

𝜎P=−1

2(𝜖0𝐸Ch ·𝐸Ch +𝜇0HCh ·HCh)1+𝜖0𝐸Ch ⊗𝐸Ch +(3.53e)

+𝑃Ch ⊗(𝐸Ch +𝜇0𝑣×HCh) + 𝜇0HCh ⊗HCh +𝑀Ch ⊗𝜇0(HCh −𝜖0𝑣×𝐸Ch),

𝐶P=𝑃Ch ⊗(𝐸Ch +𝑣×𝜇0HCh) + 𝜇0𝑀Ch ⊗(HCh −𝑣×𝜖0𝐸Ch),(3.53f)

𝑢P=−1

2(︀𝜖0𝐸Ch ·𝐸Ch +𝜇0HCh ·HCh)︀,(3.53g)

𝑞P=−𝐸Ch ×HCh + (𝑃Ch ⊗𝐸Ch +𝜇0𝑀Ch ⊗HCh)·𝑣,(3.53h)

cf., [Hutter, Ven, and Ursescu (2006), Sec. 3.3.1]. Using the low velocity conversion rules

developed in Sec. 2.2 results in:

𝑔P=1

𝑐2(ℰ−𝑣×𝜇0H)×(ℋ+𝑣×𝜖0𝐸),(3.54)

𝜎P=−1

2(𝜖0𝐸Ch ·𝐸Ch +𝜇0HCh ·HCh)1+𝜖0𝐸Ch ⊗𝐸Ch +(3.55)

+𝑃Ch ⊗(𝐸Ch +𝜇0𝑣×HCh) + 𝜇0HCh ⊗HCh +𝑀Ch ⊗𝜇0(HCh −𝜖0𝑣×𝐸Ch),

𝐶P=𝑃Ch ⊗(𝐸Ch +𝑣×𝜇0HCh) + 𝜇0𝑀Ch ⊗(HCh −𝑣×𝜖0𝐸Ch),(3.56)

𝑢P=−1

2(︀𝜖0𝐸Ch ·𝐸Ch +𝜇0HCh ·HCh)︀,(3.57)

𝑞P=−𝐸Ch ×HCh + (𝑃Ch ⊗𝐸Ch +𝜇0𝑀Ch ⊗HCh)·𝑣.(3.58)

Section 3.4. Frequently used coupling models

In particular, for a material at rest with 𝑣=0, one has

𝑔P=1

𝑐2𝐸×H,𝜏P=𝑃×𝐸+𝜇0𝑀×H,(3.59)

𝜎P=−1

2(𝜖0𝐸·𝐸+𝜇0H·H)1+D⊗𝐸+𝐵⊗H,(3.60)

𝐶P=𝑃⊗𝐸+𝜇0𝑀⊗H,(3.61)

𝑢P=−1

2(︀𝜖0𝐸·𝐸+𝜇0H·H)︀,(3.62)

𝑞P=−𝐸×H,(3.63)

where the stress tensor is the same as in the

Einstein

–

Laub

model. In this case, the coefficients

in Eq. (3.30) are given by,

𝐾P

1=1

𝜇0𝜇2

, 𝐾P

2=𝜖0, 𝑆P

1=1

𝜇0𝜇r

, 𝑆P

2=𝜖0𝜖r.(3.64)

This is the same set of coefficients that results from the model by

Einstein

and

Laub

, see

Eq. (3.39).

Chapter 3. Electromagnetic coupling models

On the electromagnetic coupling problem 35

4 Consequences of the coupling problem and

force model invalidation

Only experiments can show if a coupling model is applicable or not. The hope is that there exists

a coupling model that correctly predicts measurable quantities in all technologically relevant

scenarios. Such a coupling model could then be deemed to be “correct.” However, a more

cautious conclusion is that the corresponding model is “generally applicable,” until an experiment

is found for which the theoretical predictions disagree with the experimental evidence. In short,

experiments may only serve to invalidate a given model. Moreover, an electromagnetic coupling

model consists of the force, moment and power components, all of which are interconnected a

priori. Therefore, the thermodynamic consistency of coupling models is investigated in Sec. 4.1.

It is found that for simple material response functions the electromagnetic force model must be

found first and independently of the other coupling components. Thus, the subsequent sections

are concerned with measurable quantities resulting from electromagnetic force models. It can

be noted that the electromagnetic moment

𝜏(em)

is only discussed tangentially because it is

not allowed for classical

Euler

–

Cauchy

continua. It is used in the analysis of the motion of

ferrofluids, but there is consensus on its form, see [Shliomis (1971); Rosensweig (2013)], which is

given by 𝑀×𝐵for magnetic fluids and 𝐸×𝑃for micropolar dielectrics.

4.1 Thermodynamic consistency

The constitutive theory as introduced in rational thermodynamics is a delicate topic. This

section is not meant to be a comprehensive review of this theory. Furthermore, the thorough

exploitation of the restrictions resulting from the second law of thermodynamics is beyond the

scope of this thesis. Instead, the entropy inequality is investigated for simple material response

functions in order to observe the impact of the electromagnetic coupling. Therefore, no specific

coupling model is assumed a priori.

The method put forward by

Coleman

and

Noll

is employed in the following. For a

comprehensive review of the different techniques for the exploitation of the second law of

thermodynamics, see [Müller (1985)] or [Papenfuß and Muschik (2018)]. The entropy balance in

Eq.

(2.65e)

or rather the second law of thermodynamics, i.e.,

𝜎≥

0, poses a restriction either

on admissible processes or a restriction upon admissible material behavior. It is in the spirit of

rational thermodynamics to assume the latter. Therefore, all processes which satisfy the (other)

balance equations may occur, but the material responses are restricted. A “process” could be

defined as a temporal series of states. A state in turn is determined by the state space, which is

spanned by its state variables. These state space variables are chosen in such a way that by

knowing them, a given physical situation is completely determined. Because of the balance laws,

it is clear that a state space for a fluid (say) should have at least the following variables in a

spatial description,

𝒵={𝜌s,𝑣s,𝜔s, 𝑇, 𝐸,𝐵}.(4.1)

Then, the constitutive fields,

𝑢 , 𝑞, 𝑠 , 𝜑,𝜎,𝜇,𝑗f,𝑃,𝑀.(4.2)

are said to be functions of the given state space. However, the naive choice in Eq.

(4.1)

for the

state space must be adjusted for two reasons. First, the principle of material objectivity states

that constitutive functions should be observer-independent. Hence, non-objective state space

variables are to be removed or replaced with objective combinations. The second adjustment of

the state space results from its localization in space and time, which is achieved by replacing the

function 𝑇s(𝑥s, 𝑡)by its value 𝑇sas well as its derivatives ∇s𝑇s,˙

𝑇sand the derivatives thereof.

Due to the objectivity requirement, the velocity vector must be removed completely and the

angular velocity must be replaced with the objective combination

ℎ

𝜔s−1

2∇s×𝑣s

, see [Cowin

(1974), pg. 303]. Similarly, the velocity gradient

𝐿

𝑣s⊗∇s

is not objective, but its symmetric

part

sym

(

𝐿

) =

(

𝐿

𝐿T

)is objective. It is more convenient to express

ℎ

through a tensor of

second rank,

𝑊=−ℎ·⟨3⟩

𝜖=−𝜔s·⟨3⟩

𝜖−skw(𝐿),skw(𝐿) = 1

2(𝑣s⊗∇s−∇s⊗𝑣s).(4.3)

According to [Cowin (1974)], the gradient

𝐺

𝜔s⊗∇s

is objective and can be used without

any modification. Regarding the electromagnetic fields, it is usually argued that the magnetic

flux density

𝐵

is a low-velocity invariant, see [Kovetz (2000)] or [Liu and Müller (1972)]. The

electric field, however, must be replaced with the electromotive intensity

ℰ=𝐸+𝑣s×𝐵.(4.4)

Finally, the simplest objective state space for a viscous micropolar fluid is given by

𝒵={𝜌s,sym(𝐿),𝑊,𝐺, 𝑇, ∇s𝑇, ℰ,𝐵}.(4.5)

In order to exploit the second law of thermodynamics, all relevant balance laws are inserted

into the entropy balance. To this end, the following free energy density is introduced

𝜓=𝑢−𝑇𝑠 +𝜌−1

s𝜓(em) ⇒𝜌s𝑇˙𝑠=𝜌s˙𝑢+𝜌s(𝜌−1

s𝜓(em))∙−𝜌s(︀˙

𝜓+𝑠˙

𝑇)︀,(4.6)

where the dot indicates the substantial time derivative

/D𝑡

from Eq.

(2.47)

, which is not to

be confused with the total time derivative. Note that the newly introduced quantity

𝜓(em)

not a constitutive function itself. Rather, it is an auxiliary quantity that is introduced to ease

the calculations. For the current purposes, it may depend upon the electromagnetic fields,

𝜓(em)

(

ℰ,𝐵,𝑃,ℳ

). Upon invoking the identity in Eq.

(3.7)

, the substitution of the

entropy balance as well as the balance of internal energy from Eq. (2.65d) yields

𝑇𝜎 =−∇s·𝑘+ (𝜎+𝜓(em)1+𝜎(em)

aux )··(∇s⊗𝑣s) + 𝜇··(∇s⊗𝜔s) +

+𝜌s𝑟−𝑇𝑧 −(∇s𝑇)·𝜑−𝜎··(𝜔s·⟨3⟩

𝜖)−𝜌s(︀˙

𝜓+𝑠˙

𝑇)︀+˙

𝜓(em) +˙

𝑃·ℰ−˙

𝐵·ℳ,(4.7)

with 𝑇𝜎 ≥0and yet another auxiliary quantity

𝑘=𝑞−𝑇𝜑+𝑞(em)

aux .(4.8)

The

Clausius

–

Duhem

inequality in Eq.

(4.7)

must hold for all processes. Hence, it is regarded

as a restriction upon the material functions or equivalently the material itself. In particular,

the above inequality must hold for processes in which

𝑟≡

0and

𝑧≡

0. Since these terms are

supply terms, they can be turned off, at least in theory, and thus shall not influence the entropy

principle. In contrast, the electromagnetic quantities (

)

(em)

are regarded as production terms

and cannot be neglected.

All balance equations must be incorporated at some point. The balance of linear momentum

as well as the spin balance have already been used in order to obtain the balance of internal

energy. The latter is also already incorporated in the entropy inequality in Eq.

(4.7)

. Thus,

the mass balance as well as

Maxwell

’s equations have to be considered. The mass balance

Chapter 4. Consequences of the coupling problem and force model invalidation

On the electromagnetic coupling problem 37

˙𝜌s

−𝜌s

(

∇s·𝑣s

)enters via the chain rule applied to the time derivative of the free energy

density 𝜓=^

𝜓(𝜌s,ℰ,𝐵, . . .),

𝜓=−𝜌s1··(𝑣s⊗∇s)𝜕𝜓

𝜕𝜌s

+𝜕𝜓

𝜕ℰ·˙

ℰ+𝜕𝜓

𝜕𝐵·˙

𝐵+. . . . (4.9)

It is now tempting to try to insert

Maxwell

’s equations for the time derivatives of

ℰ

and

𝐵

However, it follows from Eq.(2.64) that

Dℰ

D𝑡=𝑐2⟨3⟩

𝜖··(∇s⊗𝐵) + 1

𝜖0

(𝐽f+𝐽b) + 𝑣s·(∇s⊗𝐸).(4.10)

Therein,

𝐽b

∇s×𝑀

𝜕𝑃

𝜕𝑡

contains the constitutive functions

𝑀

and

𝑃

, which would result

in products with other constitutive functions such as

𝜕𝜓

𝜕ℰ

. Fortunately, it can be argued that by

means of employing Eq.

(3.7)

Maxwell

’s equations were already used. Thus, upon applying

the chain rule and introducing 𝜓(em) =−ℰ·𝑃for convenience, it follows that

𝑇𝜎 =−∇s·𝑘+(︂𝜎−(ℰ·𝑃)1+𝜎(em)

aux −𝜌2

𝜕𝜓

𝜕𝜌s

1)︂··(∇s⊗𝑣s) + 𝜇··(∇s⊗𝜔s)−

−(∇s𝑇)·𝜑−𝜎··(𝜔s·⟨3⟩

𝜖)−𝜌s(︂𝜕𝜓

𝜕𝑇 +𝜂)︂˙

𝑇−𝜌s

𝜕𝜓

𝜕sym(𝐿)··sym( ˙

𝐿)−𝜌s

𝜕𝜓

𝜕𝑊·· ˙

𝑊−

−𝜌s

𝜕𝜓

𝜕𝐺·· ˙

𝐺−𝜌s

𝜕𝜓

𝜕∇s𝑇··(∇s𝑇)∙+(︂𝑃−𝜌s

𝜕𝜓

𝜕ℰ)︂·˙

ℰ−(︂ℳ+𝜌s

𝜕𝜓

𝜕𝐵)︂·˙

𝐵.

(4.11)

To simplify the arguments, a non-viscous or perfect fluid is temporarily assumed such that

𝒵={𝜌s, 𝑇, ∇s𝑇, ℰ,𝐵}(4.12)

must be employed. In textbooks like [Eringen and Maugin (1990)] and [Kovetz (2000)] it is

assumed that

𝑞

𝑇𝜑

and thus

𝑘

𝑞(em)

aux

. Hence, conclusions along the following line are drawn.

Since the bracket in front of the velocity gradient cannot depend on

𝐿

, it must vanish and one

arrives at, e.g.,

𝜎=(︂ℰ·𝑃+𝜌2

𝜕𝜓

𝜕𝜌s)︂1−𝜎(em)

aux ,

𝜎(em)

aux = (𝑃·ℰ−𝐵·ℳ)1+𝐵⊗ℳ−𝑃⊗ℰ,

(4.13)

where the auxiliary electromagnetic stress tensor for the generalized

Lorentz

model was

provided as an example. This would mean that the electromagnetic coupling model must be

incorporated directly into the constitutive relations. Otherwise, a simpler constitutive relation

𝜎=−𝑝1, 𝑝 =−𝜌2

𝜕𝜓

𝜕𝜌s

,(4.14)

would violate the second law of thermodynamics. Of course, this reasoning relies heavily on the

assumption, that the entropy flux is given by the energy flux divided by the temperature. In

[Liu and Müller (1972)], it is rigorously shown that this assumption is too restrictive and

𝜑

must depend upon the other field variables as well. Hence, if

𝜑

, and by extension

𝑘

, is assumed

to be a constitutive function, its divergence must be expanded according to the chain rule

∇s·𝑘=𝜕𝑘

𝜕𝜌s·(∇s𝜌s)+ 𝜕𝑘

𝜕𝑇 ·(∇s𝑇)+ 𝜕𝑘

𝜕∇s𝑇··(∇s⊗∇s𝑇)+ 𝜕𝑘

𝜕ℰ··(∇s⊗ℰ)+ 𝜕𝑘

𝜕𝐵··(∇s⊗𝐵).(4.15)

Section 4.1. Thermodynamic consistency

Therein, the gradient of the electromotive intensity can be rewritten

∇s⊗ℰ=∇s⊗𝐸+ (∇s⊗𝑣s)×𝐵−(∇s⊗𝐵)×𝑣s.(4.16)

Thus, the entropy inequality in Eq.(4.11) for a perfect fluid from Eq. (4.12) reads

𝑇𝜎 =−𝜕𝑘

𝜕𝜌s·(∇s𝜌s)−𝜕𝑘

𝜕𝑇 ·(∇s𝑇)−𝜕𝑘

𝜕∇s𝑇··(∇s⊗∇s𝑇)−𝜕𝑘

𝜕𝐵··(∇s⊗𝐵)−

−𝜕𝑘

𝜕ℰ··(∇s⊗𝐸)−(︂𝜕𝑘

𝜕ℰ×𝑣s)︂··(∇s⊗𝐵) +

+(︂𝜎−(ℰ·𝑃)1+𝜎(em)

aux −𝜌2

𝜕𝜓

𝜕𝜌s

1+𝜕𝑘

𝜕ℰ×𝐵)︂··(∇s⊗𝑣s) + 𝜇··(∇s⊗𝜔s)−

−(︂𝜕𝑘

𝜕𝑇 +𝜑)︂·(∇s𝑇)−𝜎··(𝜔s·⟨3⟩

𝜖)−𝜌s(︂𝜕𝜓

𝜕𝑇 +𝜂)︂˙

𝑇−𝜌s

𝜕𝜓

𝜕ℎ·˙

ℎ−

−𝜌s

𝜕𝜓

𝜕∇s𝑇··(∇s𝑇)∙+(︂𝑃−𝜌s

𝜕𝜓

𝜕ℰ)︂·˙

ℰ−(︂ℳ+𝜌s

𝜕𝜓

𝜕𝐵)︂·˙

𝐵.

(4.17)

The rigorous exploitation of this expression is complicated, as the analysis in [Liu and Müller

(1972)] goes to show, and beyond the scope of this thesis. Rather, the expression in Eq.

(4.17)

proves that simple material behavior is thermodynamically consistent even if it is independent of

the electromagnetic coupling model. For example, choosing

𝜎

−𝑝1

does not violate the second

law of thermodynamics. Instead, a particular restriction upon

𝑘

and therefore the entropy flux

𝜑results.

Of course, this was only demonstrated for perfect fluids, but the arguments stay the same for

more complicated mechanical dependencies upon

sym

(

𝐿

)(say). Furthermore, the arguments

provided in this section do not rule out material response functions such as

𝜎

(

ℰ,𝐵, . . .

). The

conclusion is that the dependencies are not required for thermodynamic consistency. Moreover,

it is shown that the electromagnetic force model is not subjected to the entropy principle.

On the contrary, it highly influences the entropy principle and the thermodynamic restriction

upon the electromagnetic energy model. Therefore, before a complicated constitutive theory is

constructed, a generally applicable electromagnetic force model is to be found. This shall be

the task for the remainder of this thesis.

4.2 Global forces and global moments on bodies

The global electromagnetic force and electromagnetic moment acting on a body arise from

electromagnetic forces in volumetric regions and from their jumps across singular interfaces.

The term “global force” is used instead of the more natural “total force,” in order to emphasize

the fact that there is another way electromagnetic fields can influence the exerted force on a

body, see Sec. 4.3. In this section, only the forces and moments caused directly by the volumetric

force density 𝑓(em) and its singular counterpart 𝑓(em)

𝐼are investigated.

Consider a body occupying the region

𝛺−

, which is free of any internal surfaces of discontinuity.

The global electromagnetic force and the corresponding moment are given by, see Eq. (2.51):

𝐹(em) =˛

𝜕𝛺−

𝑓(em)

𝐼d𝐴+ˆ

𝛺−

𝑓(em) d𝑉 ,

𝑇(em) =˛

𝜕𝛺−

𝜏(em)

𝐼d𝐴+ˆ

𝛺−

(𝜏(em) +𝑥×𝑓(em)) d𝑉 .

(4.18)

Therein, the force and moment densities may be replaced utilizing Eqs.

(3.4a)

–

(3.4b)

(3.5a)

and

Chapter 4. Consequences of the coupling problem and force model invalidation

On the electromagnetic coupling problem 39

(3.5b)

. Bearing in mind that

𝐼

𝜕𝛺−

is material, the surface velocity is given by

𝑤𝐼

𝑣

and

the distinction between the spatial and material description is omitted. Furthermore, because

the volumetric contributions can be rewritten via,

𝛺−

𝑓(em) d𝑉=˛

𝜕𝛺−

𝑛·(𝜎(em)

−+𝑣⊗𝑔(em)

−) d𝐴−d

d𝑡ˆ

𝛺−

𝑔(em) d𝑉 ,

𝛺−

𝑥×𝑓(em) d𝑉=−˛

𝜕𝛺−

𝑛·(𝜎(em)

−+𝑣⊗𝑔(em)

−)×𝑥d𝐴+

+ˆ

𝛺−

⟨3⟩

𝜖··(𝑣⊗𝑔(em) −𝜎(em)) d𝑉−d

d𝑡ˆ

𝛺−

𝑥×𝑔(em) d𝑉 ,

(4.19)

where the minus sign subscript indicates the limit from the interior, the global force and the

global moment simplify to read:

𝐹(em) =˛

𝜕𝛺−

𝑛·(𝜎(em)

++𝑣⊗𝑔(em)

+) d𝐴−d

d𝑡ˆ

𝛺−

𝑔(em) d𝑉 , (4.20a)

𝑇(em) =−˛

𝜕𝛺−

𝑛·(𝜎(em)

++𝑣⊗𝑔(em)

+)×𝑥d𝐴−d

d𝑡ˆ

𝛺−

𝑥×𝑔(em) d𝑉+(4.20b)

+ˆ

𝛺−

⟨3⟩

𝜖··(𝑣⊗𝑔(em) −𝜎(em) +𝐶(em)) d𝑉 .

Thus, the global electromagnetic force difference with respect to the generalized

Lorentz

model

is given by

Δ𝐹(em) =˛

𝜕𝛺−

𝑛·(Δ𝜎++𝑣⊗Δ𝑔+) d𝐴−d

d𝑡ˆ

𝛺−

Δ𝑔d𝑉 . (4.21)

In [Reich (2017)], the global electromagnetic force on a body immersed in a non-polarizable

and non-magnetizable medium was analyzed. It was shown that any electromagnetic force model

that obeys the pseudo balance law in Eq.

(3.4)

yields the same global force

𝐹(em)

. This becomes

evident from Eq.

(4.20a)

, because all force models yield the same value for

𝜎(em)

𝑣⊗𝑔(em)

if the

magnetization and polarization vanish. In this case, subtracting the

Lorentz

force prediction

from another force model prediction yields

𝐹(em) −𝐹gL =−d

d𝑡ˆ

𝛺−

Δ𝑔d𝑉 , (4.22)

where the force model decomposition from Eq.

(3.13)

was introduced. It is interesting to note

that the global force difference is determined by the electromagnetic momentum difference only.

The numerical values of Δ

𝑔A

and Δ

𝑔M

in Eq.

(3.19)

can be considered to be small in most

technically relevant settings. Hence, for rigid body applications, the generalized total

Lorentz

force can be used safely, even in a dynamic setting. More specifically, the force decomposition

leads to the theorem of total force equivalence:

The global electromagnetic force acting on a body immersed in a non-polarizable and non-

magnetizable medium can be computed via

𝐹(em) =˛

𝜕𝛺

𝑓(em)

𝐼d𝐴+ˆ

𝛺

𝑓(em) d𝑉(4.23)

Section 4.2. Global forces and global moments on bodies

is the same for all electromagnetic force models constructed in accordance with Eq.

(3.4)

. The

value for this force vector is then given by

𝐹(em) =˛

𝜕𝛺−(︀𝑞f

𝐼⟨𝐸⟩+𝐽f

𝐼×⟨𝐵⟩)︀d𝐴+ˆ

𝛺−

(𝑞f𝐸+𝐽f×𝐵) d𝑉 . (4.24)

From this theorem, it can be concluded that, for technical applications that meet its requirements,

the total force acting on a body is known and the coupling problem has no impact. Furthermore,

most total force experiments cannot serve to identify the correct force model. To observe a

difference between force models, an experiment must be constructed such that the test body in

question is immersed in a magnetizable or polarizable medium.

Similarly, the global electromagnetic moment in Eq. (4.18) can be decomposed according to

𝑇(em) −𝑇gL =ˆ

𝛺−

⟨3⟩

𝜖··(𝑣⊗Δ𝑔−Δ𝜎+ Δ𝐶) d𝑉−d

d𝑡ˆ

𝛺−

𝑥×Δ𝑔d𝑉 . (4.25)

where it was also assumed that the exterior is free from magnetization and polarization, and

the global moment from the Lorentz model reads

𝑇gL =−˛

𝜕𝛺−

𝑛·(𝜎gL

++𝑣⊗𝑔gL

+)×𝑥d𝐴+

+ˆ

𝛺−

⟨3⟩

𝜖··(𝑣⊗𝑔gL −𝜎gL) d𝑉−d

d𝑡ˆ

𝛺−

𝑥×𝑔gL d𝑉

(4.26)

The difference in Eq.

(4.25)

shows that, even in a static setting, the global moment acting on a

body calculated from a generic force model may differ from the

Lorentz

force model prediction,

if the associated electromagnetic stress tensor is not symmetric. In the literature, it is usually

pointed out, that non-symmetric electromagnetic stress tensors become symmetric for linear

material behavior, e.g., [Kemp (2011)] or [Mansuripur (2017)]. This can also be seen from

Eq.

(3.30)

. However, this argument does not apply to experiments involving permanent magnets

(say), where the magnetization

𝑀

is not proportional to an applied field and thus an inherently

non-symmetric electromagnetic stress tensor does not become symmetrized.

From the analyses in this section, the following three conclusions can be drawn

•

Global electromagnetic force experiments are only useful in the context of the EMCP if

the probe is immersed in a magnetizable or polarizable medium.

•

In turn, this means that applications, where this is not the case, are not affected by the

EMCP.

•

The analysis of total moments may serve to invalidate electromagnetic coupling models

even without magnetizable or polarizable surroundings.

One way of creating a magnetizable or polarizable environment is to submerge the measuring

specimen in fluid, e.g., magnetic ferrofluid or polarizable oil. However, in doing so, the total

electromagnetic force acting upon the specimen is not only given

𝐹(em)

, but additionally by

the pressure force. Therefore, it is necessary to study the pressure distribution caused by

electromagnetic fields.

Chapter 4. Consequences of the coupling problem and force model invalidation

On the electromagnetic coupling problem 41

4.3 Static electromagnetic pressure in fluids

There are different types of pressures to be distinguished. In Sec. 4.1 the thermodynamic pressure

𝑝th

naturally results from the second law of thermodynamics and is obtained from the free

energy. On the other hand, the total pressure is defined via

𝑝tot :=−1

31··𝜎(mech) .(4.27)

For a fluid at rest or for an incompressible fluid, the thermodynamic pressure and the total

pressure coincide and thus they are simply referred to as pressure

𝑝

that generates the

Cauchy

stress tensor

𝜎

−𝑝1

. However, in the case of fluids that react to external electric or

magnetic fields, the notion of an associated electric pressure or magnetic pressure, respectively,

is introduced in the literature, see for example [Kovetz (2000)].

For a fluid at rest, the electromagnetic pressure can be defined as the difference between the

total pressure and the pressure due to gravity. In particular, if the fluid is incompressible, the

electromagnetic pressure results from the balance of linear momentum

𝑝(em) :=𝑝−𝜌s𝑓·𝑥⇒ ∇𝑝(em) =∇·𝜎(em) ,(4.28)

where

𝑓

is given by gravity. In order to investigate this concept, consider a polarizable and

magnetizable fluid at rest, for which the electromagnetic stress tensor representation from

Eq. (3.30) is given by

𝜎(em) =−1

2(𝐾1𝐵·𝐵+𝐾2𝐸·𝐸)1+𝑆1𝐵⊗𝐵+𝑆2𝐸⊗𝐸.

It is assumed that there are neither free charges nor free electric currents. Thus, upon invoking

the identity,

𝑎·(∇⊗𝑎) = 1

2∇(𝑎·𝑎)−𝑎×(∇×𝑎),(4.29)

as well as the static Maxwell’s equations,

∇·𝐵= 0 ,∇×𝐵=𝜇0∇×𝑀,∇·𝐸=−1

𝜖0∇·𝑃,∇×𝐸=0,(4.30)

the static balance of linear momentum for an incompressible fluid reduces to

∇𝑝(em) =1

2∇(︀[𝑆1−𝐾1]𝐵·𝐵+ [𝑆2−𝐾2]𝐸·𝐸)︀−1

𝜖0𝑆2(∇·𝑃)𝐸+

+𝐵⊗𝐵·(∇𝑆1) + 𝜇0𝑆1(∇×𝑀)×𝐵+𝐸⊗𝐸·(∇𝑆2),(4.31)

In particular, in the case that all material parameters are homogeneous, it follows that

𝑆1

and

𝑆2

are homogeneous as well, see Eq.

(3.29)

. As a consequence,

Maxwell

’s equations reduce

further, because

∇×𝑀=0,∇·𝑃= 0 .(4.32)

Subsequently, Eq.

(4.31)

is readily solved since the electromagnetic forces are obtained by means

of the gradient of a scalar potential. That is to say, the static electromagnetic volume forces are

conservative for all models if the material parameters are constant. Therefore, the solution for

the pressure field inside a homogeneous Euler fluid is given by

𝑝=𝜌s𝑓·𝑥+𝑝(em) +𝐶 , 𝑝(em) =1

2(𝑆1−𝐾1)𝐵·𝐵+1

2(𝑆2−𝐾2)𝐸·𝐸,(4.33)

where

𝐶

is a constant. In this special case, the electromagnetic pressure vanishes for some models,

e.g., the generalized

Lorentz

model, for which the parameters

𝑆gL

𝐾gL

and

𝑆gL

𝐾gL

cancel, leading to

𝑝(em)

= 0. Moreover, the electromagnetic pressure resulting from conservative

electromagnetic forces can also be obtained from Eq.

(4.33)

for stationary or quasi-stationary

Section 4.3. Static electromagnetic pressure in fluids

processes, i.e., the fluid does not need to be at rest. However, it should be stressed that the

direct identification of the electromagnetic pressure is only possible if homogeneous material

parameters are assumed and if the particular force model is symmetric under the conditions

leading to Eq.

(3.30)

. In general, the balance of linear momentum in Eq.

(4.31)

must be solved.

The relevant coefficients for the frequently used force models from Sec. 3.4 are given in Tab. 4.1.

From the table, it can already be seen that some of the electromagnetic force models yield

Tab. 4.1: Summary of the stress tensor coefficients resulting from different force models.

model 𝐾1𝐾2𝑆1𝑆2(𝑆1−𝐾1) (𝑆2−𝐾2)

gen. Lorentz 1

𝜇0𝜖01

𝜇0𝜖00 0

Abraham 1

𝜇0𝜇r𝜖0𝜖r1

𝜇0𝜇r𝜖0𝜖r0 0

Minkowski

Einstein–Laub 1

𝜇0𝜇2

r𝜖01

𝜇0𝜇r𝜖0𝜖r𝜇r−1

𝜇0𝜇2

r𝜖0(𝜖r−1)

Pao–Hutter

Eringen–Maugin 2−𝜇r

𝜇0𝜇r𝜖01

𝜇0𝜇r𝜖0𝜖r𝜇r−1

𝜇0𝜇r𝜖0(𝜖r−1)

Kovetz

the same electromagnetic pressure. In particular, for homogeneous material parameters, only

the last two columns of Tab. 4.1 are relevant for the pressure solution in Eq.

(4.33)

. Thus, two

different electric pressure predictions can be expected. However, for the magnetic pressure, three

different expressions have to be considered. More differences arise if the material parameters

are not homogeneous.

In any case, the electromagnetic pressure must be attributed to the electromagnetic surface

force density. Recall the balance of linear momentum in singular points from Eq.

(2.66a)

together

with the definition of the electromagnetic surface force in Eq.

(3.5a)

. In view of the pressure

surface force density,

−𝑝(em)𝑛

, the effective surface force acting upon a body

𝛺−

immersed inside

a fluid 𝛺+is defined as

𝑓(em)

𝐼,eff :=𝑓(em)

𝐼−𝑝(em)

+𝑛,(4.34)

Thus, the electromagnetic force

𝐹(em)

must be supplemented by the electromagnetic pressure

force. As a result, the total electromagnetic force acting on a body is given by

𝐹=𝐹(em) +˛

𝜕𝛺−

(−𝑝(em)

+𝑛) d𝐴 . (4.35)

Therein, the electromagnetic pressure can have a significant impact on the total electromagnetic

force, as will be demonstrated in Sec. 6.1 and in Sec. 7.3.

4.4 Local motion and force model equivalence

In [Hutter, Ven, and Ursescu (2006)], the equivalence between various force models expressed in

different formulations of

Maxwell

’s equations is investigated. To this end,

Hutter

analyzes

the impact of different force models on the resulting boundary value problems. Thus, two

force models are referred to as equivalent if they cannot be distinguished experimentally, i.e., if

their (initial) boundary value problems are identical. The models analyzed by

Hutter

are all

equivalent if the corresponding free energy densities are chosen appropriately. That is to say,

the disagreement between the two models can be compensated by means of the constitutive

relations for (say) the

Cauchy

stress tensor. This reasoning is problematic for three reasons.

Chapter 4. Consequences of the coupling problem and force model invalidation

On the electromagnetic coupling problem 43

Firstly, it breaks down for the very simplest material behavior, in which the free energy density

is not a function of the electromagnetic fields. Secondly, even if two models are equivalent, it

does not necessarily mean that they are correct. Thirdly, and most importantly, there is no

general procedure for finding the appropriate free energy densities that correct a potentially

incorrect electromagnetic coupling model. Finally, it could be argued that this state of affairs is

just unsatisfactory from the point of view of the user of the theory.

At least for the electromagnetic force model, it can be shown that only the electromagnetic

stress tensor affects the motion of the material under consideration, see Chap. 7. The corre-

sponding analysis of the electromagnetic force model in the local balance of linear momentum is

complicated due to the ambiguity of the electromagnetic force, stress and momentum terms.

Furthermore, in order to study the effect of a given force model one has to investigate a specific

initial boundary value problem, which introduces the surface force in the boundary conditions.

This analysis can be simplified conceptually utilizing the principle of virtual power. It is

obtained by dot multiplying the local balance of linear momentum with a virtual velocity

𝛿𝑣

and subsequent integration over a control volume. The resulting scalar equation is a global

statement that is equivalent to the original balance of linear momentum if and only if it is

satisfied for all virtual velocities

𝛿𝑣

. The principle of virtual power is therefore a variational

principle. The resulting equation is beneficial for the force model investigation as it readily

contains all information compactly. In particular, the surface forces enter explicitly and the

pressure forces are also contained.

The electromagnetic force density in the local momentum balance in Eq.

(2.65b)

is replaced

with the electromagnetic stress tensor and the electromagnetic momentum via Eq.

(3.4a)

Subsequent dot multiplication with a virtual velocity

𝛿𝑣

while observing the product rule yields

𝜌s

d𝑣s

d𝑡·𝛿𝑣=∇s·([𝜎+𝜎(em)]·𝛿𝑣)−(∇s⊗𝛿𝑣)··(𝜎+𝜎(em)) + 𝜌s𝑓·𝛿𝑣−𝜕𝑔(em)

𝜕𝑡 ·𝛿𝑣.(4.36)

Subsequent integration and application of the generalized

Gauß

theorem from Eq.

(C.16)

yields

𝑊s

𝜌s

d𝑣s

d𝑡·𝛿𝑣d𝑊=ˆ

𝛤

𝑛·(𝜎+𝜎(em))·𝛿𝑣d𝐴−ˆ

𝑊s

(∇s⊗𝛿𝑣)··(𝜎+𝜎(em)) d𝑊+

+ˆ

𝑊s

𝜌s𝑓·𝛿𝑣d𝑊−ˆ

𝑊s

𝜕𝑔(em)

𝜕𝑡 ·𝛿𝑣d𝑊−ˆ

𝐼

𝑛·J𝜎+𝜎(em)K·𝛿𝑣d𝐴 ,

(4.37)

where it was assumed that

J𝛿𝑣K

,i.e., the test function is continuous everywhere. The

interface

𝐼

is assumed to be material with

𝜌𝐼

= 0 as well as

𝑤𝐼

𝑣s

. Hence, the insertion of the

singular momentum balance from Eq. (2.66a) yields

𝑊s

𝜌s

d𝑣s

d𝑡·𝛿𝑣d𝑊=ˆ

𝛤N

(𝑡0+𝑛·𝜎(em))·𝛿𝑣d𝐴−ˆ

𝑊s

(∇s⊗𝛿𝑣)··(𝜎+𝜎(em)) d𝑊+

+ˆ

𝑊s

𝜌s𝑓·𝛿𝑣d𝑊−ˆ

𝑊s

𝜕𝑔(em)

𝜕𝑡 ·𝛿𝑣d𝑊+ˆ

𝐼(︀𝑡𝐼+𝑛·J𝑣s⊗𝑔(em)K)︀·𝛿𝑣d𝐴 ,

(4.38)

Therein, the boundary set

𝛤

𝜕𝑊s∖𝐼

was divided into

𝛤D

and

𝛤N

, on which

Dirichlet

boundary conditions or

Neumann

boundary conditions are prescribed, respectively. On the

Dirichlet

part the test function vanishes, i.e.,

𝛿𝑣

and on the

Neumann

part dynamical

boundary conditions are prescribed via

𝑛·𝜎

𝑡0

, with the known and prescribed traction

vector 𝑡0. Furthermore, the surface traction vector is given by

𝑡𝐼=∇𝐼·(1𝐼·𝜎𝐼).(4.39)

Section 4.4. Local motion and force model equivalence

The Eq.

(4.38)

can be referred to as the principle of virtual power. Since it incorporates all

boundary conditions directly, the electromagnetic surface forces enter. The impact of the parts

of the electromagnetic force model becomes apparent to their full extent.

In Sec.

(3.3)

as well as in [Reich, Rickert, and Müller (2017)] it was pointed out that, due to

Maxwell

’s equations, there are infinitely many possibilities for the mathematical reshuffling

of terms between the electromagnetic force

𝑓(em)

and the electromagnetic stress tensor

𝜎(em)

For a particular boundary value problem it was found that the local deformation behavior only

differs between force models if their electromagnetic stress tensors differ. From Eq.

(4.38)

can be seen that two force models will always yield the same local predictions, provided the

electromagnetic momentum

𝑔(em)

can be neglected or has no influence. This was the case in

the study conducted in [Reich et al. (2016)] and thus their findings were in accordance with

this statement. As it turns out, all suggestions for the electromagnetic momentum incorporate

products of electric and magnetic fields and thus

𝑔(em)

vanishes in purely electric or purely

magnetic settings. Therefore, the mathematical ambiguity between

𝜎(em)

and

𝑓(em)

does not

affect many technically relevant applications with only electric or magnetic effects. There are,

however, exceptions to this statement. For example, in applications involving electromagnetic

waves, both electric and magnetic effects are observed. Even if the material under consideration

only has either polarization or magnetization, one of the momentum deviations from Eq.

(3.19)

is triggered, since both

and

are present. These types of applications have to be considered

separately.

In many technical applications, the electromagnetic momentum can be neglected. Subsequently,

it can be concluded from the principle of virtual power that electromagnetic influence on the

mechanical motion of bodies is governed by the electromagnetic stress tensor. That encompasses

also the local deformation of bodies. Therefore, any experiment that aims to invalidate an

electromagnetic coupling model, must trigger differences in its electromagnetic stress tensor.

Otherwise, the predictions of this coupling model will agree with those of other models and they

can be referred to as equivalent.

4.5 Invalidation of electromagnetic coupling models

In the preceding sections, the impact of electromagnetic force models on measurable quantities

was investigated. The following was found:

•

For a body immersed in a non-magnetizable and non-polarizable medium, the total

stationary force acting upon the body can be calculated without considering the EMCP.

•

The same is true for total dynamic forces if the electromagnetic momentum can be

neglected.

•

Total electromagnetic moments on bodies are subject to the EMCP even if their surrounding

is free of 𝑀and 𝑃.

•

The difference between static electromagnetic moment predictions of force models is

governed by their electromagnetic stress tensor difference.

•

For quasi-stationary conditions, the effect on the local motion of deformable bodies due to

electromagnetic fields is governed by the electromagnetic stress tensor.

•

This is also true for non-stationary processes if the electromagnetic momentum can be

neglected.

These results motivate the choice of experiments used for the invalidation process. However,

there is an additional requirement on the employed experiments. On the one hand, the

Chapter 4. Consequences of the coupling problem and force model invalidation

On the electromagnetic coupling problem 45

experiments should be designed such that the electromagnetic force model difference has a

significant influence on the measured quantities. On the other hand, the experiments should be

simple enough, such that the measured quantities are directly accessible. In that regard, static

experiments are to be favored over dynamical ones, as in the latter case additional uncertainties

are introduced due to the field transformations, see e.g., [Weber (1997)]. Therefore, the following

three experiments are considered:

The total electromagnetic force between two cylindrical permanent magnets is measured.

In order to trigger force model differences, one of the magnets is immersed in a ferrofluid.

The total moment between magnets is considered both in a gedankenexperiment and in a

real experiment. For the latter, the moment between two non-coaxial permanent magnets

is measured.

The local deformation of a drop of silicone oil immersed in a bath of castor oil is evaluated.

In the experiment conducted by [Torza, Cox, and Mason (1971)], the container with the

two oils is subjected to an external electric field by means of a plate condenser, which

causes the droplet to deform. Since viscous processes are involved in this experiment, the

temporal evolution of the droplet’s interface must be calculated.

In order to investigate these experiments theoretically, electromagnetic field solutions must be

obtained. These have to be computed numerically due to the nonlinear material behavior of the

ferrofluid and the complicated geometries involved in the other two experiments. However, it is

not suitable to use commercial for the numerical computations, because the force model analysis

requires full control over the equations that are solved. Thus, the open-source software FEniCS

is used. In order to perform the numerical evaluation of the experiments efficiently, a finite

element method in curvilinear coordinates is developed. This enables the exact implementation

of symmetry conditions and reduces the computational costs significantly. Therefore, the

developed method and the required mathematical preliminaries are provided in Chap. 5.

Section 4.5. Invalidation of electromagnetic coupling models

On the electromagnetic coupling problem 47

5 Numerical field calculation with a reduced

finite element method

In this chapter, the mathematical foundations for the following chapters are presented compactly.

Most importantly, the numerical details are presented and guidelines for accurate computation

of electromagnetic fields and reliable post-processing are developed. In addition, an efficient

finite element scheme for boundary-value problems with an inherent symmetry is derived

and its application to the open source software FEniCS [Logg, Mardal, and Wells (2012)] is

demonstrated, which is frequently used in the subsequent chapters.

5.1 Finite element method in curvilinear coordinates

In a conventional finite element method, Cartesian coordinates are used for several reasons,

one of which being that the associated natural basis is global, i.e., it does not depend upon

the position vector. This enables software packages like FEniCS to provide a high level of

abstraction and generalization when it comes to solving partial differential equations, rendering

the software versatile. The user is barely required to perform numerical mathematics like matrix

assembly manually, rather the weak formulation of the boundary-value problem is implemented

in abstract notation and the geometry is to be provided after which the software takes care of

the subsequent steps required for a finite element analysis. The key to this versatility lies in the

usage of Cartesian coordinates as they allow for easily accessible abstractions. However, there

are at least three shortcomings of the Cartesian coordinates and therefore to the corresponding

discretizations:

•

Curved boundaries cannot

be captured exactly with tetrahedral or triangular elements

rather these boundaries are approximated with flat surfaces or straight lines.

•This also entails reduced accuracy in curvature computations.

•Symmetries can usually be exploited approximately only.

In many cases, these shortcomings can be overcome if instead of Cartesian coordinates curvilinear

coordinates are used. However, these are associated with a local natural basis and thus, in

connection with differential operators,

Chrisotffel

symbols arise, which complicate the

streamlined process of obtaining a finite element solution with FEniCS. Not only that, but

the software does neither provide support for other coordinates nor does its high-level user

interface grant direct access to all required geometrical quantities. Most importantly, customarily

used curvilinear coordinates such as cylindrical and spherical coordinates introduce apparent

singularities, e.g.,

/𝜌

/(︀𝑟sin(𝜗))︀

, in their derivatives. Note that the incorporation of curvilinear

coordinates into a finite element method is nothing new in general, see e.g., [De Santis, Geraci,

and Guardone (2012)]. However, the required mathematical perquisites together with the

methodology to implement them into FEniCS do not seem to be represented in the literature.

1There are indeed curved finite element types, see [Zlamal (1973)], but they are not as versatile as the generic

tetrahedral elements.

In the following, the procedure for a finite element analysis in curvilinear coordinates is

explained. The idea is to replace the original domain

𝛺

in the

space with a transformed

domain 𝛺⋆defined in another coordinate space S, see Fig. 5.1.

2π

𝑥

𝑧

𝑥

𝑦

𝛺

𝑛

𝜏

𝑧3

⋆

𝑥⋆

𝑧1

⋆

𝑧2

⋆

𝑛⋆

𝜏⋆

𝛺⋆

Fig. 5.1: Coordinate space transformation

In the original space

, the position vector is either represented by means of Cartesian

coordinates or using some curvilinear coordinates

𝑧𝑖

that results in the component functions

𝑓𝑖

thus

𝑥=𝑥𝑒𝑥+𝑦𝑒𝑦+𝑧𝑒𝑧=^

𝑓𝑖(𝑧1, 𝑧2, 𝑧3)^

𝑔𝑖(𝑧1, 𝑧2, 𝑧3).(5.1)

For instance, in spherical coordinates

𝑧𝑖

= [

𝑟, 𝜗, 𝜙

]

𝑖

most of the

𝑓𝑖

are zero, as

𝑥

𝑟^

𝑔1

with

𝑔1

𝑒𝑟

. In the transformed space

, however, the position vector is given by a pseudo-Cartesian

representation according to

𝑥⋆=𝑧1

⋆𝑔⋆

1+𝑧2

⋆𝑔⋆

2+𝑧3

⋆𝑔⋆

3.(5.2)

Algebraically, one can demand the correspondence 𝑥↔𝑥⋆and thus it is possible to identify

𝑧𝑖

⋆↔^𝑥𝑖(𝑧1, 𝑧2, 𝑧3),𝑔⋆

𝑖↔^

𝑔𝑖(𝑧1, 𝑧2, 𝑧3) = ^

𝑆𝑖𝑗(𝑧1, 𝑧2, 𝑧3)𝑒𝑗,(5.3)

with the transformation matrix

𝑆

and the self dual Cartesian basis [

𝑒𝑖

] = [

𝑒𝑖

]. However, it is

important to note that the pseudo-Cartesian base vectors 𝑔⋆

𝑖do not depend upon coordinates,

i.e., they constitute a global basis system in

, in contrast to the original local basis vectors

𝑔𝑖

(

𝑧1, 𝑧2, 𝑧3

)in

. As a consequence, the differentiations which entail

Christoffel

symbols

have to be performed first and only then the pseudo-Cartesian representation can be identified.

Furthermore, the connections between original geometric quantities and their pseudo-Cartesian

counterparts are required, as for example the normal vector

𝑛

to the original boundary

𝜕𝛺

is not

equal to the normal vector of

𝜕𝛺⋆

,i.e.,

𝑛

𝑛⋆

, and rather the following set of transformation

formulae are to be determined:

𝑛=^

𝑛(𝑛⋆),𝜏=^

𝜏(𝜏⋆),

d𝑉= d ^

𝑉(d𝑉⋆,𝑥⋆),d𝐴= d ^

𝐴(d𝐴⋆,𝑥⋆,𝑛⋆),dℓ= d^

ℓ(dℓ⋆,𝑥⋆,𝜏⋆),(5.4)

where the hat indicates the functional dependencies of the normal vector, the tangent vector and

the differential elements for volumes, surfaces and lines, respectively. The normal and tangent

vectors to

𝜕𝛺⋆

are naturally represented with respect to the pseudo-Cartesian basis, i.e.,

𝑛⋆=𝑛𝑖

⋆𝑔⋆

𝑖and 𝜏⋆=𝜏𝑖

⋆𝑔⋆

𝑖. Similarly, the differential elements d𝑉⋆,d𝐴⋆and dℓ⋆are generated

Chapter 5. Numerical field calculation with a reduced finite element method

On the electromagnetic coupling problem 49

from the computational domain in a Cartesian manner:

d𝑉⋆= d𝑧1

⋆d𝑧2

⋆d𝑧3

⋆,d𝐴⋆=𝑛⋆d𝐴⋆= d𝑧2

⋆d𝑧3

⋆𝑔⋆

1+ d𝑧1

⋆d𝑧3

⋆𝑔⋆

2+ d𝑧1

⋆d𝑧2

⋆𝑔⋆

dℓ⋆=𝜏⋆dℓ⋆= d𝑧1

⋆𝑔⋆

1+ d𝑧2

⋆𝑔⋆

2+ d𝑧3

⋆𝑔⋆

3.(5.5)

5.1.1 Conversion procedure

Any (initial) boundary value problem to be solved numerically should be normalized before

numerical calculations are performed. To this end, the normalized position vector

𝑥

ℓ−1

ref 𝑥

introduced together with the reference length

ℓref

. In the same fashion, any physical quantity

𝜓

may be replaced with a normalized one which is indicated by a superimposed tilde, i.e.,

𝜓

𝜓ref ˜

𝜓

. The starting point for finite element analysis in curvilinear coordinates is given by:

Derive a coordinate free formulation of the weak form

ℱ

(

𝑢, 𝛿𝑢

) = 0 where

𝑢

is the wanted

field and

𝛿𝑢

is the corresponding test function. In the following it is assumed that the

aforementioned normalization procedure was already performed such that both

𝑢

and the

domain 𝛺are dimensionless.

If these requirements are met, the following steps must be performed in order to obtain the

weak formulation of a given boundary value problem in the curvilinear coordinate system:

Transform the original (normalized) domain

𝛺

into the computational domain

𝛺⋆

according

to the underlying coordinate transformation.

Modify the integrals contained in the weak form by means of replacing the differential

elements with the scaling functions 𝑠𝑉,𝑠𝐴and 𝑠ℓfor the respective elements:

d𝑉=𝑠𝑉(𝑥⋆)d𝑉⋆,d𝐴=𝑠𝐴(𝑥⋆,𝑛⋆)d𝐴⋆,dℓ=𝑠ℓ(𝑥⋆,𝜏⋆)dℓ⋆.(5.6)

Replace the test function with a scaled version that eliminates apparent singularities

arising from the nabla operator,

𝛿𝑢→𝑠𝛿(𝑥⋆)𝛿𝑢⋆.(5.7)

Identify a scaled nabla operator

∇⋆

scaled

that is free from apparent singularities. Additionally,

specify the component matrix 𝑈⋆=^

𝑈⋆(𝑢,𝑥⋆)of the scaled gradient:

∇⋆

scaled :=𝑠∇(𝑥⋆)∇,𝑢⊗∇⋆

scaled =𝑈𝑖𝑗

⋆𝑔⋆

𝑖⊗𝑔⋆

𝑗.(5.8)

Therein, the gradient is obtained from its original definition with all relevant

Christoffel

symbols but expressed with respect to the new pseudo-Cartesian basis 𝑔⋆

𝑖.

4. After the scaled nabla operator is evaluated, perform the component identification

^𝑢𝑖→𝑢𝑖

⋆,(5.9)

where the respective components correspond to the following vectors,

𝑢= ^𝑢𝑖(𝑧1, 𝑧2, 𝑧3)^

𝑔𝑖(𝑧1, 𝑧2, 𝑧3),𝑢⋆=𝑢𝑖

⋆(𝑧1

⋆, 𝑧2

⋆, 𝑧3

⋆)𝑔⋆

𝑖.(5.10)

Apply the same procedure to the test function to obtain

𝛿𝑢⋆

and the corresponding

representation of its gradient similar to Eq.(5.8).

Solve the resulting system

ℱ⋆

(

𝑢⋆, 𝛿𝑢⋆

)=0. Subsequently, apply the inverse coordinate

transformation in two steps. First, transform the vector components such that a proper

Section 5.1. Finite element method in curvilinear coordinates

Cartesian representation results, thus

𝑢=𝑢𝑖

⋆(𝑧1

⋆, 𝑧2

⋆, 𝑧3

⋆)^

𝑆𝑖𝑗(𝑧1

⋆, 𝑧2

⋆, 𝑧3

⋆)𝑒𝑗.(5.11)

Secondly, apply the inverse coordinate transformation from

𝛺⋆

𝛺

where the correspond-

ing coordinate transformation entails also the transformation of the underlying function

spaces for the finite element functions, e.g.,𝑢𝑖

⋆.

In order to evaluate this procedure for a given coordinate system, the set of scaling functions

and the transformations

𝑠𝑉

𝑠ℓ

𝑠𝛿

𝑠∇

𝑛

𝜏

, and

𝑆

must be given. In the following, the

required functions for spherical and cylindrical coordinates are derived.

5.1.2 Spherical coordinates

For dimensionless spherical coordinates

𝑧𝑖

= [

𝑟, 𝜗, 𝜙

]

𝑖

with the local but orthonormal basis system

𝑒𝑟

(

𝜗, 𝜙

)

𝑒𝜗

(

𝜗, 𝜙

)

𝑒𝜙

(

𝜙

)

}

the original normalized position vector and its pseudo-Cartesian

counterpart are given by:

𝑥=𝑟^

𝑒𝑟(𝜗, 𝜙),𝑥∘=𝑟𝑒∘

𝑟+𝜗𝑒∘

𝜗+𝜙𝑒∘

𝜙.(5.12)

Therein,

𝑔⋆

𝑖

𝑒∘

𝑖

are the pseudo-Cartesian orthonormal basis vectors. In the following, the

functional dependencies of the

𝑒𝑖

vectors will be omitted. The transformation function for the

volume element is readily obtained from the definition as

d𝑉=𝑟2sin(𝜗)d𝑟d𝜗d𝜙⇒𝑠𝑉(𝑟, 𝜗) = 𝑟2sin(𝜗).(5.13)

The other two differential elements require further analysis as their vectorial nature demands

an explicit basis vector transformation. In vectorial form, they read

d𝐴=𝑆∘(𝑟, 𝜗, 𝜙)·d𝐴∘,dℓ=𝐿∘(𝑟, 𝜗, 𝜙)·dℓ∘,(5.14)

with the spatially dependent two-point transformation tensors:

𝑆∘(𝑟, 𝜗, 𝜙) = 𝑟2sin(𝜗)^

𝑒𝑟⊗𝑒∘

𝑟+𝑟sin(𝜗)^

𝑒𝜗⊗𝑒∘

𝜗+𝑟^

𝑒𝜙⊗𝑒∘

𝜙,

𝐿∘(𝑟, 𝜗, 𝜙) = ^

𝑒𝑟⊗𝑒∘

𝑟+𝑟^

𝑒𝜗⊗𝑒∘

𝜗+𝑟sin(𝜗)^

𝑒𝜙⊗𝑒∘

𝜙.(5.15)

Upon noting the relations between the vectorial and scalar differential elements

d𝐴=𝑛d𝐴 , d𝐴∘=𝑛∘d𝐴∘,dℓ=𝜏dℓ , dℓ∘=𝜏∘dℓ∘,(5.16)

it follows immediately that:

d𝐴=‖𝑆∘·𝑛∘‖d𝐴∘=𝑟√︁sin2(𝜗)[𝑟2(𝑛∘

𝑟)2+ (𝑛∘

𝜗)2]+(𝑛∘

𝜙)2d𝐴∘,

dℓ=‖𝐿∘·𝜏∘‖dℓ∘=√︁(𝜏∘

𝑟)2+𝑟2(𝜏∘

𝜗)2+𝑟2sin2(𝜗)(𝜏∘

𝜙)2dℓ∘.

(5.17)

These relations are helpful for determining the normal and tangent vector transformation

formulae, thus:

𝑛=𝑆∘·𝑛∘

‖𝑆∘·𝑛∘‖=𝑟sin(𝜗)𝑛∘

𝑟^

𝑒𝑟+ sin(𝜗)𝑛∘

𝜗^

𝑒𝜗+𝑛∘

𝜙^

𝑒𝜙

√︁sin2(𝜗)[𝑟2(𝑛∘

𝑟)2+ (𝑛∘

𝜗)2]+(𝑛∘

𝜙)2,

𝜏=𝐿∘·𝜏∘

‖𝐿∘·𝜏∘‖=𝜏∘

𝑟^

𝑒𝑟+𝑟𝜏∘

𝜗^

𝑒𝜗+𝑟sin(𝜗)𝜏∘

𝜙^

𝑒𝜙

√︁(𝜏∘

𝑟)2+𝑟2(𝜏∘

𝜗)2+𝑟2sin2(𝜗)(𝜏∘

𝜙)2.

(5.18)

Chapter 5. Numerical field calculation with a reduced finite element method

On the electromagnetic coupling problem 51

The scaling functions are given by

𝑠𝛿

sin

(

𝜗

)and

𝑠∇

𝑟sin

(

𝜗

), respectively. As a result,

the components

𝑈𝑖𝑗

∘

of the scaled gradient

𝑢⊗ ∇∘

scaled

of a vector field

𝑢

^𝑢𝑟

(

𝑟, 𝜗, 𝜙

)

𝑒𝑟

^𝑢𝜗(𝑟, 𝜗, 𝜙)^

𝑒𝜗+ ^𝑢𝜙(𝑟, 𝜗, 𝜙)^

𝑒𝜙reads:

𝑈∘=⎡

⎢

⎣

𝑟sin(𝜗)𝜕^𝑢𝑟

𝜕𝑟 sin(𝜗)(︁𝜕^𝑢𝑟

𝜕𝜗 −^𝑢𝜗)︁𝜕^𝑢𝑟

𝜕𝜙 −sin(𝜗)^𝑢𝜙

𝑟sin(𝜗)𝜕^𝑢𝜗

𝜕𝑟 sin(𝜗)(︁𝜕^𝑢𝜗

𝜕𝜗 + ^𝑢𝑟)︁𝜕^𝑢𝜗

𝜕𝜙 −cos(𝜗)^𝑢𝜙

𝑟sin(𝜗)𝜕^𝑢𝜙

𝜕𝑟 sin(𝜗)𝜕^𝑢𝜙

𝜕𝜗

𝜕^𝑢𝜙

𝜕𝜙 + cos(𝜗)^𝑢𝜗+ sin(𝜗)^𝑢𝑟

⎤

⎥

⎦

.(5.19)

In particular, in the case of azimuthal symmetry, no component function may depend upon

𝜙

e.g.,

𝑛∘

𝑟

(

𝑟, 𝜗

), and no vector field has a

𝜙

-component. As a result, no scalar integral kernel

depends upon

𝜙

and the corresponding integration is readily performed yielding a factor of

. Furthermore, a dimensional reduction arises for the computational domain as depicted in

Fig. 5.2.

𝑥

𝑦

𝑧

𝛺

𝑦

𝑧

𝑟

𝜗

𝒪

𝛺sym

azimuthal symmetry spherical coordinates

𝛤=𝜕𝛺

𝜗

ℓ∘

𝑧−

𝑛∘

𝜏∘

Ssym

𝑟

ℓ∘

𝒪ℓ∘

𝑧+

𝛺∘

sym

2π

4π

𝛤∘

d𝐴

d𝐴∘

s= d𝑟∘d𝜗∘

𝑛

𝜏

Fig. 5.2: Exemplary domain transformation for the exploitation of azimuthal symmetry.

In summary, a simplified system of replacements is provided.

Spherical coordinates with azimuthal symmetry

The differential element transformations are given by:

d𝑉= 2π𝑟2sin(𝜗) d𝐴∘

sym ,d𝐴= 2π𝑟sin(𝜗)√︁𝑟2(𝑛∘

𝑟)2+ (𝑛∘

𝜗)2dℓ∘

sym ,(5.20)

where d

𝐴∘

sym

and d

ℓ∘

sym

are the surface and line element to the domain

𝛺∘

sym

, see Fig. 5.2,

to be calculated according to Eq.

(5.5)

. The normal vector in terms of

𝑛∘

𝑟𝑒∘

𝑟

𝑛∘

𝜗𝑒∘

𝜗

and the tangent vector are given by

𝑛=𝑟𝑛∘

𝑟^

𝑒𝑟+𝑛∘

𝜗^

𝑒𝜗

√︁𝑟2(𝑛∘

𝑟)2+ (𝑛∘

𝜗)2,𝜏∘=𝑄·𝑛∘with 𝑄=𝑒∘

𝑟⊗𝑒∘

𝜗−𝑒∘

𝜗⊗𝑒∘

𝑟,(5.21)

and the component matrix of the scaled gradient (^𝑢𝑖^

𝑒𝑖)⊗∇∘

scaled reduces to

𝑈sym

∘=⎡

⎢

⎣

𝑟sin(𝜗)𝜕^𝑢𝑟

𝜕𝑟 sin(𝜗)(︁𝜕^𝑢𝑟

𝜕𝜗 −^𝑢𝜗)︁0

𝑟sin(𝜗)𝜕^𝑢𝜗

𝜕𝑟 sin(𝜗)(︁𝜕^𝑢𝜗

𝜕𝜗 + ^𝑢𝑟)︁0

0 0 cos(𝜗)^𝑢𝜗+ sin(𝜗)^𝑢𝑟

⎤

⎥

⎦

.(5.22)

The test function

sin

(

𝜗

)

𝛿𝑢

should be used and the scaled nabla operator is given by

∇scaled =𝑟sin(𝜗)∇.

Section 5.1. Finite element method in curvilinear coordinates

It should be noted that even though the computations are performed on the two-dimensional

domain

𝛺∘

sym

, the resulting solutions correspond to a three-dimensional problem in which the

symmetry requirements are fulfilled exactly.

5.1.3 Cylindrical coordinates

For dimensionless cylindrical coordinates

𝑧𝑖

= [

𝜌, 𝜙, 𝑧

]

𝑖

with the local but orthonormal basis

system

𝑒𝜌

(

𝜙

)

𝑒𝜙

(

𝜙

)

,𝑒𝑧}

the original normalized position vector and its pseudo-Cartesian

counterpart are given by:

𝑥=𝜌^

𝑒𝜌(𝜙) + 𝑧𝑒𝑧,𝑥

∪

∘

=𝜌𝑒

∪

∘

𝜌+𝜙𝑒

∪

∘

𝜙+𝑧𝑒

∪

∘

𝑧.(5.23)

Therein,

𝑔⋆

𝑖

𝑒

∪

∘

𝑖

are the pseudo-Cartesian orthonormal basis vectors. The scaling functions

are obtained analogously to the derivations in Sec. 5.1.2 and are omitted here. For an example

of how the domain transformation is performed see Fig. 5.5. Only the important case of the

simplification for azimuthal symmetry is summarized in the following.

Cylindrical coordinates with azimuthal symmetry

The differential elements and normal vector are given by:

d𝑉= 2π𝜌d𝐴

∪

∘

sym ,d𝐴= 2πdℓ

∪

∘

sym ,𝑛=𝑛

∪

∘

.(5.24)

Both the test function and the nabla operator are scaled by

𝜌

,i.e.,

𝜌𝛿𝑢

and

∇

∪

∘

scaled

𝜌∇

Hence, the component matrix of the scaled gradient (^𝑢𝜌^

𝑒𝜌+ ^𝑢𝑧^

𝑒𝑧)⊗∇

∪

∘

scaled is given by

𝑈sym

∪

∘

⎡

⎢

⎣

𝜌𝜕^𝑢𝜌

𝜕𝜌 0𝜌𝜕^𝑢𝜌

𝜕𝑧

0 ^𝑢𝜌0

𝜌𝜕^𝑢𝑧

𝜕𝜌

𝜕^𝑢𝑧

𝜕𝜙 𝜌𝜕^𝑢𝑧

𝜕𝑧

⎤

⎥

⎦

.(5.25)

5.2 Static magnetic field computation

The accurate computation of magnetic fields is crucial for an electromagnetic force analysis. A

finite element analysis is not perfectly suited for this task as no boundary conditions are known

and rather attenuation conditions at infinity are to be imposed. However, as the magnetic field

of any magnet attenuates with the distance to the magnet cubed, a reasonable approximation is

to put a large enough sphere around the magnet at which the attenuation condition is to be

prescribed. In the following, the numerical scheme for the magnetic field calculation of a magnet

is accompanied by a proper determination of the sphere’s size based on an error analysis. As

an example that is directly relevant to Chap. 6, the magnetic field of a cylindrical permanent

magnet is computed.

5.2.1 Governing equations and the weak formulation

Consider the cylindrical magnetic domain

𝛺m

equipped with a magnetization

𝑀

, that is

embedded in an external region

𝛺0

without any magnetization,

𝑀≡0

, such that the total

computational domain is given by

𝛺

𝛺m∪𝛺0

and the magnet’s interface results from

𝐼

𝛺m∩𝛺0

. In a static setup without any free charges or free currents, the

Maxwell

’s

Chapter 5. Numerical field calculation with a reduced finite element method

On the electromagnetic coupling problem 53

equations in regular points 𝑥∈𝛺m∪𝛺0and singular points 𝑥∈𝐼reduce to:

∇×H=0,∇·𝐵= 0 ,𝑛×JHK=0,𝑛·J𝐵K= 0 .(5.26)

From the first equation, it follows that the magnetic field can be expressed via a potential, i.e.,

−∇𝑉m

. By means of the space-time relation

𝐵

𝜇0

(

𝑀

), the magnetic flux density

can be eliminated from the system of equations in order to obtain a

Poisson

equation for the

magnetic potential

𝑉m

. However, since the gradient of the magnetic potential is discontinuous in

the normal direction at the interface

𝐼

, the corresponding weak problem involves the evaluation of

these discontinuities in terms of jump operations. The formulation of the weak problem simplifies

significantly, if 𝑛·J𝐵K= 0 is utilized in the generalized Gauß theorem from Eq. (C.16),

∇·𝐵= 0 ⇒˛

𝜕𝛺

𝑛·𝐵𝛿𝑉 d𝐴−ˆ

𝛺

(∇𝛿𝑉 )·𝐵d𝑉= 0 .(5.27)

Subsequent insertion of the magnetic flux density in terms of the magnetic potential yields

−˛

𝜕𝛺0

𝑛·(∇𝑉m)𝛿𝑉 d𝐴+ˆ

𝛺0

(∇𝛿𝑉 )·(∇𝑉m) d𝑉−ˆ

𝛺m

(∇𝛿𝑉 )·(𝑀−∇𝑉m) d𝑉= 0 .(5.28)

Upon normalizing the system by means of

∇=1

ℓref

∇,d𝑉=ℓ2

refd˜

𝑉 , d𝐴=ℓ2

refd˜

𝐴 , 𝑉m=𝑉ref ˜

𝑉m(5.29)

cylindrical coordinates with azimuthal symmetry can be introduced following the scheme laid

out in Sec. 5.1 with the aid of the formulas provided in Sec. 5.1.3. After dropping the factor 2

this results in

0 = −ˆ

𝛤

∪

∘

sym

𝑛

∪

∘

·(∇

∪

∘

scaled𝑉m)𝛿𝑉 𝜌 dℓ

∪

∘

sym −ˆ

𝛺

∪

∘

sym,m

(𝜌𝑀−∇scaled𝑉m)·(∇scaled𝛿𝑉 +𝛿𝑉 ^

𝑒𝜌) d𝐴

∪

∘

sym +

+ˆ

𝛺

∪

∘

sym

(∇

∪

∘

scaled𝑉m)·(∇

∪

∘

scaled𝛿𝑉 +𝛿𝑉 ^

𝑒𝜌) d𝐴

∪

∘

sym ,(5.30)

where the domain

𝛺

∪

∘

sym

and its corresponding geometrical relations are depicted in Fig. 5.3a.

Note that although the computational domain is now two-dimensional, the underlying problem

is still three-dimensional.

The weak form in Eq.

(5.30)

is supplemented by boundary conditions and transition conditions.

The transition condition

𝑛×JHK

is satisfied if the potential itself is continuous, see [Reich

(2017)] for a proof,

0=𝑛×JHK=−𝑛×J∇𝑉mK⇔J𝑉K= 0 .(5.31)

Besides these transition conditions,

Dirichlet

boundary conditions are required. Unfortunately,

the continuous magnetic potential

𝑉m

is only restricted by its regularity and it is supposed to

vanish at infinity, i.e.,

lim

‖𝑥‖→∞

𝑉m= 0 .(5.32)

Such a condition can only be incorporated in a finite element method if it is combined with a

boundary element method, which can account for conditions at infinity. However, a boundary

element method is not only mathematically intricate but also has limited numerical precision.

Therefore, the vanishing potential

𝑉m

= 0 is imposed at the finite boundary

𝜕𝛺0

, which needs to

be far away enough from the domain of interest

𝛺1

. This is intuitively feasible, as any permanent

Section 5.2. Static magnetic field computation

magnet has a static magnetic potential of

𝑉m≈cos(𝜗)

/𝑟2

if the distance

𝑟

is sufficiently large. To

prove that the chosen boundary

𝜕𝛺0

is far enough away, a convergence analysis is performed in

Sec. 5.2.2.

Finally, the function space choice is discussed. Let

𝒫𝑘

be a piece-wise polynomial function space

of order

𝑘

, which generates continuous functions. Similarly,

𝒟𝑘

is a function space constructed

from discontinuous polynomials. If the magnetic potential is constructed from

𝑉m∈ 𝒫𝑘

, then

the current potential and thus the magnetic flux density are given by

H,𝐵∈ 𝒟𝑘−1

. For the

convergence analysis in Sec.5.2.2,

𝑘

= 2 is used for simplicity. However, a higher polynomial

degree 𝑘= 3 will prove to be more appropriate for the analysis in Sec.5.3.

5.2.2 Analytical solution and convergence analysis

As a reference the magnetic field of one cylindrical permanent magnet with respect to its own

cylindrical coordinate system

{𝜉, 𝜙, 𝑧}

is analytically by, see [Reich, Stahn, and Müller (2016)]:

𝐵(𝜉, 𝜙, 𝑧) = 𝜇0𝑀0

∑︁

𝛾=0

(−1)𝛾

2𝜉√𝑎𝛾[︁{︁(2𝑅𝜉 −𝑎𝛾)K(𝑚𝛾) + 𝑎𝛾E(𝑚𝛾)}︁𝑒𝜉(𝜙) +

+𝜉(𝑧+ [−1]𝛾𝐻){︁(𝑅−𝜉)

(𝑅+𝜉)Π(𝑛, 𝑚𝛾) + K(𝑚𝛾)}︁𝑒𝑧]︁,

(5.33)

with the auxiliary functions:

𝑎𝛾= (𝑅+𝜉)2+ (𝑧+ [−1]𝛾𝐻)2, 𝑚𝛾=4𝑅𝜉

𝑎𝛾

, 𝑛 =4𝑅𝜉

(𝑅+𝜉)2.(5.34)

For the occurring complete elliptic integrals, the following definitions are used:

K(𝑚) =

𝜙=0

√︀1−𝑚sin2𝜙d𝜙 , E(𝑚) =

𝜙=0 √︁1−𝑚sin2𝜙d𝜙 ,

Π(𝑛, 𝑚) =

𝜙=0

(1 −𝑛sin2𝜙)√︀1−𝑚sin2𝜙d𝜙 .

(5.35)

Note that in the literature the elliptic moduli are squared, see [Wolfram Research, Inc. (2016)].

The definitions in Eq.

(5.35)

follow the implementations in the

Mathematica

software, see

[Wolfram Research, Inc. (2016)]. This analytical solution contains several apparent singularities

when the contained functions are evaluated separately, e.g., for

𝜉

𝑅

and an arbitrary

𝑧

one has

𝑛

= 1 such that

(

𝑛→

, 𝑚

)

→ ∞

. However, most of these singularities are only

apparent, as in the previous example the term

𝑅−𝜉

regulates this behavior for most values

𝑧

. An implementation of the analytical solution in FEniCS and therefore in python is very

tedious since the singularity checks need to be performed manually. However, the commercial

software

Mathematica

performs these checks automatically, such that it is more convenient

to use

Mathematica

to export the components (

𝐵𝑖

ana,𝜌, 𝐵𝑖

ana,𝑧

)for a point specified by the

coordinates

𝑝𝑖

= (

𝜌𝑖, 𝑧𝑖

)for all desired grid points

𝑝𝑖

outside of the magnet as indicated in

Fig. 5.3a. Note that at the points with coordinates (

𝑅, ±𝐻

), real singularities occur, which

is why these particular points are excluded from the following analysis. The values for the

magnetic flux density are exported using a numerical accuracy of ten digits after the comma.

For a given mesh density an appropriate amount

𝑁

of grid points is used to evaluate an absolute

Chapter 5. Numerical field calculation with a reduced finite element method

On the electromagnetic coupling problem 55

as well as a relative error via

𝑒abs =1

𝑁

∑︁

𝑖=1 √︁(𝐵𝑖

ana,𝜌 −𝐵𝑖

sim,𝜌)2+ (𝐵𝑖

ana,𝑧 −𝐵𝑖

sim,𝑧)2,

𝑒rel =∑︀𝑁

𝑖=1 √︁(𝐵𝑖

ana,𝜌 −𝐵𝑖

sim,𝜌)2+ (𝐵𝑖

ana,𝑧 −𝐵𝑖

sim,𝑧)2

∑︀𝑁

𝑖=1 √︁(𝐵𝑖

ana,𝜌)2+ (𝐵𝑖

ana,𝑧)2,

(5.36)

with the numerical solution from the finite element analysis (

𝐵𝑖

sim,𝜌, 𝐵𝑖

sim,𝑧

)evaluated at the

grid point

𝑝𝑖

. In Fig. 5.3b both the convergence analysis for the mesh refinement process as

well as the error caused by the pseudo infinity boundary distance

𝑑ext

are shown. In Fig. 5.3b

𝛤

∪

∘

sym

𝛺

∪

∘

sym,m𝜉

𝑧

𝑑ext

𝑃𝑖

(a) grid value sketch

103104105

10−6

10−4

10−2

number of elements

eabs

erel

0 50 100 150 200

10−9

10−5

10−1

dext =dext/R

eabs

erel

(b) error plots

Fig. 5.3:

Depiction of the grid points and different error plots for the magnetic field. In (b) a distance of

𝑑ext/𝑅 ≈

10 was used for the mesh density analysis. For the distance error plot the maximum number of

elements was used.

two different error dependencies are presented. The upper plot shows the convergence of the

magnetic field solution as the mesh is successively refined and the number of elements increases.

For the analyzed grid points, the relative error reaches a satisfactory level of below 10

−6

. Note

that this error measure represents a global error of the post-processed quantity

𝐵

, which is

obtained by means of differentiation of the primary potential field

𝑉m

. In the plot at the

bottom of Fig. 5.3b the same error is shown for different values of the pseudo boundary

𝛤

∪

∘

sym

‖𝑥‖

𝑑ext

. The relative error of 10

−6

obtained for the maximum number of elements analyzed

can be reduced significantly by increasing

𝑑ext

. For an external distance of

𝑑ext ≈

𝑅

the

relative error reaches a minimum value of 10

−9

, which hardly can be improved any further. This

goes to show that the developed numerical method has high accuracy. However, it should be

noted that these results are obtained by explicitly excluding the critical corners of the magnet,

which will affect the errors in the subsequent analyses.

5.3 Numerical magnetic force computation

Regardless of the electromagnetic coupling model used, any force computation that relies on

numerically obtained electromagnetic fields is affected by the error cascade as the electromagnetic

fields are error-prone and their subsequent numerical evaluation may accumulate these errors. It

Section 5.3. Numerical magnetic force computation

is therefore crucial to perform a proper error analysis which in the following will lead to design

recommendations for the discretization of electromagnetic problems as well as guidelines as to

how the post-processing is best to be performed.

The following two exemplary problems are analyzed, (a) the electromagnetic eigen force

𝐹(em)

1,eigen

a cylindrical permanent magnet

𝛺1

with axial magnetization exerts upon itself and

(b) the electromagnetic force

𝐹(em)

1,2

due to another coaxial cylindrical permanent magnet

𝛺2

The latter was analyzed analytically in [Reich, Rickert, and Müller (2017)]. Note that both

scenarios are free from the electromagnetic coupling problem, i.e., all force models will yield the

same result and therefore the generalized

Lorentz

model may be used without restricting the

generality of the following analysis. The first problem of calculating the force

𝐹(em)

1,eigen

serves as

an error analysis in itself as this force is known to be zero analytically. For the second problem,

the solution derived in [Reich, Rickert, and Müller (2017)] is used in order to compare the

numerical results and to assess their quality. For both problems only the total force along the

cylinder’s main axis 𝑒𝑧is calculated, i.e.,𝐹1=𝐹(em)

1·𝑒𝑧, see the sketch in Fig.5.7.

There are three different approaches to the calculation of the static total force acting on the

magnet

𝛺1

that can be envisioned:

(I)

force calculation by means of the original force densities,

(II)

employing the

Maxwell

stress tensor concept and

(III)

using the enclosing surface integral,

𝐹(em)

(I)

=˛

𝜕𝛺

𝑓(em)

𝐼d𝐴+ˆ

𝛺

𝑓(em) d𝑉(II)

=˛

𝜕𝛺

𝑛·𝜎(em)

+d𝐴(III)

=˛

𝑆

𝑛·𝜎(em)

+d𝐴 . (5.37)

Therein,

𝑆

is a closed surface that fully contains the magnet

𝛺1

and does not cut through any

permeable matter. The approach

(III)

is commonly used in commercial numerical software

because it avoids evaluating the magnetic field at edges where singularities are known to be

located, see [Kim (2019); Humphries (2015); Humphries (2020)]. The surface

𝑆

is then drawn

approximately two to three cell diameters away from the actual interface. Both approaches

(II)

and

(III)

are more convenient to use because it is neither required to calculate jumps and mean

values of electromagnetic fields nor is specific knowledge of the magnet’s magnetization required

once the magnetic field outside of the magnet is known. However, the approach

(I)

enables

further analytical preparation of the corresponding integrals which may yield better numerical

results at the cost of a more problem-specific implementation.

For both scenarios analyzed, the volumetric force density vanishes,

𝑓(em)

, and only a

surface force density

𝑓(em)

𝐼

= (

𝑛×J𝑀K

)

×⟨𝐵⟩

is to be calculated, where

J𝑀K

−𝑀0𝑒𝑧

. Due

to the special geometry of the cylindrical magnet,

𝑛×𝑒𝑧

vanishes at the top and bottom surface

of the magnet such that only a mantle integral with

𝑛

𝑒𝜌

remains which further reduces to a

single line integral due to azimuthal symmetry,

𝐹(I)

1=−𝑀0𝑒𝑧·ˆ

𝛤mantle

(𝑛×𝑒𝑧)×⟨𝐵⟩d𝐴=−2π𝑀0𝑅

𝐻

𝑧=−𝐻

⟨𝐵𝜌⟩d𝑧 . (5.38)

For the other two approaches the required integrals read

𝐹(II)/(III)

1=2π

𝜇0ˆ

𝒞

∪

∘

(︂(𝑛·𝐵)(𝑒𝑧·𝐵)−1

2(𝐵·𝐵)(𝑛·𝑒𝑧))︂𝜌

∪

∘

dℓ

∪

∘

sym ,(5.39)

where

𝒞

∪

∘

is a contour in

𝛺

∪

∘

sym

that either follows the surface of the magnet for the approach

(II)

or is arbitrarily shaped around the magnet for

(III)

. At least for the integration along the surface

of the magnet the product

𝑛·𝑒𝑧

can be simplified analytically such that the mantle surface is

excluded. However, this product should be numerically zero in the FEniCS implementation and

Chapter 5. Numerical field calculation with a reduced finite element method

On the electromagnetic coupling problem 57

the idea is to not insert any of the geometry information manually in order to keep the method

convenient to use.

At this point is already clear why the expressions in Eqs.

(5.38)

–

(5.39)

are suspected to yield

numerically different accuracies. Firstly, for the approach

(I)

the integration domain is smaller

and thus fewer numerical errors contained in the magnetic flux density are accumulated. Secondly,

in Eq.

(5.38)

only one component of the magnetic flux density occurs whereas products of the

components, e.g.,

𝐵𝜌𝐵𝑧

, are required in the contour integral approaches which may lead to error

amplification. Note that the computations are performed with normalized quantities and only

after the values for the integrals are obtained the dimensional factors are used in order to obtain

a force value in the unit of Newton. It is important to analyze the non-normalized (real) force

value as for example the factor of dimensionality in Eq.

(5.39)

is given by

𝜇−1

0ℓ2

ref𝐵2

ref ≈180 N

Hence, the numerical error in the force calculation may be amplified by two orders of magnitude.

In the end, only the force value with correct units is relevant for comparison with experimental

data. In Fig. 5.4 the self-force error is plotted against the number of degrees of elements for

103104105

10−5

10−3

10−1

number of elements

|f1,eigen|/N

I (k= 2)

I (k= 3)

II (k= 2)

II (k= 3)

III (k= 2)

III (k= 3)

Fig. 5.4:

Self-force error

𝐹1,eigen

over mesh refinement for the different formulations of the force integral

(I)

–

(III)

. The polynomial degree

𝑘

of the function space

𝒫𝑘

for the magnetic potential

𝑉m

is indicated in

the brackets.

the three different force calculation methods. Additionally, for each method, the self-force is

depicted using polynomial spaces

𝒫𝑘

of with both

𝑘

= 2 and

𝑘

= 3 for the magnetic potential

𝑉m

, which yield a discontinuous polynomial approximation

𝐵∈ 𝒟𝑘−1

. In Fig. 5.4 it can be seen

that the second approach, i.e., the

Maxwell

stress tensor concept

(II)

performs better than

the other two approaches. However, upon increasing the mesh density, the resulting error does

not decrease monotonically. Furthermore, increasing the polynomial degree

𝑘

for the magnetic

potential does not necessarily reduce the error. The reason for this undesired behavior is that

the region around the corners of the magnet is discretized differently in each refinement step as

a naive generation of the mesh using unstructured triangulation does not restrict the orientation

of the triangles. As a result, the top and bottom corners are triangulated differently, see Fig. 5.5.

Therefore, the singularities located at these corners may be approximated differently as well.

Thus, a larger error for a finer mesh compared to a symmetric coarse mesh arises, even if the

cell diameter is smaller. In order to circumvent this behavior, the mesh is partially structured

as shown in Fig. 5.5.

Note that implementing such a partially structured triangulation is only feasible due to the

two-dimensional computational domain arising from using cylindrical coordinates and assuming

azimuthal symmetry. In the meshing software GMSH, [Geuzaine, C. and Remacle, J.-F. (2022)],

the partially structured grid is achieved by adding additional points and lines to the model. In

addition to the rectangle that represents the magnet, another rectangle that is inflated by one

cell diameter is added. Similarly, a deflated rectangle inside the magnet is added, such that a

well-structured triangulation is enforced, see the most right-hand side in Fig. 5.5.

Section 5.3. Numerical magnetic force computation

𝑧

𝑦

𝑥azimuthal

symmetry

𝑧

𝜌

naive mesh partially structured

Fig. 5.5: Mesh structuring rule in corner regions.

If the rules established from the analysis in Fig. 5.5 are observed during the mesh construction

process, the resulting force computations are significantly more accurate. Even for a compara-

tively coarse mesh with a few thousand elements, the numerical error reaches satisfactory levels

for both of the approaches

(I)

and

(II)

, see Fig. 5.6. Most importantly, the results are more

103104105

10−5

10−4

10−3

10−2

10−1

number of elements

|f1,eigen|/N

I (k= 2)

I (k= 3)

II (k= 2)

II (k= 3)

III (k= 2)

III (k= 3)

Fig. 5.6:

Self-force error

𝐹1,eigen

over mesh refinement for the different formulations of the force integral

(I)–(III) using the partially structured mesh from Fig. 5.5.

consistent, i.e., a refinement of the mesh by means of increasing the mesh density results in a

smaller force computation error, which is a feature that is missing in the analysis leading to

Fig. 5.4. Additionally, the error of one approach decreases now consistently if the polynomial

degree 𝑘of the magnetic potential is increased from two to three.

Furthermore, the difference between the two approaches

(I)

and

(II)

are less pronounced,

such that they may be deemed equally good. In contrast, the relative error of the force between

two cylindrical magnets presented in Fig.5.7 shows that the

Maxwell

stress tensor approach

(II)

is worse than the other two. For small distances,

𝑑

between the magnets, the errors for the

approaches

(I)

and

(III)

are nearly identical. For larger distances, the exterior contour approach

starts to develop inconsistencies. However, the obtained results suggest that the relative error

for all approaches remains well below one percent for relevant distances between the magnets.

From the above analyses, it can be concluded that, upon using an appropriate mesh generation

technique, all the approaches

(I)

and

(II)

are numerically equally good, but the approach using

an external contour should be avoided. In the following, the approach

(II)

is favored over

(I)

because requires less manipulation of a given force model and is thus more suitable for a force

model comparison.

Chapter 5. Numerical field calculation with a reduced finite element method

On the electromagnetic coupling problem 59

𝑓1,2

𝛺1

𝛺2

𝑑

2𝑅

0.511.522.5 3

0.1

0.2

0.3

normalized distance d/R

relative error (%)

|fsim

1,2−fana

1,2|

|fana

1,2|

III

Fig. 5.7:

Relative error for the force between two cylindrical magnets against the face-to-face distance

𝑑

In all three approaches, a polynomial degree of 𝑘= 3 was used.

5.4 Two-phase flow

In this section, the mathematical prerequisites for the analysis of the oil droplet deformation

in Chap. 7 are derived. In the literature, a commonly used approach is the introduction of a

level-set function, which acts as an indicator function for the two domains and subsequently

for the interface itself, see [Olsson and Kreiss (2005); Habera and Hron (2017)]. The evolution

of the level set function is governed by a transport equation. This equation is unstable when

solved with the finite element method and thus requires numerical stabilization. In order to

prevent both, the high computational costs as well as the introduction of a smooth indicator

function, an interface tracking method by means of appropriate meshing is used. To this end,

only azimuthally symmetric problems are considered and thus the methods described in Sec. 5.1

can be used. As a result, the computational domain is reduced to a two-dimensional region.

The interface itself reduces its apparent dimension from two to one, i.e., in the new coordinate

space it becomes a line. This line can be represented discretely in terms of a set of points,

which is convenient. Note that such an approach comes at the cost of nonlinear coordinate

dependencies,

Christoffel

symbols and several modifications to the classical finite element

analysis, see Sec. 5.1. However, the dimension of the computational domain is reduced to two,

while the problem still retains its three-dimensional structure, i.e., the obtained solutions are

still three-dimensional. Furthermore, the description of the interface in terms of a point set

allows for a blunt interface remeshing technique described in Sec. 5.4.3. In each time step of the

simulation, the underlying mesh is regenerated such that the interface points represent actual

nodes of the mesh. This way, the interface line is linearly approximated by the element facets

between the discrete interface points.

5.4.1 Problem formulation and governing equations

In this section, the governing equations and boundary conditions for a general immiscible

two-phase fluid flow are derived. Subsequently, the coordinate-free weak form of a two-phase

fluid flow problem is presented and an appropriate normalization is introduced. Consider a

container

𝛺

filled with two different fluids that do not mix. The fluids are set up such that

one fluid is completely immersed inside the other. Thus, an interface

𝐼

𝛺+∩𝛺−

separates

both fluids, which occupy the domains

𝛺+

and

𝛺−

, respectively. The situation is schematically

depicted in Fig. 5.8.

The mechanical motion is governed by the balance of linear momentum in regular and singular

Section 5.4. Two-phase flow

𝑛

𝜏

𝑛∘

𝜏∘

𝑛∘

𝜏∘

Ssym

azimuthal symmetry spherical coordinates

𝑥

𝑦

𝑧

𝑦

𝑧

𝑟

𝜗

𝛤

𝒪

𝛺sym

2π

4π

𝜗

𝑟

𝛤∘

ℓ∘

𝒪ℓ∘

𝑧+

ℓ∘

𝑧−

𝛺∘

sym

Fig. 5.8:

Two immiscible fluids in a spherical container and the domain transformation into a spherical

coordinate space.

points, respectively:

𝜌𝜕𝑣s

𝜕𝑡 +𝜌𝑣s·(∇s⊗𝑣s) = ∇s·𝜎+𝜌𝑓,𝑥s∈(𝛺+∪𝛺−),

∇𝐼·𝜎𝐼=−𝑛·𝜎,𝑥s∈𝐼 .

(5.40)

Both fluids are assumed to behave like

Newton

ian fluids. According to [Dziubek (2011)], the

surface tension of an interface with negligible thickness can be assumed to be proportional to

the curvature of the interface,

𝜎=−𝑝1+𝜇(𝑣⊗∇s+∇s⊗𝑣),∇𝐼·𝜎𝐼=−𝜎

𝐼(∇s·𝑛)𝑛.(5.41)

The following normalization is used:

𝑥=ℓref ˜

𝑥,∇s=1

ℓref

∇, 𝑡 =𝑡ref ˜

𝑡 , 𝑓=𝑔0˜

𝑓,𝑣=𝑣ref ˜

𝑣,

𝜎=𝜎ref ˜

𝜎, 𝑝 =𝑝ref ˜𝑝 , 𝜎𝐼=𝜎𝐼

ref ˜

𝜎𝐼, 𝜌 =𝜌ref ˜𝜌 , 𝜇 =𝜇ref ˜𝜇 .

(5.42)

Therein, quantities with a tilde are dimensionless as the reference quantities contain both the

unit as well as the order of magnitude of the fields. Of course, not all of these reference values

are independent. With the special choices

𝑡ref =ℓref

𝑣ref

, 𝜎ref =𝑝ref =𝜎(em)

ref =𝜌ref𝑣2

ref , 𝜎𝐼

ref =𝜎

𝐼,(5.43)

the following dimensionless numbers arise:

Re :=𝜌ref 𝑣refℓref

𝜇ref

,We :=𝜌ref𝑣2

refℓref

𝜎

𝐼

,Fr :=𝑣2

ref

𝑔0ℓref

.(5.44)

Therein,

is the

Reynolds

number,

is the

Weber

number and

is the

Froude

number.

The normalized system of equations reads

⎧

⎪

⎨

⎪

⎩

˜𝜌𝜕˜

𝑣

𝜕˜

𝑡+ ˜𝜌˜

𝑣·(˜

∇⊗ ˜

𝑣) = ˜

∇· ˜

𝜎+1

Fr ˜𝜌˜

𝑔,˜

∇· ˜

𝑣= 0 ,𝑥∈(𝛺−∪𝛺+),

𝑛·˜

𝜎=1

We (˜

∇·𝑛)𝑛,˜

𝑣=0,𝑥∈𝐼 .

(5.45)

which has to be supplemented by the initial conditions

𝑣(˜

𝑥,˜

𝑡= 0) = 0,˜𝑝(˜

𝑥,˜

𝑡= 0) = 0 ,(5.46)

Chapter 5. Numerical field calculation with a reduced finite element method

On the electromagnetic coupling problem 61

as well as the normalized constitutive relation

𝜎=−˜𝑝1+˜𝜇

Re (˜

𝑣⊗˜

∇+˜

∇⊗ ˜

𝑣).(5.47)

5.4.2 Weak form and adaptive time stepping

For the spatial discretization, the Finite Element Method using FEniCS is employed. Following

the procedure outlined in Sec. 5.1.1, the weak formulation of the dimensionless system is derived.

Each differential equation is multiplied with its respective test function

𝛿𝑣

and

𝛿𝑝

. Subsequently,

each equation is integrated over the whole domain 𝛺=𝛺−∪𝛺+, which results in:

𝛺

˜𝜌𝜕˜

𝑣

𝜕˜

𝑡·𝛿𝑣d𝑉=−ˆ

𝛺

˜𝜌𝛿𝑣·(˜

𝑣⊗˜

∇)·˜

𝑣d𝑉+ˆ

𝛺

(˜

∇· ˜

𝜎)·𝛿𝑣d𝑉+1

Fr ˆ

𝛺

˜𝜌˜

𝑓·𝛿𝑣d𝑉 ,

0 = ˆ

𝛺

𝛿𝑝(˜

∇· ˜

𝑣) d𝑉 .

(5.48)

By means of the product rule and the generalized

Gauss

one can rewrite a generic weak flux

form via

𝛺

[˜

∇·𝐴]·𝑏d𝑉=ˆ

𝜕𝛺∖𝐼

𝑛·𝐴·𝑏d𝐴−ˆ

𝐼

𝑛·J𝐴·𝑏Kd𝐴−ˆ

𝛺

𝐴··(˜

∇⊗𝑏) d𝑉(5.49)

For the interface integral the singular balances can be inserted if

𝑏

is a continuous function. In

the equations above,

𝑏

is one of the test functions respectively, and as such, continuous, i.e.,

J𝐴·𝑏K

J𝐴K·𝑏

. Furthermore, for the momentum balance the integral over

𝜕𝛺 ∖𝐼

vanishes as

only

Dirichlet

boundary conditions are prescribed and

𝛿𝑣

vanishes there. Hence, the weak

formulation of the balance of linear momentum becomes

𝛺

˜𝜌𝜕˜

𝑣

𝜕˜

𝑡·𝛿𝑣d𝑉=−ˆ

𝛺

˜𝜌𝛿𝑣·(˜

𝑣⊗˜

∇)·˜

𝑣d𝑉−1

We ˆ

𝐼

(˜

∇·𝑛)𝛿𝑣·𝑛d𝐴−

−ˆ

𝛺

𝜎··(˜

∇⊗𝛿𝑣) d𝑉+1

Fr ˆ

𝛺

˜𝜌˜

𝑓·𝛿𝑣d𝑉 .

(5.50)

The first equation is essentially the principal of virtual power as described in Eq.

(4.38)

. Both

weak forms transformed and subsequently solved in spherical coordinates using the method

developed in Sec. 5.1.1. To this end, the computational domain

𝛺

is transformed into

𝛺∘

sym

depicted in Fig. 5.8. Furthermore, the weak form in spherical coordinates reads

0 = ˆ

𝛺∘

sym

˜𝑟2sin2(𝜗)˜𝜌𝜕˜

𝑣

𝜕˜

𝑡·𝛿𝑣∘d𝐴∘

s+ˆ

𝛺∘

sym

˜𝑟sin(𝜗)˜𝜌𝛿𝑣∘·(˜

𝑣⊗˜

∇sph)·˜

𝑣d𝐴∘

+ˆ

𝛺∘

sym

˜𝑟sin(𝜗)˜

𝜎··(˜

∇sph ⊗𝛿𝑣∘+ cos(𝜗)𝑒∘

𝜗⊗𝛿𝑣∘) d𝐴∘

We ˆ

𝐼∘

sym

𝜅(˜𝑟, 𝜗)(︀˜𝑟𝑛∘

𝑟𝛿𝑣∘

𝑟+𝑛∘

𝜗𝛿𝑣∘

𝜗)︀dℓ∘

s−1

Fr ˆ

𝛺∘

sym

˜𝑟2sin2(𝜗)˜𝜌˜

𝑓·𝛿𝑣∘d𝐴∘

(5.51)

with the scaled curvature function

𝜅= ˜𝑟sin2(𝜗)( ˜

∇·𝑛).(5.52)

Section 5.4. Two-phase flow

Note that for the mass density, there is no second spatial derivative that requires a replacement

𝛿𝑝 →sin

(

𝜗

)

𝛿𝑝∘

. However, it turns out that symmetries with respect to the equatorial plane

(𝜗=π

2) are better reproduced if

0 = ˆ

𝛺

˜𝑟sin(𝜗)𝛿𝑝∘(˜

∇sph ·˜

𝑣) d𝐴∘

s(5.53)

is used as the weak form of the mass balance.

Before the weak form can be implemented and subsequently solved, a choice for the underlying

finite element function spaces must be made. For incompressible fluid flow problems a saddle

point problem for the pressure field

𝑝

and the velocity field

𝑣

arises and thus the function space

choice is crucial. The finite element analysis of this problem type is known to be unstable

unless the so-called

Ladyzhenskaya

–

Babuška

–

Brezzi

condition is observed, see [John (2016),

pg. 34]. Roughly speaking, for the stability of the finite element analysis, the discrete function

space for

𝑣

is required to be sufficiently large when compared to the function space of

𝑝

Therefore, a common choice is the

Taylor

–

Hood

element with (

𝑣, 𝑝

)

∈

(

𝒫𝑘,𝒫𝑘−1

)for

𝑘≥

Therein,

𝒫𝑘

is a piece-wise polynomial function space of order

𝑘

, which generates continuous

functions. However, in the present problem, the pressure function is discontinuous at the

interface

𝐼

. Therefore, a function space

𝒟𝑘

of discontinuous polynomials is used for the pressure.

A stable pair in two dimensions is then given by, e.g., the

Scott

–

Vogelius

element (

𝒫𝑘,𝒟𝑘−1

)

for

𝑘≥

4. However, for the

Reynolds

number analyzed in this work, a much smaller but still

stable choice is given by (

𝒫3,𝒟1

). This pair is used in the following. A summary of all the finite

element functions involved in the simulation as well as their function spaces is listed in Tab. 5.1.

Note that the space for the scaled curvature is artificially enhanced, see the last paragraph in

Tab. 5.1: Finite element functions and their function spaces used for the two-phase flow simulations

name symbol FEM function space

velocity 𝑣𝒫2

pressure 𝑝𝒟0

geometric interface normal 𝑛∘

h𝒫1

physical interface normal 𝑛h𝒫2

scaled curvature 𝜅𝒫2

Sec. 5.4.3.

The time derivative in the balance of linear momentum is discretized by means of a finite

difference, i.e.,

𝜕˜

𝑣

𝜕˜

𝑡≈˜

𝑣𝑛−˜

𝑣𝑛−1

Δ˜

𝑡𝑛,˜

𝑣𝑛=˜

𝑣(˜

𝑥, 𝑡 =𝑡𝑛).(5.54)

Apart from the CFL criterion, there are two limiting mechanisms for the time step size Δ

𝑡𝑛

: the

general convection and the interface evolution. Since the

Reynolds

number in this problem is

so small, the nonlinear convection term may be linearized via

𝑣·(˜

∇sph ⊗˜

𝑣)≈˜

𝑣𝑛−1·(˜

∇sph ⊗˜

𝑣𝑛), 𝑖 ∈ {1, . . . , 𝑁}.(5.55)

Furthermore, the small

Reynolds

number indicates that the interface evolution is more critical

than the convection in regular points. Hence, after each time step the maximum interface

velocity is calculated,

˜𝑣𝑛,max

𝐼= max{‖˜

𝑣𝑛(𝑥1)‖,...,‖˜

𝑣𝑛(𝑥𝑁)‖} ,𝑥𝑖∈𝐼 , (5.56)

Chapter 5. Numerical field calculation with a reduced finite element method

On the electromagnetic coupling problem 63

and the time step is subsequently adjusted

Δ˜

𝑡𝑛+1 =(︂d𝑢tol

d𝑢𝑛

max )︂𝑝𝑢tol

˜𝑣𝑛,max

𝐼

.(5.57)

Therein,

𝑢tol

is a fraction of the minimal cell size of the mesh,

ℎmin

, and represents the desired

interface displacement magnitude for a given time step. Furthermore, d

𝑢𝑛

max

is the maximum of

the differences between the displacements of neighboring interface points and represents the

tangential surface displacement gradient. Both, the surface displacement as well as its tangential

gradient are compared to their tolerated values

𝑢tol

and d

𝑢tol

, respectively. These values are

given by fractions of the smallest cell size

ℎmin

. It turns out, that the rather extreme choice of

𝑢tol

= d

𝑢tol

= 0

ℎmin

and

𝑝

= 3 results in a good convergence behavior. Over time and when

the solution is close to being stationary, the time step increases, which reduces computational

costs. With the above naive update procedure for the time step, overshooting can occur, i.e.,

the time step increase may be too significant. This can lead to an oscillation between a large

and a small time step. In order to circumvent this, the error tangential displacement gradient

error 𝑒is calculated and subsequently used to restart the simulation with another step size

𝑒=d𝑢𝑛+1

max

d𝑢tol

<1⇒Δ˜

𝑡𝑛→𝑒𝑛+1˜

𝑡𝑛.(5.58)

This way, each time step is solved twice, but an appropriate time step is used. Note that no

remeshing is required for restarting the simulation for a given time step.

5.4.3 Interface tracking method and curvature computation

Since the interface between both fluid phases is not necessarily aligned with a naive unstructured

mesh, a numerical description of the interface defined on the given mesh is challenging. In

literature, the level-set formulation is usually employed, [Olsson and Kreiss (2005)], in which a

smooth indicator function is used to identify the fluid phase at a given location by assuming

predefined values (say) one and zero for both respective phases. The interface is then defined as

the region in which the level-set function assumes a given value in between these values (say)

Subsequently, the level-set function and consequently the interface are convected by the velocity

field. There are a number of numerical challenges arising from such a formulation, [Sethian and

Smereka (2003)].

In this thesis, a different approach is followed, namely the interface tracking by means of

mesh alignment, which is feasible on the two-dimensional domain

𝛺∘

sym

only. Consider the

interface

𝐼∘

sym

𝛺∘,+

sym ∪𝛺∘,−

sym

, that is given initially and can be approximated by a set of

points

{𝑥∘

1,...,𝑥∘

𝑁}

. Subsequently, the interface evolution is obtained from the no-slip condition

J𝑣K

. Once a temporal discretization is employed, the interface points are advanced in each

time step. Over time the quality of the interface approximation by means of

𝑁

points may

become insufficient. Therefore, in each time step the distance between neighboring points is

calculated and compared with some tolerance value

𝑑max

. In order to evaluate such a condition

properly, a sophisticated ordering of the discrete interface points is required in order to keep

track of the property of points being neighboring. However, in the special case analyzed here,

the two-dimensional interface can be represented by a line in the spherical coordinate space.

Then, in a list of ordered points, the list index can easily be used in order to identify if points

are neighboring.

Adopting the notation from Sec. 5.1.2 the interface is approximated via

𝐼∘

sym(𝑡)≈ {𝑝∘

1(𝑡), . . . , 𝑝∘

𝑁(𝑡)}={𝑥∘

1(𝑡),...,𝑥∘

𝑁(𝑡)}, 𝑁 ∈N.(5.59)

Section 5.4. Two-phase flow

where the identification of a point

𝑝∘

𝑖

by its position vector

𝑥∘

𝑖

is for convenience. The interface

evolution is obtained from the no-slip condition 𝑣=0, by solving

𝜕𝑥𝐼

𝜕𝑡 =𝑣(𝑥𝐼, 𝑡)∀𝑥𝐼∈𝐼with 𝑥𝐼=𝑟𝐼(𝑡)𝑒𝑟(︀𝜗

𝐼(𝑡), 𝜙)︀(5.60)

Hence, the evolution equation for the interface points expressed in terms of the pseudo-Cartesian

representation 𝑥∘=𝑟𝑒∘

𝑟+𝜗𝑒∘

𝜗reads:

𝜕𝑥∘

𝜕𝑡 =𝑣∘

𝑟(𝑥∘, 𝑡)𝑒∘

𝑟+𝑣∘

𝜗(𝑥∘, 𝑡)

𝑟𝑒∘

𝜗∀𝑥∘∈𝐼∘

sym .(5.61)

Naturally, the time derivative is discretized analogously to Eq.

(5.54)

. Using the Python library

SciPy by [Virtanen et al. (2020)], a cubic spline interpolation 𝑠∘(𝛼)is constructed such that

𝑠∘(𝑘)!

=𝑥∘

𝑘∀𝑘∈ {1, . . . , 𝑁}.(5.62)

The parameter 0

≤𝛼≤𝑁

of the spline interpolation is not restricted to whole numbers. In fact

for any real

𝑘 < 𝛼 <

(

𝑘

+ 1) the interpolation returns the position vector of a point

“in between”

both points

𝑝∘

𝑘

and

𝑝∘

𝑘+1

, as the interpolation itself is continuous. If the distance between two

points becomes too large, i.e.,

‖𝑥∘

𝑘+1 −𝑥∘

𝑘‖ ≥ 𝑑max

, another point is added to the (ordered) list

of interface points in accordance with the spline interpolation. Hence, the discrete interface

is refined by inserting

𝑥∘

𝑘+1

/2:

𝑠∘

(

𝑘

)to the list. The total number of discrete interface

points

𝑁

is then updated accordingly. The interface refinement and the spline interpolation

are depicted in Fig. 5.9a. The spline interpolation 𝑠(𝛼)can also be used to obtain the tangent

𝑝∘

𝑘

𝑝∘

𝑘+1

𝑝∘

𝑘+1

𝛺−

∘

𝛺+

∘𝑠∘

(a) Discrete interface refinement

𝑝∘

𝑘+1

𝜏∘

𝑘+1

𝑛∘

𝑘+1

𝑥∘

𝑘

𝑛∘

𝑘

𝜏∘

𝑘−1

𝜏∘

𝑘

𝑠∘

𝑥∘

𝑘−1

𝑛∘

𝑘−1

𝑒∘

𝑟

𝑒∘

𝜗

(b) Normal approximation

Fig. 5.9:

Spline interpolation of the interface and normal vector approximation. Note that the curvature

of the spline interpolation is exaggerated in order to make it visible.

vector to the curve and also the curve’s normal vector via:

𝜏∘=𝑠′

∘

‖𝑠′

∘‖,𝑛=𝑠′′

∘

‖𝑠′′

∘‖,(·)′=d(·)

d𝛼.(5.63)

The latter representation is convenient from an analytical point of view, but may or may not

give the correct sign for an outward normal, depending on the curvature. Additionally, it

introduces another derivative, which may lead to reduced accuracy in the numerical method.

Given the tangent vector, it is actually only an algebraic task to find the normal vector to a

curve in a two-dimensional setting. Consider the tangent vectors in Fig. 5.9b, which follow a

Chapter 5. Numerical field calculation with a reduced finite element method

On the electromagnetic coupling problem 65

counterclockwise sense of rotation if the discrete interface points are ordered counterclockwise.

The corresponding outward normal vectors are then obtained by a 90

∘

rotation in the clockwise

direction:

𝑛∘=𝑄·𝜏∘=

𝑠′

∘

−1𝑄·𝑠′

∘with 𝑄=𝑒∘

𝑟⊗𝑒∘

𝜗−𝑒∘

𝜗⊗𝑒∘

𝑟.(5.64)

Since the spline interpolation is obtained from the interface point list, it is not reasonable to

evaluate the interpolation more frequently than the original points. In order to post-process the

normal vector function, e.g., for the curvature computation, the normal vector is interpolated

onto a continuous finite element function space 𝒫𝑘.

Note that a continuation trick is required because the normal vector is only defined on the

interface

𝐼∘

sym

and not on the whole domain, on which the function space is constructed, i.e.,

𝒫1

(

𝛺∘

sym

). In the FEniCS software this is achieved by means of first creating an empty vector

function with zero value everywhere. Subsequently, the degrees of freedom located at the

interface points are set with the values from the spline normal function in Eq.

(5.64)

. As a result

a finite element function 𝑛∘

h∈ 𝒫1(𝛺∘

sym)is available for post-processing.

Once the interface normal vector

𝑛∘

is available, it can be used to obtain the real normal

vector to the original domain 𝑛hvia Eq.(5.21), thus,

𝑛h= ^𝑛𝑟𝑒∘

𝑟+ ^𝑛𝜗𝑒∘

𝜗=˜𝑟𝑛∘

𝑟𝑒∘

𝑟+𝑛∘

𝜗𝑒∘

𝜗

√︁˜𝑟2(𝑛∘

𝑟)2+ (𝑛∘

𝜗)2.(5.65)

In order to allow for higher accuracy in the curvature calculation, the resulting expression

from Eq.

(5.65)

is projected onto a higher order finite element space, i.e.,

𝒫2

. Subsequently, an

intermediate scaled curvature similar to 𝜅from Eq. (5.52) can be calculated via

𝜅

sin(𝜗)= ˜𝑟sin(𝜗)˜

∇·𝑛h= ˜𝑟sin(𝜗)𝜕^𝑛𝑟

𝜕˜𝑟+ 2 sin(𝜗)^𝑛𝑟+ sin(𝜗)𝜕^𝑛𝜗

𝜕𝜗 + cos(𝜗)^𝑛𝜗.(5.66)

Naturally, the result is a function in

𝒟1

. However, the required scaled curvature function is

obtained from another multiplication of the expression from Eq.

(5.66)

with

sin

(

𝜗

). Therefore,

additional continuous information is gained, which allows for a projection of the resulting

function onto a second order function space, i.e.,

𝜅∈ 𝒫2

. Without implementing this trick,

the simulations become unstable. In particular, for an insufficient curvature function space,

the interface points at the poles, i.e.,

𝜗

= 0 and

𝜗

, experience an unbound acceleration

over time. This is due to non matching degrees of freedom between the pressure force and the

surface tension force at the interface. Hence, for the chosen function space for the velocity and

the pressure, the curvature must be interpolated onto a higher order function space 𝒫2.

5.4.4 Consistency analysis

In order to analyze the consistency of the model, a boundary initial value problem with a known

stationary state is investigated. To this end, a droplet of ellipsoidal shape is placed inside

another fluid, see Fig. 5.10. The corresponding surface parametrization is given by

𝑟

𝐼(𝜗) = √︃𝑎2𝑏2(︀1 + tan2(𝜗))︀

𝑎2+𝑏2tan2(𝜗).(5.67)

Due to surface tension, the ellipsoidal droplet “snaps” into a spherical shape. The incompress-

ibility condition enforces volume conservation. Hence, the final radius

𝑅

of the sphere results

from equating the initial volume

𝑉0

3π𝑎𝑏2

with the final volume

𝑉

3π𝑅3

. However, in a

normalized setting with

ℓref

𝑅

, the target radius is given by

˜𝑟

= 1 and the semi-axes are

related by

˜𝑎˜

𝑏2

= 1. In Fig. 5.11 the resulting shapes for different time steps are shown. From the

Section 5.4. Two-phase flow

2𝑏

2𝑎

𝑅

Fig. 5.10:

Two-dimensional schematic depiction of the initial state of the deformed droplet and its final

spherical state.

0π

23π

0.9

0.95

1.05

r/R

Fig. 5.11: The interface of a droplet in the snap test for different time steps.

figure, it can be seen, that the stationary solution represents a sphere of dimensionless radius

one. This shows that the solution obtained by the developed method obeys the prescribed

volume conservation.

Chapter 5. Numerical field calculation with a reduced finite element method

On the electromagnetic coupling problem 67

6 Forces and moments between permanent

magnets

In this chapter, the total electromagnetic forces and moments between permanent magnets

are investigated. To this end, different geometries of the permanent magnets are analyzed,

namely spherical magnets as well as cylindrical ones. In Sec. 6.1, the total force between two

cylindrical magnets, where one is immersed in the ferrofluid, is investigated theoretically and

experimentally

. Subsequently, a theoretical analysis of the moments acting on permanent

magnets in a homogeneous external magnetic fields is performed in Sec. 6.2. Based on the

developed insights, a gedankenexperiment for spherical magnets is used to dismiss a whole class

of electromagnetic coupling models. In addition, the analysis is not restricted to magnets but

is readily extended to electrets. Finally, in Sec. 6.3 the total moment between two cylindrical

magnets is analyzed theoretically and experimentally.

6.1 Global forces on magnets immersed in ferrofluid

From Eq.

(4.20)

it follows that the global electromagnetic force on a body

𝛺−

in a static setting

at rest results from the electromagnetic stress tensor

𝜎(em)

from the exterior domain

𝛺+

. Hence,

the global force predictions of different force models will only differ if the stress tensor expressions

in the exterior domain are different. As is required in Eq.

(3.14)

one has Δ

𝜎+

𝑀+≡0

and

𝑃+≡0

. Subsequently, a global force measurement experiment has to incorporate an

exterior domain 𝛺+with 𝑀+=0.

A readily available substance is ferrofluid, which is a suspension of ferromagnetic nanoparticles,

see [Rosensweig (2013)]. It is magnetizable and exhibits both magnetic relaxation as well as

magnetic saturation. The macroscopic magnetization of the ferrofluid results from the micro-

magnetizations of the nanoparticles. When an external magnetic field is applied, a magnetic

moment acts on the ferromagnetic particles, which in turn rotate to be aligned with the external

field. During their rotation, they have to overcome friction with the carrier fluid. Thus, the

alignment with the external field is not instantaneous, which is why in the static experiment

holding times need to be observed. The relaxation times provided in the literature are usually

of the order of nanoseconds, see [Rosensweig (2013)]. Hence, a conservative holding time of

several seconds is employed.

In order to measure the force acting on a magnet immersed in a ferrofluid, a non-magnetic

assembly is constructed. Both a schematic and a photograph of the setup are given in Fig. 6.1.

The whole assembly is elevated from the table by means of aluminum profiles. The glass

container for the ferrofluid is secured in a 3D-printed mount made from the material polylactide

(PLA). On the bottom of this mount, there is a groove in which the second magnet is placed,

such that its position is centered with respect to the first magnet. In turn, the first magnet is

connected to the load cell by means of a non-magnetic high-grade steel rod that is threaded

on both ends. A brass nut is inserted in a plastic adapter that is subsequently glued to the

first magnet in order to realize a connection to the stainless steel rod. The connection is locked

by another brass nut. The load cell is placed in another 3D-printed mount that ensures the

alignment of the magnets. In the right photograph of Fig. 6.1, the connection of the load cell to

1Margarita Dementeva assisted in the preparation of the experiment as part of her bachelor thesis.

aluminum

plastic

load cell

magnet 1

magnet 2

plastic

stainless

glass

brass nuts

container

mount

proﬁle

steel rod

adapter

Fig. 6.1: Schematic and photograph of the non-magnetic assembly for the ferrofluid experiment.

the mount is not yet fastened such that the alignment is not perfect. Of course, this was fixed

during the experiment.

Recently, a similar experiment was performed in [Yu et al. (2019)]. Therein, the authors

analyzed the magnetic levitation of a permanent magnet immersed in a ferrofluid. However,

they only investigated one particular force model provided in [Rosensweig (2013)]. In contrast

to the experiment in this thesis, only one magnet was present in the investigation in [Yu et al.

(2019)].

6.1.1 Experimental procedure

There are several mechanisms that affect the force reading at the load cell. Besides the weight of

the connection rod and the first magnet itself, the magnet experiences buoyancy. Furthermore,

the ferrofluid sticks to the surface of the glass container and also forms a bubble around the

upper part of the magnet, which in turn influences the pressure distribution inside the ferrofluid.

All of these effects are obstructive to the analysis of the impact of electromagnetic coupling

models. Therefore, one could zero the load cell reading before the second magnet is added such

that the mentioned effects are not included.

Fig. 6.2: Photographs of the first attempt of the ferrofluid assembly.

However, upon the first naive assembly attempt without the second magnet, which is depicted

in Fig. 6.2, an unexpected upwards force was measured. First, the ferrofluid was poured into the

glass container. Subsequently, the magnet was connected to the load cell and inserted into the

glass. The ferrofluid splashed as it was attracted by the magnet. Moreover, the magnet resisted

insertion as the ferrofluid bubble around it adhered to the glass surface. Upon swiftly forcing

it into the container a bubbling sound was audible. As the load cell was secured in place, an

Chapter 6. Forces and moments between permanent magnets

On the electromagnetic coupling problem 69

upward force reading of approximately

8 N

was obtained. A rough calculation suggests that this

value can neither be explained by buoyancy nor by electromagnetic forces. It is most likely, that

during the assembly air bubbles were forced into the glass container. Due to the magnetically

enhanced surface tension between the ferrofluid bubble and the glass container, the air bubbles

are confined at the bottom of the container. Thus, by further compressing the bubbles, the

pressure rises which in turn results in an upward force. Hence, the resulting force measurements

in presence of the second magnet cannot be used for the subsequent analysis.

In order to avoid both the splashing of the ferrofluid as well as the insertion of air bubbles,

the connection rod is shortened and another assembly procedure is followed, see Fig. 6.3. First,

the magnet is inserted into the glass container without a ferrofluid. Then, the load cell is zeroed

such that the combined weight of the magnet and the connection assembly is erased. In view of

the first assembly attempt, the ferrofluid is added by means of a plastic syringe (stage I).

Fig. 6.3: Photographs of the second assembly attempt.

Furthermore, the force measurement is recorded over time, which increases confidence in

the measurement result. The measured force over the duration of the experiment is shown in

Fig. 6.4. Note that the injection stage I took twelve minutes and is depicted on a reduced time

interval.

During stage I, the ferrofluid is gradually inserted in bursts of

20 mL

, which increases the

measured weight. This injection process is performed until the ferrofluid bubbles around the top

and bottom surface of the magnet reach the glass container (stage II). There is an air bubble

beneath the magnet, see Fig. 6.3, which contributes to the upward force reading, see the force

value in Fig. 6.4 for stage II. After three injections of ferrofluid, the upward force reaches a value

of 1.5 N.

Subsequently, in stage III the second magnet is placed beneath the assembly such that both

the ferrofluid as well as the first magnet are pulled downwards. Two more injections of ferrofluid

fill the container completely and a resulting force of

−8.2 N

is measured, where the minus

sign indicates a downward force. In order to ensure that all air bubbles escaped the fluid, the

second magnet is released and put beneath multiple times in stage IV. In doing so, the force

measurement of

−0.65 N

(without the second magnet) is reliably obtained. This value represents

the effects of mechanical buoyancy and the magnetic interaction with the ferrofluid. Finally, in

stage V the second magnet is placed beneath the assembly at different distances using plastic

spacers. The distance between the two magnets is thereby decreased discreetly. The second

magnet is completely removed before the number of spacer plates is reduced. The resulting

force values are depicted in Fig. 6.6.

Section 6.1. Global forces on magnets immersed in ferrofluid

0 . . . 12 14 17 19 24

-10

-8.2

-0.65

1.5

I II III IV V

−7.55 N

time (min)

Fexp

z(N)

Fig. 6.4: Force measurement over the course of the experiment.

6.1.2 Theoretical analysis

The assembly in Fig. 6.1 possesses azimuthal symmetry and can be reduced to a two-dimensional

representation, see Fig. 6.5. Therein, the magnet

𝛺1

is immersed in the ferrofluid domain

𝛺F

, which in turn is located inside the exterior domain

𝛺ext

. Note that both the adapter

and the rod have a relative permeability very close to one, such that they are modeled as

part of the exterior domain without magnetic properties. Furthermore, the second magnet

𝛺2

is located beneath the container and the total computational domain is given by

𝛺

∪

∘

,sym

tot

𝛺

∪

∘

,sym

1∪𝛺

∪

∘

,sym

F∪𝛺

∪

∘

,sym

2∪𝛺

∪

∘

,sym

ext

, where the superscript index indicates the reduced domains

from Fig. 6.5.

2𝐻

ℓtop

𝑑

2𝐻

𝐻F𝛤

∪

∘

,sym

𝛤

∪

∘

,sym

𝛤

∪

∘

,sym

𝛤

∪

∘

,sym

𝑧

𝛺1𝛺F

𝛺ext

𝛺2

𝑧

𝜌

𝑅𝑅F

𝑅

Fig. 6.5:

Global force experiment between permanent magnets. Note that in the cylindrical coordinate

space,

𝛤

∪

∘

,sym

denotes the boundary of the

𝛺

∪

∘

,sym

domain and

𝛤

∪

∘

,sym

represents the internal boundary

of the ferrofluid domain 𝛺

∪

∘

,sym

The goal is to compute the total force acting on the magnet 𝛺1by means of

𝐹tot =𝐹g+𝐹b+𝐹p+𝐹(em) .(6.1)

with the gravitational force, the buoyancy force, the pressure force and the electromagnetic

Chapter 6. Forces and moments between permanent magnets

On the electromagnetic coupling problem 71

force:

𝐹g=−𝑚1𝑔𝑒𝑧,𝐹b=𝜌F𝑉1𝑔𝑒𝑧,𝐹p=ˆ

𝛤

∪

∘

,sym

(−𝑝(em)𝑛) d𝐴 , 𝐹(em) =ˆ

𝛤

∪

∘

,sym

𝑛·𝜎(em)

+d𝐴 . (6.2)

Therein,

𝑚1

is the mass of the magnet,

𝑉1

its volume,

𝑝(em)

is the electromagnetic pressure

inside of the ferrofluid,

𝜌F

its density and

𝑔

9.81 m/s2

. It is important to note that

𝑛

𝑛1

the outward normal to

𝛺1

and that

𝜎(em)

is the electromagnetic stress tensor in the limit from

the exterior, i.e., from

𝛺2

. For the numerical calculation, the force computation by means of the

electromagnetic stress tensor concept is employed in accordance with the findings in Sec. 5.3.

In view of the experimental procedure, it is not the total force from Eq.

(6.1)

that is required

for the comparison to the measured force. Rather, the shifted force

𝐹*

is to be calculated, which

is related to the total force via

𝐹*=𝑒𝑧·(𝐹tot −𝐹g).(6.3)

In order to calculate the electromagnetic force and the pressure force, the magnetic fields

and

𝐵

are computed. Usually, whenever magnetizable matter is present, its magnetization

vector is to be incorporated into Eq.

(5.30)

as a part of the weak formulation. However, this is

not the case for ferrofluid as its magnetic behavior is completely determined by the external

magnetic field

generated by the permanent magnets. For a ferrofluid consisting of spherical

nanoparticles, the magnetization 𝑀obeys the following relaxation law, [Rosensweig (2013)],

d𝑀

d𝑡=𝜔×𝑀+1

𝜏r

(𝑀eq −𝑀),(6.4)

where

𝜏r

is the relaxation time and

𝑀eq

is the equilibrium magnetization. The latter is a

function of the magnetic field strength as well as the temperature,

𝑀eq

(

H0, 𝑇

), and

represents the material response in the steady state condition. The dependence upon the

magnetic field is nonlinear and is usually modeled with the well-known Langevin equation:

‖𝑀eq‖=𝑀s(︂coth(𝛼m)−1

𝛼m)︂, 𝛼m=𝜇0𝑚n‖H0‖

𝑘𝑇 ,(6.5)

Therein,

𝑚n

is the magnetic moment of one nanoparticle,

𝑘

1.38 ×10−23 J/K

Boltzmann

’s

constant,

𝑇

is the absolute temperature and

𝑀s

is the saturation magnetization. However, in

technical applications, the saturation flux density

𝐵s

𝜇0𝑀s

is preferably used and is given

in the unit

. The direction of

𝑀eq

is prescribed by the direction of

. Of course, in a

static setting Eq.

(6.4)

naturally reduces to

𝑀

𝑀eq

and the constitutive equation can be

summarized as follows

𝑀=𝜒(H0)H0, 𝜒(H0) = 𝑀s

‖H0‖(︂coth(𝛼m)−1

𝛼m)︂.(6.6)

In Tab. 6.1 selected properties of the used ferrofluid are provided. Using these values, the

Langevin parameter in Eq. (6.5) is approximated

𝛼m≈22𝜇0‖H0‖m

A.(6.7)

Section 6.1. Global forces on magnets immersed in ferrofluid

Tab. 6.1:

Selected properties of ferrofluid EFH1 according to the datasheet by the supplier [Laborladen

Online Shop (2022)] in the first part and estimated values in the second part. 𝜇b≈9.27 ×10−24 J/Tis

the Bohr magneton.

property symbol value

particle diameter 𝐷n10 nm

saturation flux density 𝐵s44 ±9 mT

density of the mixture 𝜌F1210 kg/m3

particle magnetic moment 𝑚n104𝜇b

The following steps are performed in the simulation in order to obtain the total magnetic flux

density 𝐵:

Calculation of the magnetic field

(

𝑥

)for

𝑥∈𝛺

∪

∘

,sym

without ferrofluid using Eq.

(5.30)

i.e., only with the source magnetizations

𝑀1

𝑀1𝑒𝑧

and

𝑀2

𝑀2𝑒𝑧

of the permanent

magnets.

Using the obtained field

, the magnetization

𝑀F

(

𝑥

)for

𝑥∈𝛺

∪

∘

,sym

is computed

according to Eq. (6.5).

The total magnetic flux density is given by

𝐵

𝜇0

(

𝑀

)with the piece-wise function

𝑀=⎧

⎪

⎨

⎪

⎩

𝑀0𝑒𝑧,𝑥∈(𝛺1∪𝛺2),

𝑀F(𝑥),𝑥∈𝛺F,

0,𝑥∈𝛺ext .

(6.8)

6.1.3 Model dependent electromagnetic force calculation

After all fields

𝑀

, and

𝐵

outside of the magnet

𝛺1

are determined, the resulting electro-

magnetic force can be computed. To this end, the following relations inside the ferrofluid are

used:

𝑀=𝜒H0,𝐵=𝜇0𝜇rH0, 𝜇r= 1 + 𝜒 , (6.9)

where the dependence

𝜒

^𝜒

(

)from Eq.

(6.6)

has to be observed throughout this section.

Upon noting that

∇×H0

and by using the identity in Eq.

(4.29)

, it follows that the balance

of linear momentum in Eq. (4.31) simplifies significantly

𝜇2

0∇𝑝(em) =1

2∇(︁[𝑆1−𝐾1][1 + ^𝜒]2H0·H0)︁+ (1 + ^𝜒)2(H0·[∇𝑆1])H0+

+ (1 + ^𝜒)𝑆1(︁(︀[∇^𝜒]·H0)︀H0−(H0·H0)(∇^𝜒))︁,

(6.10)

Therein, the model-dependent coefficients are gathered in Tab. 4.1. Similarly, the global

electromagnetic force from Eq.

(4.21)

can be expressed in terms of these coefficients. In particular,

upon introducing, e.g.,Δ

𝐾(em)

1−𝐾gL

, the electromagnetic stress tensor difference of

the form as in Eq. (3.30) can be represented via

Δ𝜎=−1

2𝜇2

0Δ𝐾1(1 + ^𝜒)2(H0·H0)1+𝜇2

0Δ𝑆1(1 + ^𝜒)2H0⊗H0,(6.11)

from which the global force difference is readily expressed as

Δ𝐹(em) = 2πˆ

𝛤

∪

∘

,sym

𝑛1·Δ𝜎(em)

+·𝑒𝑧𝜌

∪

∘

dℓ

∪

∘

sym ,(6.12)

Chapter 6. Forces and moments between permanent magnets

On the electromagnetic coupling problem 73

where

𝑛1

is the outward normal to the magnet in domain

𝛺1

and Δ

𝜎(em)

is to be evaluated at

the interface

𝛤

∪

∘

,sym

, but in the limit from the exterior region, i.e., from

𝛺

∪

∘

,sym

ext ∪𝛺

∪

∘

,sym

. From

the values provided in Tab. 4.1, the following coefficient differences are obtained:

(II) Δ𝐾(II)

1:= Δ𝐾A

1= Δ𝐾M

1=1−𝜇r

𝜇0𝜇r

,Δ𝑆(II)

1:= Δ𝑆A

1= Δ𝑆M

1=1−𝜇r

𝜇0𝜇r

(III) Δ𝐾(III)

1:= Δ𝐾EL

1= Δ𝐾P

1=1−𝜇2

𝜇0𝜇r

,Δ𝑆(III)

1:= Δ𝑆EL

1= Δ𝑆P

1=1−𝜇r

𝜇0𝜇r

(IV) Δ𝐾(IV)

1:= Δ𝐾Er

1= Δ𝐾K

1=2(1 −𝜇r)

𝜇0𝜇r

,Δ𝑆(IV)

1:= Δ𝑆Er

1= Δ𝑆K

1=1−𝜇r

𝜇0𝜇r

(6.13)

Due to the nonlinearities involved, the pressure functions are not decomposed with respect

to the

Lorentz

model. Finally, from Eq.

(4.20a)

the global electromagnetic force in vertical

direction acting on

𝛺1

and the electromagnetic pressure equation according to the

Lorentz

force model constitute the model (I). It is given by:

𝐹gL = 2π𝜇0ˆ

𝛤

∪

∘

,sym

(1 + ^𝜒)2[︀(𝑛1·H+

0)(H+

0·𝑒𝑧)−1

2(H+

0·H+

0)(𝑛1·𝑒𝑧)]︀𝜌

∪

∘

dℓ

∪

∘

sym ,

∇𝑝(em)

gL = (1 + ^𝜒)𝜇0(︁(︀[∇^𝜒]·H0)︀H0−(H0·H0)(∇^𝜒))︁,

(6.14)

6.1.4 Experimental results and conclusion

From stage V in Fig. 6.4 the force measurements between the two magnets in presence of the

ferrofluid can be obtained for several discrete distances between the magnets. The resulting

force measurements are depicted in Fig. 6.6. The measurement tolerance is indicated by the

thick blue curve. Furthermore, the force model predictions are shown. It can be seen that

60 65 70 75 80 85

−8

−6

−4

d(mm)

F∗(N)

exp.

(I)

(II)

(III)

(IV)

Fig. 6.6:

Force measurement of the magnet immersed in ferrofluid and the comparison the model

predictions by (I) – generalized

Lorentz

, (II) –

Abraham

Minkowski

, (III) –

Einstein

–

Laub

Pao–Hutter, (IV) – Kovetz,Eringen–Maugin

the generalized

Lorentz

model touches the upper measurement values. The predicted force

values by the other models are barely distinguishable from each other. However, the difference

between the generalized

Lorentz

as well as the experimental data is visible. In particular, for

the closest distance 𝑑= 59 mm between the magnets, the following force values are predicted:

𝐹*

(I) =−8.47 N , 𝐹*

(II) =−7.80 N , 𝐹*

(III) =−7.72 N , 𝐹 *

(IV) =−7.72 N ,(6.15)

which are to be compared to the experimental value

𝐹*

exp.

−

13)

. This shows

that the difference between the force model predictions is measurable. However, the difference

Section 6.1. Global forces on magnets immersed in ferrofluid

is comparatively small due to the low saturation magnetization of the ferrofluid. It can be

estimated, that with stronger ferrofluids a more significant deviation from the experimental

data will be observed.

From the total force measurement in this section, it can be concluded that the generalized

Lorentz

model predicts the experimental data best when compared to the other analyzed

models. However, the error produced by the other force models is not large enough to invalidate

them. That is to say, they can be used as well and result in reasonable accuracy. Hence, this

investigation cannot serve to dismiss any of the analyzed models.

6.2 Moments acting on rigid permanent magnets and electrets

The goal of this section is to investigate the dependence of the total electromagnetic moment

on the force density

𝑓(em)

and the moment density

𝜏(em)

. Therefore, the moments acting on an

arbitrarily shaped magnet and electrets subjected to external fields are analyzed analytically.

In Fig. 6.7a, a permanent magnet with an axial permanent magnetization

𝑀0

is placed in an

external and homogeneous magnetic field H0=H0𝑒𝑧. Similarly, in Fig. 6.7b

𝑀0

𝒪

𝑒𝑥𝑒𝑦

𝑒𝑧

𝛼

𝐵0=𝜇0H0

(a)

𝑃0

𝐸0

𝑒𝑥𝑒𝑦

𝑒𝑧

𝒪

𝛼

(b)

Fig. 6.7: Schematics of magnets and electrets in external electromagnetic fields.

6.2.1 Uniform external fields

Consider the scenario depicted in Fig.6.7a. Therein, a permanent magnet with homogeneous

and axial magnetization

𝑀0

is embedded in an external magnetic field

, which is also

homogeneous:

H0=H0𝑒𝑧,𝑀0=𝑀0(︀cos(𝛼)𝑒𝑧+ sin(𝛼)[︀cos(𝛽)𝑒𝑥+ sin(𝛽)𝑒𝑦]︀)︀.(6.16)

In order to calculate the electromagnetic force as well as the electromagnetic moment act-

ing upon the sphere, the simplified expressions in Eqs.

(4.20a)

–

(4.20b)

solely in terms of the

electromagnetic stress tensor could be used, which reduce in a magnetostatic scenario to

𝐹(em) =˛

𝜕𝛺

𝑛·𝜎(em)

+d𝐴 , 𝑇(em) =−˛

𝜕𝛺

𝑛·𝜎(em)

+×𝑥d𝐴+ˆ

𝛺

(𝜏(em) −⟨3⟩

𝜖··𝜎(em)

+) d𝑉 . (6.17)

However, they require the evaluation of the magnetic fields outside of the magnet, i.e.,

𝐵+

and

H+

, which only depend upon the magnetization vector

𝑀0

implicitly. For the following

analysis, it is convenient to start from the original expressions for the global force and global

moment from Eq.

(4.18)

. For the

Lorentz

force model, the volumetric force density vanishes

inside the sphere as no gradients are present. Furthermore, in the magnetostatic setting, the

Chapter 6. Forces and moments between permanent magnets

On the electromagnetic coupling problem 75

surface force density reduces to

𝑓gL

𝐼= (𝑛×J𝑀K)×⟨𝐵⟩,(6.18)

Hence, the total force and the total moment are given by

𝐹gL =ˆ

𝜕𝛺

(𝑛×J𝑀K)×⟨𝐵⟩d𝐴 , 𝑇gL =ˆ

𝜕𝛺

𝑥𝒪×[︀(𝑛×J𝑀K)×⟨𝐵⟩]︀d𝐴 . (6.19)

At the boundary of the magnet,

𝜕𝛺

, one has

J𝑀K

−𝑀0

. Since the magnet’s magnetic field

is permanent, the total magnetic field is a superposition of the external field,

𝐵ext

𝜇0H0

, and

the field of the magnet, 𝐵M,

𝐵=𝜇0H0+𝐵M.(6.20)

Furthermore, permanent magnets do not accelerate on their own. Therefore, the total force and

the total moment must vanish in the absence of the external field:

𝐹gL(H0=0) = −ˆ

𝜕𝛺

(𝑛×𝑀0)×⟨𝐵M⟩d𝐴!

=0,

𝑇gL(H0=0) = −ˆ

𝜕𝛺

𝑥×[︀(𝑛×𝑀0)×⟨𝐵M⟩]︀d𝐴!

=0.

(6.21)

Hence, for the total quantities, integrals over products containing only “magnet intrinsic”

quantities must vanish. With the continuous external field ⟨𝜇0H0⟩=𝜇0H0one has

𝐹gL =−ˆ

𝜕𝛺

(𝑛×𝑀0)×𝜇0H0d𝐴 , 𝑇gL =−ˆ

𝜕𝛺

𝑥×[︀(𝑛×𝑀0)×𝜇0H0]︀d𝐴 . (6.22)

Since both vectors

𝑀0

and

are constant, they may be extracted from the integrals. Further-

more, since

𝑥×[︀(𝑛×𝑀0)×𝜇0H0]︀=𝜇0H0·(𝑥⊗𝑛)×𝑀0−𝜇0H0⊗𝑀0·(𝑛×𝑥),(6.23)

one has

𝐹gL =(︂𝑀0×ˆ

𝜕𝛺

𝑛d𝐴)︂×𝜇0H0,

𝑇gL =𝜇0H0⊗𝑀0·ˆ

𝜕𝛺

𝑛×𝑥d𝐴−𝜇0H0·ˆ

𝜕𝛺

𝑥⊗𝑛d𝐴×𝑀0.

(6.24)

By means of Gauss’ theorem the following identities are readily proved:

𝜕𝛺

𝑛d𝐴=0,ˆ

𝜕𝛺

𝑛×𝑥𝒪d𝐴=0,ˆ

𝜕𝛺

𝑥𝒪⊗𝑛d𝐴=𝑉1,(6.25)

where

𝑉

is the volume of the domain

𝛺

. Hence, one obtains a vanishing total force and the

following total moment:

𝐹gL =0,𝑇gL =𝑉𝑀0×𝜇0H0.(6.26)

Therein, the product of the volume and the constant magnetization of the magnet is sometimes

referred to as magnetic moment, see [Maugin (1976)]. The resulting electromagnetic moment

is therefore given by the cross-product of the external field and the magnetic moment. This

is in analogy to the mechanical moment formula for a dipole in an external field. However, it

Section 6.2. Moments acting on rigid permanent magnets and electrets

should be emphasized that the correspondence between a point dipole and an extensive magnet

is only possible in the case of a homogeneous external field. The analogy fails if a local field

distribution is considered.

Note that no mechanical constitutive relation was used in the derivation, because of the

assumption of a rigid body. Furthermore, the underlying electromagnetic stress tensor is

symmetric. Hence, the obtained result strongly opposes the general statement made by

Maugin

in [Maugin (2013)]: “[...] the description of stresses. The latter are not symmetric a priori

since there exists an applied couple (12.28), something that cannot be denied as otherwise there

would not exist such an evident effect as the compass alignment with a magnetic field.” Clearly,

this statement is not correct as is demonstrated by the above derivation for an even simpler

geometry than a compass needle.

According to Eq. (6.16) the cross product can be evaluated to read

𝑇gL =𝑉𝑀0×𝜇0H0=𝑉 𝜇0H0𝑀0sin(𝛼)[︀sin(𝛽)𝑒𝑥−cos(𝛽)𝑒𝑦]︀.(6.27)

This result is reasonable if the following is considered. The alignment with the external field,

i.e.,

𝛼

= 0 or

𝛼

, are equilibrium positions. Then, as the moment always points in direction

of 𝑑=[︀sin(𝛽)𝑒𝑥−cos(𝛽)𝑒𝑦]︀,

𝑇gL ·𝑑=−𝑉 𝜇0H0𝑀0sin(𝛼),(6.28)

from which the two equilibrium positions are readily obtained. The alignment with the external

field, i.e.,

𝛼

= 0 is a stable configuration while the antialigned equilibrium solution is unstable.

This result can be compared to the prediction of other coupling models. From the coupling

models analyzed in Sec. 3.4, the models with Δ

𝜎··⟨3⟩

𝜖

or with

𝐶(em) 

result in different

moment predictions when compared to the generalized

Lorentz

model. The relevant quantities

are given by

Δ𝜎A··⟨3⟩

𝜖=𝐵×𝑀,𝜏A=0,

Δ𝜎M··⟨3⟩

𝜖=𝐵×𝑀,𝜏M=0,

Δ𝜎EL ··⟨3⟩

𝜖=𝑀×𝐵,𝜏EL =𝜇0𝑀×H,

Δ𝜎P··⟨3⟩

𝜖=𝑀×𝐵,𝜏Er =𝜇0𝑀×H,

Δ𝜎Er ··⟨3⟩

𝜖=𝑀×𝐵,𝜏Er =𝑀×𝐵,

Δ𝜎K··⟨3⟩

𝜖=𝑀×𝐵,𝜏Er =0.

(6.29)

A comment regarding the moment densities is in order. A classical

Euler

–

Cauchy

continuum

does not support distributed moment densities that do not result from force densities, [Bertram

and Glüge (2015)]. However, it can be seen from Eq.

(6.29)

that the differences resulting from

Einstein

–

Laub

and

Pao

–

Hutter

are the same and it turns out that it is zero, because,

e.g.,

𝜏EL −

𝜎EL ··⟨3⟩

𝜖

. Similarly, for

Eringen

–

Maugin

the difference to the generalized

Lorentz

model vanishes because the moment density cancels the skew-symmetric part of the

stress tensor deviation. However, this is not the case for the model by

Kovetz

. Thus, upon

observing Eq. (4.25) two differences arise:

Δ𝑇A=ˆ

𝛺

𝑀×𝐵d𝑉 , Δ𝑇K, mod =ˆ

𝛺

𝐵×𝑀d𝑉 . (6.30)

The difference due to the model by

Kovetz

is specialized with the superscript “mod” in order

to emphasize, that this is the result of embedding

Kovetz

’ electromagnetic stress tensor in

the current framework. If the expression for the total electromagnetic moment by

Kovetz

directly taken from his textbook, the difference would be zero, Δ

𝑇K

,cf., [Kovetz (2000),

pg. 267]. This was discussed in Sec. 3.4.4. Note that the differences in Eq.

(6.30)

do not vanish,

Chapter 6. Forces and moments between permanent magnets

On the electromagnetic coupling problem 77

because 𝑀is independent of the external field. For the current scenario, it follows that

Δ𝑇A=−Δ𝑇K, mod =𝜇0𝑀0×H0𝑉 . (6.31)

This means that the

Abraham

model predicts twice the amount of moment when compared to

the

Lorentz

model and the modified

Kovetz

model does predict no moment at all. The last

result is of course unreasonable and shows the incompatibility of

Kovetz

’ theory the mechanical

theory developed in Chap. 2. Nonetheless, it will be used in the following sections, because it

could also result from the force model modifications proposed in Sec. 3.3. Furthermore, the idea

that a force model predicts a vanishing electromagnetic moment in the scenario described in

this section, is further investigated in Sec. 6.2.2.

For the sake of completeness, consider now the equivalent electric scenario depicted in Fig. 6.7b,

which has a similar mathematical structure. For an arbitrarily shaped electret with constant

axial polarization 𝑃0in a constant external electric field 𝐸0one has:

𝐸0=𝐸0𝑒𝑧,𝑃0=𝑃0(︀cos(𝛼)𝑒𝑧+ sin(𝛼)[︀cos(𝛽)𝑒𝑥+ sin(𝛽)𝑒𝑦]︀)︀.(6.32)

In this electrostatic setting, the Lorentz force densities reduce to

𝑓gL =0,𝑓gL

𝐼=𝑛·J𝑃K⟨𝐸⟩,(6.33)

where the volumetric force density vanishes. In the considered scenario the jump of the

polarization and the mean electric field are given by

J𝑃K=−𝑃0,⟨𝐸⟩=𝐸0+⟨𝐸E⟩,(6.34)

where

𝐸E

is the electric field produced by the electret. By similar arguments as in the magnetic

case, self-acceleration cannot occur, i.e.,(

𝑛·𝑃0

)

⟨𝐸E⟩

does not contribute to the total force

and a reduced force density is given by 𝑓gL

𝐼=−(𝑛·𝑃0)𝐸0. Hence, the total force vanishes

𝐹gL =−ˆ

𝜕𝛺

(𝑛·𝑃0)𝐸0d𝐴=−𝑃0·ˆ

𝜕𝛺

𝑛d𝐴⊗𝐸0=0,(6.35)

and the total moment is given by

𝑇gL =−ˆ

𝜕𝛺

(𝑛·𝑃0)𝑥×𝐸0d𝐴=−𝑃0·ˆ

𝜕𝛺

𝑛⊗𝑥d𝐴×𝐸0=−4

3π𝑅3𝑃0×𝐸0

=−4

3π𝑅3𝑃0𝐸0sin(𝛼)(︀sin(𝛽)𝑒𝑥−cos(𝛽)𝑒𝑦)︀,

(6.36)

which is in complete analogy to the result in the magnetostatic setting from Eq. (6.27).

6.2.2 A gedankenexperiment

All solutions for the mechanical moment from Sec. 6.2.1 are obtained for arbitrary shapes of

the magnet or the electret, respectively. If the scenario is specialized to spherical geometries, it

becomes possible to derive a simple expression for the moment in terms of the electromagnetic

stress tensor. Consider the scenarios depicted in Fig. 6.7, where a spherical magnet of constant

magnetization

𝑀0

is spatially fixed in an external homogeneous magnetic field

. In the

figure, the electric equivalent, namely a permanent spherical electret in an external field

𝐸0

, is

considered as well. In both scenarios, there are no free charges or free currents present. Consider

Section 6.2. Moments acting on rigid permanent magnets and electrets

an electromagnetic stress tensor of the form similar to Eq. (3.27),

𝜎(em) =𝜂1+

∑︁

𝛼=1

∑︁

𝛽=1

𝜎𝛼𝛽𝑞𝛼⊗𝑞𝛽,𝑞𝛼={𝐵,𝐸,𝑀,𝑃}𝛼.(6.37)

Then, for a spherical permanent magnet with constant magnetization

𝑀0

in a homogeneous

external field

one has

𝐵

𝜇0H0

𝐵M

with the magnetic field

𝐵M

of the magnet itself.

Subsequently, the electromagnetic moment is given by (if any distributed moment densities are

neglected)

𝑇(em) =−ˆ

𝜕𝛺

𝑛·J𝜎11𝐵⊗𝐵+𝜎13𝐵⊗𝑀+𝜎31𝑀⊗𝐵+𝜎33𝑀⊗𝑀K×𝑥d𝐴

=−𝜎11𝜇0H0·ˆ

𝜕𝛺

𝑛⊗J𝐵MK×𝑥d𝐴−

−𝜎11 ˆ

𝜕𝛺

𝑛·J𝐵M⊗𝐵MK×𝑥d𝐴+ (𝜎31 −𝜎13)𝜇0𝑉H0×𝑀0−

−𝜎13 ˆ

𝜕𝛺

(𝑛·𝐵+

M)𝑥×𝑀0d𝐴−𝜎31 ˆ

𝜕𝛺

(𝑛·𝑀0)𝑥×𝐵−

Md𝐴 .

(6.38)

As before, the integrals not containing H0must vanish. Therefore, one has

𝑇(em) =−𝜎11𝜇0H0·ˆ

𝜕𝛺

𝑛⊗J𝐵MK×𝑥d𝐴+ (𝜎31 −𝜎13)𝜇0𝑉H0×𝑀0.(6.39)

For a spherical permanent magnet of radius

𝑅

the first integral vanishes and the following

expression remains

𝑇(em) = (𝜎31 −𝜎13)𝜇0

3π𝑅3H0×𝑀0.(6.40)

Since this moment shall only vanish if

and

𝑀0

are aligned, the following condition must

hold

𝜎31 =𝜎13 .(6.41)

This severs as a general restriction upon force models. However, another conclusion is that

𝜏(em)

cannot be zero if

𝜎31

𝜎13

. In any case, the resulting argument can be used in order to

sanity-check a given model and can be applied to more models than just the ones analyzed in

this thesis. A similar analysis can be performed for an electret with constant polarization

𝑃0

in a homogeneous electric field

𝐸0

. However, the resulting expression is not relevant for the

subsequent analysis.

Chapter 6. Forces and moments between permanent magnets

On the electromagnetic coupling problem 79

6.3 Moments between cylindrical permanent magnets

In [Reich, Rickert, and Müller (2017)] the total force between two coaxial cylindrical permanent

magnets was calculated using the

Lorentz

force model. In such a scenario all force models

yield the same total force as is demonstrated in Sec. 4.2. The total moments, however, may vary

between different force models, which is the reason this experiment is considered theoretically

and experimentally in this section. If the two cylindrical permanent magnets from the last section

are rotated against each other, i.e., they are not coaxial anymore, they exert an electromagnetic

moment on each other. The measurement

of this moment and the subsequent comparison with

the theoretical prediction of different electromagnetic coupling models is the aim of this section.

The measurement of moments is more complicated than the measurement of forces if essentially

the same equipment as in Sec. 6.1 is to be used: A tension testing machine paired with a load

cell. The basic idea is to use a turntable that is connected to the force-measuring apparatus via

a connecting rod, such that a resulting moment is converted into an axial force. Similar to the

last section a non-magnetic assembly is required, which is depicted schematically in Fig. 6.8.

The whole assembly is mounted on a wooden board, in order to be easily removed from the

Fig. 6.8: Schematic of the turntable assembly.

tensile testing machine. By means of aluminum profiles, the mount for the turntable is elevated

from the board. Both, the second magnet as well as the turntable with its shaft are located on

the mount. The first magnet is fixed on the turntable disk by means of a plastic bearing that

was 3D printed. The turntable is connected by means of a lever to another connection rod that

is mounted inside a needle bearing. Finally, the connection rod is inserted into the load cell

that is mounted onto the tensile testing machine.

Fig. 6.9: Photographs of the turntable assembly.

Almost all parts of the assembly are made of aluminum, indicated by the gray color, except

the magnet retainers as well as the ceramic ball bearings. The assembly is constructed such

2Mario Phuc Anh Vu assisted in the preparation of the experiment as part of his master thesis.

Section 6.3. Moments between cylindrical permanent magnets

that both magnets’ axes are in the same plane with the midpoints of the magnets. The specific

dimensions of the parts are chosen in accordance with the considerations in Sec. 6.3.2.

In Fig. 6.9 photographs of different views of the assembly are shown. Note that in the first

image, a 3d printed connection rod made of plastic is shown, which is replaced with a stiffer

variant in the subsequent pictures. Therein, the connection rod is made of aluminum and the

pins are implemented by means of brass screws.

6.3.1 Theoretical electromagnetic moment computation

Consider the free body diagram in Fig. 6.10. Therein, the two initially coaxial cylindrical

permanent magnets are depicted, where the magnet

𝛺′

with the midpoint

𝒪′

is rotated against

the stationary magnet

𝛺

𝒪

by an angle of

𝛼

. As a result, both magnets experience an equal

but opposite moment

𝑇(em)

. Both magnets are assumed to be permanently magnetized in the

axial direction and rigid. The goal is to calculate the moment

𝑇(em)

as a function of the relative

𝑧′

𝐹(em)

𝑧

𝑇(em)

𝑥

𝐹(em)

𝑦

𝐹machine

𝒪′

𝛿𝑢

Δ𝜓

𝒪′

𝜓

ℓ

𝛼

𝛼0

𝑑

𝑦0

𝑥0

𝒪

𝑧0

𝑅t

ℓ

ℓ+𝑅t−𝑢

𝜓

𝐷

𝑢

Fig. 6.10: Drawing of the turntable setup (right) and an incomplete free body diagram (left).

rotation angle

𝛼

for different electromagnetic force models. In particular, due to the planar

setup, only a moment

𝑇(em)

𝑇(em) ·𝑒𝑥

is expected. According to

Newton

’s third law, the

resulting moments that the magnets exert on each other are equal and opposite. Using the

generalized

Lorentz

force model from Eq.

(3.11c)

one obtains the following expression for the

Chapter 6. Forces and moments between permanent magnets

On the electromagnetic coupling problem 81

total force and the total moment acting on the rotated magnet:

𝐹gL =ˆ

𝜕𝛺′(︀𝑛×J𝑀′K)︀×𝐵d𝐴′,𝑇gL =ˆ

𝜕𝛺′

𝑥′×[︀(︀𝑛×J𝑀′K)︀×𝐵]︀d𝐴′.(6.42)

Therein, the magnetic field can be decomposed via

𝐵

𝐵0

𝐵′

. However, since the magnet

cannot accelerate on its own, all integrals containing 𝐵′must vanish and only 𝐵0contributes.

Since this setup has no azimuthal symmetry, the numerical method from Sec. 6.1 is not feasible

for the magnetic field calculation. Instead, the analytical solution for the magnetic field from

Eq.

(5.33)

is used. The boundary

𝜕𝛺

of the rotated cylinder may be decomposed into the top

surface, the mantle and the bottom surface:

𝜕𝛺′

𝛤t∪𝛤m∪𝛤b

. Due to the corresponding

surface normals and

J𝑀′K

−𝑀′

0𝑒′

𝑧

the integrals over the top and bottom surface vanish,

resulting in

𝑇gL =ˆ

𝛤m

(𝑅𝑒′

𝜉+𝑧′𝑒′

𝑧)×[︀(︀𝑒′

𝜉×J𝑀′K)︀×𝐵0]︀d𝐴′

=𝑀′

0𝑅3

2π

𝜙=0

𝐻

𝑧=−˜

𝐻(︁[𝑒′

𝜉·𝐵0] + ˜𝑧[𝑒′

𝑧·𝐵0])︁𝑒′

𝜙d˜𝑧d𝜙′,

(6.43)

where

˜𝑧

𝑧′/𝑅

𝐻

𝐻/𝑅

and

𝑒𝜉

is the radial basis vector in the cylindrical eigensystem

{𝜉′, 𝜙′, 𝑧′}

of the rotated magnet. Furthermore, due to the planar setup only the dimensionless

moment in 𝑥-direction is of interest, where 𝑒′

𝜙·𝑒0

𝑥=−sin(𝜙′):

𝑇gL

𝑥=𝑇gL ·𝑒0

𝑥

𝜇0𝑀0𝑀′

0𝑅3=−

2π

𝜙=0

𝐻

𝑧=−˜

𝐻

[𝑒′

𝜉·˜

𝐵0+ ˜𝑧𝑒′

𝑧·˜

𝐵0] sin(𝜙′) d˜𝑧d𝜙′.(6.44)

The magnetic field solution is given in [Reich, Stahn, and Müller (2016)] and was discussed

in Sec. 5.2.2. It is naturally represented in the unrotated system, i.e.,

𝐵0

= (

𝜇0𝑀0

)

−1𝐵0

𝐵0

𝜉

(

𝜉0, 𝜙0, 𝑧0

)

𝑒0

𝜉

𝐵0

𝑧

(

𝜉0, 𝜙0, 𝑧0

)

𝑒0

𝑧

. Hence, both the coordinate transformations as well as the

base vector transformations are required, e.g.,

𝜉0

(

𝜉′, 𝜙′, 𝑧′

)as the integration is performed

with respect to the dashed coordinates. In App. C.3 the required coordinate transformations

are presented. The integral in Eq.

(6.44)

is integrated numerically using

Mathematica

, see

[Wolfram Research, Inc. (2016)], resulting in the graphs displayed in Fig. 6.11. Note that for

this calculation it was assumed that both magnets are identical.

5π1

2π4

5ππ

0.1

0.2

0.3

0.4

𝛼

|˜

𝜏L·𝑒𝑥|

𝑑

/𝐻

2.5

3.5

4.5

Fig. 6.11:

Predicted dimensionless moment according to the

Lorentz

force model plotted against the

rotation angle 𝛼, for different end to end distances 𝑑=𝐷−𝐻−𝐻′, see Fig. 6.10.

Section 6.3. Moments between cylindrical permanent magnets

The relevant moment differences with respect to the

Lorentz

prediction can be obtained

from Eq. (6.30). It follows that two different variations must be considered

Δ𝑇±=±ˆ

𝛺−

𝑀×𝐵d𝑉(6.45)

By means of the same arguments as before, only the 𝑥′-component is relevant to the analysis,

𝑒′

𝑥·Δ𝑇±=±ˆ

𝛺−

𝑒′

𝑥·(𝑀′×𝐵0) d𝑉=∓𝑀′

𝐻′

𝑧′=−𝐻′

2π

𝜙′=0

𝑅′

𝜌′=0

𝑒′

𝑦·𝐵0𝜌′d𝜌′d𝜙′d𝑧′(6.46)

and thus since 𝑒′

𝑥=𝑒0

𝑥

Δ˜

𝑇±

𝑥=Δ𝑇±·𝑒0

𝑥

𝜇0𝑀0𝑀′

0𝑅3=∓

𝐻′

˜𝑧′=−˜

𝐻′

2π

𝜙′=0

˜𝜌′=0

𝑒′

𝑦·˜

𝐵0˜𝜌′d˜𝜌′d𝜙′d˜𝑧′.(6.47)

6.3.2 Force to moment conversion

The theoretical moment prediction is readily obtained as a function of the rotation angle

𝛼

However, as the measuring device is given by a load cell paired with a tension testing machine,

the moment

𝑇(em)

(

𝛼

)is not measured directly. Instead, the force

𝐹machine

(

𝑢

)is measured as a

function of the connecting rod’s displacement. This force function is converted into

𝑇(em)

(

𝛼

)by

means of the free body diagram depicted in Fig. 6.10, where it is assumed that all parts of the

assembly are rigid.

The kinematical relation between the rotation angle of the table and the machine displacement

is obtained by evaluating the cosine theorem for the triangle drawn in Fig. 6.10 for

𝜓∈

(

−π,π

𝜓= arccos (︂2(𝑅t+ℓ)(𝑅t−𝑢) + 𝑢2

2𝑅t(𝑅t+ℓ−𝑢))︂,

𝑢=ℓ+𝑅t[1 −cos(𝜓)] −√︁ℓ2−𝑅2

tsin2

(𝜓).

(6.48)

or in dimensionless form

𝜓(˜𝑢) = arccos (︂2(1 + 𝜅)(1 −˜𝑢) + ˜𝑢2

2(1 + 𝜅−˜𝑢))︂,˜𝑢=𝑢

𝑅t

^𝑢(𝜓) = 𝜅+ 1 −cos(𝜓)−√︁𝜅2−sin2(𝜓), 𝜅 =ℓ

𝑅t

(6.49)

The maximal displacement is given by twice the table radius, i.e.,

˜𝑢max

= 2, which is reached

for

𝜓

±π

. However, due to the magnet in the middle of the turntable only

𝜓max ≈π

can be

achieved, which corresponds to

˜𝑢max

= 1 +

𝜅−√𝜅2−1

. In order to convert the force

𝐹machine

to the moment

𝑇(em)

𝑥

the principle of virtual work is used. An additional virtual displacement

𝛿𝑢 to a configuration with the displacement ˜𝑢results in the following angle difference

Δ𝜓=^

𝜓(˜𝑢+𝛿𝑢)−^

𝜓(˜𝑢)≈1

𝑅t

𝜓

d˜𝑢𝛿𝑢 (6.50)

for small

𝛿𝑢

. The virtual work resulting from the machine force and the electromagnetic moment

Chapter 6. Forces and moments between permanent magnets

On the electromagnetic coupling problem 83

must vanish for a static setting such that

𝐹machine𝛿𝑢 −𝑇(em)

𝑥Δ𝜓!

= 0 ⇒𝐹machine =𝑇(em)

𝑥

𝑅t

𝑓(˜𝑢),(6.51)

with the transmission function

𝑓(˜𝑢) = d^

𝜓

d˜𝑢=2(1 + 𝜅)(𝜅−˜𝑢) + ˜𝑢2

|1 + 𝜅−˜𝑢|√︀˜𝑢(2 + 2𝜅−˜𝑢)(˜𝑢−2)(˜𝑢−2𝜅).(6.52)

From this expression, it follows that the effective lever arm decreases with decreasing table

radius

𝑅t

and increasing connection rod length

ℓ

. Bearing in mind that the measured machine

force should be as large as possible, this leads to two opposing requirements: The lever arm

needs to be small and the electromagnetic moment needs to be large. However, a larger moment

is only achieved by means of a larger magnet, which restricts the smallest possible lever arm.

Since there is only a limited amount of sizes of cylindrical magnets available, two exemplary

magnets are compared. A list of selected properties of these magnets is compiled in Tab. 6.2.

Tab. 6.2:

Selected properties of two different magnets from [Bar magnet STM-20x34-N (2021)] and [Bar

magnet STM-30x50-N (2021)], respectively. The contact forces are specified by the data sheets. The

remanence value marked * star is given as

(1.40–1.46) T

in the datasheet, see App. A for the measurement

procedure.

property magnet type 1 magnet type 2

total axial length 2𝐻34.00 mm 50.00 mm

diameter 2𝑅20.00 mm 30.00 mm

geometrical tolerance ±0.10 mm ±0.10 mm

remanence 𝜇0𝑀0(1.32–1.37) T 1.3 T*

contact holding force 186.31 N 637.36 N

From Fig. 6.11 it can be seen that the distance between the magnets is a crucial parameter

for the development of the moment as a function of the rotation angle. For the closest distance

plotted, the maximal magnitude occurs for

𝛼

. Hence, the force

𝐹machine

arising at the

machine side is analyzed for this rotation angle. In Fig.6.12 the forces for both magnets are

5 10 15 20 25 30 35

d(mm)

|Fmachine|(N)

magnet 1

magnet 2

Fig. 6.12:

Absolute value of the predicted force on the machine side according to Eq.

(6.51)

for the two

different magnets from Tab. 6.2 in a configuration with

𝛼

plotted against the end to end distance

𝑑=𝐷−𝐻0−𝐻′, see Fig. 6.10.

plotted against their respective end-to-end distances of the magnets

𝑑

𝐷−𝐻0−𝐻′

. Of course,

this distance cannot be decreased to zero for every angle

𝛼

as the magnets would intersect.

The minimal distance is given by

𝑑min

𝑅′sin

(

𝛼

). Hence, the end-to-end distance

𝑑

has the

conservative lower bound 𝑑≥𝑅′for each respective magnet type.

Section 6.3. Moments between cylindrical permanent magnets

6.3.3 Measurement results and comparison to the theoretical prediction

The turntable assembly can only rotate in the range of

𝜓∈

[

−π

2,π

]. In order to obtain the

results for the rotation angle range

𝛼∈

,π

], the measurement is performed in two steps. In

the first step, the magnets are initially coaxial and a turntable rotation up to

𝛼

𝜓

performed. For the second step, the magnet is removed from the turntable and the turntable

assembly is put back into its initial state. Subsequently, the magnet is mounted again onto the

turntable but rotated about the angle

𝛼0

, see Fig. 6.10. Therefore, upon rotating the table until

𝜓

is reached, the measured moment correspond to the angle

𝛼

𝛼0

𝜓

and thus to the

range

𝛼∈

[

2,π

]. However, since the transmission function

𝑓

(

˜𝑢

)from Eq.

(6.52)

is singular for

˜𝑢→

0or equivalently

𝜓→

0, the measurement result at this angle becomes void. This is no

problem in the first step, as the moment vanishes anyway for coaxial magnets. However, the

uncertainty for

𝜓

= 0 affects the second step of the experiment. Therefore, the second angle

interval is split into two parts, i.e.,

𝛼0∈ {−π

4,π

. The resulting measurements are subsequently

assembled into one data set.

The measurement results as well as the theoretical predictions of various electromagnetic

coupling models are depicted in Fig. 6.13. From the figure, it can be seen that the

Lorentz

4π1

2π3

4ππ

−1

−0.5

τ(em) ·ex(N m)

exp.

(I)

(II)

(III)

Fig. 6.13:

Comparison of the theoretical moment predictions with the experimental data. (I) – generalized

Lorentz and all others that are not (II) – Abraham,Minkowski or (III) – modified Kovetz.

model agrees with the experimental data. The agreement is not perfect, but that may be due to

measurement errors, that may be amplified by the transmission function

𝑓

(

˜𝑢

). Most importantly,

the other force models show significantly different behavior. Not only do the numerical values of

the predicted moment differ from the experimental data, but not even the qualitative behavior is

captured correctly. For example, the

Minkowski

model and the

Abraham

model even predict

positive moments in a scenario where the magnet’s magnetizations are perpendicular. Thus,

both models are identified as not-applicable in general.

Chapter 6. Forces and moments between permanent magnets

On the electromagnetic coupling problem 85

7 The oil drop experiment

In contrast to the solid mechanics investigations from the last chapters, the analysis of fluid

flow or the deformation of fluid interfaces has the advantage of visible local effects. This comes

at the price of more complex mathematical problems. In particular, the fluid-fluid interaction,

which involves the effect of surface tension, requires almost always a numerical analysis. A

three-dimensional fluid flow analysis is computationally cost-intensive on its own. Additionally,

when a fluid-fluid interface is to be resolved, a high-resolution discretization is required, which

further increases the computational cost. The interface results in another problem for the

numerical analysis because it needs to be discretized as well. In Sec. 5.4 a numerical method is

developed that enables a high interface resolution while keeping the computational costs at a

minimum.

In the following, the developed method is extended by both the electric field computation as

well as the electromagnetic force computation such that the oil-drop experiment by [Torza, Cox,

and Mason (1971)] can be analyzed numerically. To this end, the coupled electromechanical

system is derived and subsequently solved for different electromagnetic coupling models. The

results are compared to the experimental data given in [Torza, Cox, and Mason (1971)].

It should be noted that this experiment was also analyzed in [Datsyuk and Pavlyniuk (2015)].

Therein, the electromagnetic force distribution for a spherical droplet is analyzed for several

force models. However, the authors refrained from calculating the resulting displacement and

thus they could only vaguely compare their results to the experimental data provided in [Torza,

Cox, and Mason (1971)]. In [Reich, Rickert, and Müller (2017)], this analysis was further

advanced.

Reich

et al. employed the concept of surface elasticity, which enabled them to derive

an analytical solution of the droplet’s surface displacement for small deformations. However,

the mechanical forces involved are the viscous shear forces during the deformation as well as the

surface tension force, none of which are elastic. Therefore, an investigation that is closer to the

realistic conditions is performed.

7.1 Governing equations and boundary conditions

Consider the scenario in depicted in Fig. 7.1, which is similar to Sec. 5.4. Therein a droplet of

silicone oil,

𝛺−

, is submerged in oxidized castor oil,

𝛺+

. The container’s top and bottom plates

are connected to a voltage source. Therefore, an external electric field is applied, to which

the oils react by means of polarization. Since both oils have different relative permittivities, a

surface force arises at the interface

𝐼

, causing it to deform. Hence, the interface evolves over

time, 𝐼=^

𝐼(𝑡).

Both fluids are considered to be incompressible, i.e., their mass densities are constant. As the

fluids do not mix, there exists surface tension between them. Furthermore, they are considered to

be polarizable and a linear relation of the form

𝑃

𝜖0

(

𝜖r−

𝐸

𝜖0𝜖r𝐸

is employed. Note

that both fluids have constant mass densities, relative permittivities and shear viscosities, but

the corresponding values can still be different for both fluids such that the material parameters

are piece-wise constant functions:

𝜓𝛼=^

𝜓𝛼(𝑥s, 𝑡) = {︃𝜓+

𝛼,𝑥s∈𝛺+(𝑡),

𝜓−

𝛼,𝑥s∈𝛺−(𝑡),(7.1)

𝐼

𝑏

electric field 𝐸0

𝑅

𝑎

initial state

𝑥

𝑦

𝑧

𝛺+

𝛺−

Fig. 7.1:

Schematic depiction of an oil-droplet submerged in another oil. The radius of the initial sphere

is not to scale.

where

𝜓𝛼∈ {𝜌s, 𝜇, 𝜖r}

. In order to simulate the temporal evolution of the interface, the balance

of linear momentum is solved alongside the mass balance according to the methods presented

in Sec. 5.4. To this end, the domain in Fig. 7.1 is transformed, resulting in the shape shown

in Fig. 7.2. It must be noted that in order for the symmetry condition to be applicable, the

walls of the container must be modified. However, this has virtually no impact on the droplet’s

deformation, since the droplet is much smaller than the container. In the real experiment,

𝑅/𝐻 ≈

4 % was used. In contrast to the equations presented in Sec. 5.4, the electromagnetic

𝑛∘

𝜏∘

𝑛∘

𝜏∘

R3Ssym

𝑛

𝜏

𝑥

𝑦

𝑧

azimuthal symmetry spherical coordinates

𝑦

𝑧

𝑟

𝜗

2π

4π

𝜗

𝑟

𝛤3

𝛤1

𝛤2

𝛤∘

𝛺

𝛤∘

𝒪

ℓ∘

𝒪ℓ∘

𝑧+

ℓ∘

𝑧−

𝛺∘

sym

𝛺sym

Fig. 7.2:

Schematic depiction of the container for the oil droplet experiment and its transformation in

spherical coordinates. The dimensions are not to scale.

influence due to the plate condenser must be considered. Therefore, the electromagnetic force

has to be implemented into the system of equations, which results in a weak form that is

essentially given by Eq.

(4.38)

. Furthermore, the electric field solution is required for the

electromagnetic force computation. Thus, the system of equations must be supplemented by

Maxwell’s equations.

Without any magnetic effects, i.e.,

𝐵≡0

𝑀≡0

and

H≡0

, as well as without any free

charges or free currents it follows that:

∇s×𝐸=0,−𝜕D

𝜕𝑡 = 0 ,∇s·D= 0 ,𝑥s∈(𝛺+∪𝛺−),

𝑛·D= 0 ,𝑛×𝐸=0,𝑥s∈𝐼 .

(7.2)

Since the electric field is curl-free, it can be represented by means of a scalar potential,

𝐸

−∇s𝑉

Chapter 7. The oil drop experiment

On the electromagnetic coupling problem 87

It is interesting to note that the conditions of electrostatics are enforced by the absence of both,

free charges and free currents as well as magnetic effects. Thus, the potential does only depend

upon the position,

𝑉

(

𝑥s

). Of course, since the permittivity is a function of time, i.e.,

𝜖r

(

𝑥s, 𝑡

), the electric potential will also evolve over time, but this time dependence is implicit.

Since the droplet is small compared to the container, the boundary conditions of the electric

field result solely from the plate condenser. Hence, the electric potential is to be prescribed

at the top and bottom plates. Without a dielectric, the corresponding potential difference

leads to a nearly homogeneous electric field in between the plates. Therefore, a shifted field

𝐸

𝐸0−∇s𝑉

is proposed, where

𝐸0

𝐸ref𝑒𝑧

and

𝐸ref

is obtained from the applied voltage

difference. For this newly introduced potential, vanishing

Dirichlet

boundary conditions are

used.

By means of the constitutive relation for the polarization, it follows that

𝜖0𝜖r

(

𝐸0−∇s𝑉

Therefore, the electric problem can be rewritten:

∇s·(𝜖r𝐸0−𝜖r∇s𝑉)=0,𝑥s∈(𝛺+∪𝛺−),

𝑛·J𝜖r𝐸0−𝜖r∇s𝑉K= 0 ,𝑛×J∇s𝑉K=0,𝑥s∈𝐼 . (7.3)

Note that according to Eq.

(7.1)

the relative permittivity is discontinuous at the interface and

hence the gradient of the electric potential is not continuous in the normal direction of the

interface as well. While the transition condition in the normal direction is later weakly imposed

by means of the weak formulation of the system, the tangential continuity of the electric field

results from the continuity of the scalar potential, see [Reich (2017)],

J𝑉K= 0 ⇒𝑛×J∇s𝑉K=0.(7.4)

The coupled system of equations is to be normalized appropriately before any numerical

analysis can be performed. The following normalization is used:

𝑥=ℓref ˜

𝑥,∇s=ℓ−1

ref ˜

∇, 𝑡 =𝑡ref ˜

𝑡 , 𝑉 =𝑉ref ˜

𝑉 , 𝑓=𝑔0˜

𝑓

𝑣=𝑣ref ˜

𝑣,𝜎=𝜎ref ˜

𝜎, 𝑝 =𝑝ref ˜𝑝 , 𝜎𝐼=𝜎𝐼

ref ˜

𝜎𝐼,

𝜎(em) =𝜎(em)

ref ˜

𝜎(em) , 𝜌s=𝜌ref ˜𝜌 , 𝜇 =𝜇ref ˜𝜇 .

(7.5)

Therein,

𝑔0≈9.81 m/s2

is the gravitational acceleration and quantities with a tilde are dimen-

sionless as the reference quantities contain both the unit as well as the order of magnitude of

the fields. Of course, not all of these reference values need to be independent. With the special

choices

𝑡ref =ℓref

𝑣ref

, 𝜎ref =𝑝ref =𝜎(em)

ref =𝜌ref𝑣2

ref , 𝜎𝐼

ref =𝜎

𝐼, 𝑉ref =𝐸refℓref ,(7.6)

the following dimensionless numbers arise:

Re :=𝜌ref 𝑣refℓref

𝜇ref

,We :=𝜌ref𝑣2

refℓref

𝜎

𝐼

,Weel :=𝜌ref 𝑣2

ref

𝜖0𝐸2

ref

,Fr :=𝑣2

ref

𝑔0ℓref

.(7.7)

Therein,

is the

Reynolds

number,

is the

Weber

number which represents the influence

of the surface tension coefficient

𝜎

𝐼

Weel

is the electrical

Weber

number and

is the

Froude

number. For the analyzed problem, the values for the reference quantities are given in Tab. 7.1.

The mass densities of both oils are identical, which eliminates the effect of buoyancy.

Note that there is no natural choice for the reference velocity

𝑣ref

since the flow is not

displacement controlled, i.e., there is no non-zero

Dirichlet

boundary condition for the

velocity. Instead, in order to emphasize the most relevant effect of surface tension, the

Weber

Section 7.1. Governing equations and boundary conditions

Tab. 7.1:

Reference parameters for the oil drop experiment as performed by [Torza, Cox, and Mason

(1971)].

name symbol value in 𝛺+(castor oil) value in 𝛺−(silicone oil)

mass density 𝜌9.8×102kg/m39.8×102kg/m3

shear viscosity 𝜇6.5 Pa s 5.39 Pa s

relative permittivity 𝜖r7.3 3.772

number is put to one, which results in

We !

= 1 ⇒𝑣ref =√︂𝜎

𝐼

𝜌refℓref

.(7.8)

From the employed values for the system parameters extracted from [Torza, Cox, and Mason

(1971)],

ℓref =𝑅= 6 ×10−4m, 𝐸ref = 4.95 ×105V

m, 𝜎

𝐼= 5.5×10−3N

m,(7.9)

the values for the dimensionless numbers in Tab. 7.2 result. The boundary-value problem is an

Tab. 7.2: Dimensionless numbers for the two-phase flow problem

name symbol value

Reynolds number Re 1.06 ×10−2

Froude number Fr 1.56

Weber number We 1

electric Weber number Weel 4.226

extended version of the one presented in Sec. 5.4 and is given by

⎧

⎪

⎨

⎪

⎩

˜𝜌𝜕˜

𝑣

𝜕˜

𝑡+ ˜𝜌˜

𝑣·(˜

∇⊗ ˜

𝑣) = ˜

∇·(˜

𝜎+˜

𝜎(em)) + 1

Fr ˜𝜌˜

𝑔,˜

𝑥∈(𝛺−∪𝛺+), (7.10a)

∇· ˜

𝑣= 0 ,˜

𝑥∈(𝛺−∪𝛺+), (7.10b)

∇·(𝜖r𝑒𝑧−𝜖r˜

∇˜

𝑉)=0,˜

𝑥∈(𝛺−∪𝛺+), (7.10c)

𝑛·J˜

𝜎+˜

𝜎(em)K=1

We (˜

∇·𝑛)𝑛,˜

𝑥∈𝐼, (7.10d)

𝑛·J𝜖r𝑒𝑧−𝜖r˜

∇˜

𝑉K= 0 ,˜

𝑥∈𝐼, (7.10e)

J˜

𝑣K=0,J˜

𝑉K= 0 ,˜

𝑥∈𝐼, (7.10f)

which has to be supplemented by the initial conditions

𝑣(˜

𝑥,˜

𝑡= 0) = 0,˜𝑝(˜

𝑥,˜

𝑡= 0) = 0 ,˜

𝑉(˜

𝑥,˜

𝑡= 0) = 0 ,(7.11)

as well as the constitutive relations

𝜎=−˜𝑝1+˜𝜇

Re (˜

𝑣⊗˜

∇+˜

∇⊗ ˜

𝑣),˜

𝑃= (1 −𝜖r)˜

𝐸.(7.12)

The boundary conditions at the walls of the container are omitted in Eq.

(7.10)

. Of course,

𝑣

is employed at each wall and homogeneous

Dirichlet

boundary conditions are applied

Chapter 7. The oil drop experiment

On the electromagnetic coupling problem 89

at the top and bottom plate. Thus, the weak of the system in spherical coordinates is given by:

0 = ˆ

𝛺∘

sym

˜𝜌𝜕˜

𝑣∘

𝜕˜

𝑡·𝛿𝑣∘˜𝑟2sin2(𝜗) d𝐴∘

s+ˆ

𝛺∘

sym

˜𝜌𝛿𝑣·(˜

𝑣⊗˜

∇sph)·˜

𝑣˜𝑟sin(𝜗) d𝐴∘

+ˆ

𝛺∘

sym

˜𝑟sin(𝜗)(˜

𝜎+˜

𝜎(em))··(˜

∇sph ⊗𝛿𝑣∘+ cos(𝜗)𝑒∘

𝜗⊗𝛿𝑣∘) d𝐴∘

s+(7.13a)

We ˆ

𝐼∘

sym

𝑡∘

𝐼dℓ∘

s−1

Fr ˆ

𝛺∘

sym

˜𝑟2sin2(𝜗)˜𝜌˜

𝑓·𝛿𝑣∘d𝐴∘

0 = ˆ

ℓ∘

sym∖𝐼∘

sym

sin(𝜗)𝛿𝑉 ∘(˜𝑟𝑛∘

𝑟𝑒∘

𝑟+𝑛∘

𝜗𝑒∘

𝜗)·(𝜖r˜

∇sph ˜

𝑉) dℓ∘

s−(7.13b)

−ˆ

𝛺∘

sym

𝜖r(˜

∇sph𝛿𝑉 ∘+ cos(𝜗)𝛿𝑉 ∘𝑒∘

𝜗)·(˜

∇sph ˜

𝑉) d𝐴∘

0 = ˆ

𝛺∘

sym

˜𝑟sin(𝜗)𝛿𝑝∘(˜

∇sph ·˜

𝑣) d𝐴∘

s,(7.13c)

see Sec.

(5.4)

for details of the used transformations. Note that in the analyzed scenario, the

electromagnetic momentum vanishes for all models,

𝑔(em) ≡0

. Thus, only the electromagnetic

stress tensor

𝜎(em)

is contained in the system. Finally, a force model has to be specified. In view

of the normalization introduced in Eq.

(7.5)

, together with the dimensionless numbers from

Eq.

(7.7)

, the normalized electromagnetic stress tensor is obtained from its original version via

𝜎(em) =1

Weel𝜖0𝐸2

ref

𝜎(em) .(7.14)

From the analyzed models in Chap. 3, only three different electromagnetic stress tensors arise:

𝜎(em)

(M1) :=˜

𝜎gL ,˜

𝜎(em)

(M2) :=˜

𝜎A=˜

𝜎M,

𝜎(em)

(M3) :=˜

𝜎K=˜

𝜎Er =˜

𝜎EL =˜

𝜎P,(7.15)

which are given by the following expressions with subtle differences:

𝜎(em)

(M1) =1

Weel [︀˜

𝐸⊗˜

𝐸−1

2(˜

𝐸·˜

𝐸)1]︀,(7.16a)

𝜎(em)

(M2) =𝜖r

Weel [︀˜

𝐸⊗˜

𝐸−1

2(˜

𝐸·˜

𝐸)1]︀,(7.16b)

𝜎(em)

(M3) =1

Weel [︀𝜖r˜

𝐸⊗˜

𝐸−1

2(˜

𝐸·˜

𝐸)1]︀.(7.16c)

Section 7.1. Governing equations and boundary conditions

7.2 Electromagnetic force distribution

The electric field at the interface is decomposed into its normal and tangential parts via

𝐸=˜

𝐸n𝑛+˜

𝐸t𝜏.(7.17)

Upon introducing

𝑓𝑚

𝐼:

𝑛·J˜

𝜎(em)

𝑚K

, the electromagnetic surface forces in Eq.

(7.16)

reduce to,

respectively

𝑓(M1)

𝐼·𝑛=1

Weel J1

2˜

𝐸2

nK,˜

𝑓(M1)

𝐼·𝜏=1

Weel

𝐸tJ˜

𝐸nK,(7.18a)

𝑓(M2)

𝐼·𝑛=1

Weel J1

2𝜖r˜

𝐸2

n−1

2𝜖r˜

𝐸2

tK,˜

𝑓(M2)

𝐼·𝜏≡0,(7.18b)

𝑓(M3)

𝐼·𝑛=1

Weel J(𝜖r−1

2)˜

𝐸2

nK,˜

𝑓(M3)

𝐼·𝜏≡0.(7.18c)

where

J˜

𝐸tK

= 0 and

J𝜖r˜

𝐸nK

= 0 were used. The surface forces in Eqs.

(7.18a)

and

(7.18c)

were

analyzed in [Reich, Rickert, and Müller (2017)] for a spherical droplet. However, the authors did

not consider the fact, that the electromagnetic pressure for the model (M3) (and for the second

model as well) is non-zero. Hence, their force computation was incomplete and the resulting

deformation solution was not comparable to the experiment. In view of the pressure analysis in

Sec. 4.3, the electromagnetic parts of the pressure solutions are given by

˜𝑝(em)

(M1) = ˜𝑝(em)

(M2) = 0 ,˜𝑝(em)

(M3) =1

Weel

𝜖r−1

2˜

𝐸·˜

𝐸.(7.19)

Using the analytical solution for the electric field of the spherical droplet from, e.g., [Reich,

Rickert, and Müller (2017)], and the material parameters from Tabs. 7.1–7.2, the initial effective

electromagnetic forces can be evaluated from

𝑓𝐼,eff =˜

𝑓(em)

𝐼−˜𝑝(em)𝑛.(7.20)

It turns out, that the electromagnetic pressure from the third model cancels the difference to

the second one, cf., Eq.

(7.24)

. This can be seen in Fig. 7.3, where the force distributions for the

three models are depicted. In particular, the equal force distributions in Fig. 7.3b and 7.3c are

qualitatively different from the results shown in

[Reich, Rickert, and Müller (2017), Fig.13b]. It

(a)

𝑓(M1)

𝐼,eff (b)

𝑓(M2)

𝐼,eff (c)

𝑓(M3)

𝐼,eff

Fig. 7.3:

Effective electromagnetic surface force for the three different coupling model types analyzed.

For each plot, the arrows are scaled differently, such that the maximum lengths are all equal.

1Note that the plots therein are rotated about 90 degrees.

Chapter 7. The oil drop experiment

On the electromagnetic coupling problem 91

can be seen that for both, (M2) and (M3), the pressure is positive everywhere and its maximum

is located at the equator of the droplet. Due to the incompressibility constraint, it can already be

deduced that the models (M2) and (M3) will result in a prolate shape of the droplet, which does

not agree with the experimental results in [Torza, Cox, and Mason (1971)]. Thus, these force

models appear to be not admissible. Note that a similar analysis was performed in [Datsyuk

and Pavlyniuk (2015)]. However, the authors refrained from the calculation of the steady state

deformation. In order to make the argument more convincing, the stationary deformation of

the droplet is calculated and compared to the experimental results.

From Eq.

(7.18)

it can be seen that at least for the models (M2) and (M3), the tangential

surface force is zero. This means, that for these models a stationary droplet at rest is admissible,

i.e.,

𝑣≡0

. In contrast, the tangential part of the surface force is not zero for the generalized

Lorentz

model (M1). Thus, it follows from Eq.

(7.22b)

that this tangential force must be

counteracted by viscous force due to velocity gradients. As a result, the generalized

Lorentz

model does not admit a stationary droplet that is at rest. Before the numerical solutions are

presented, some analytical investigations are in order.

7.3 Steady-state deformation and its semi-analytical solution

Due to the external electric field from the plate condenser, the initially spherical droplet of

radius

𝑅

deforms into an oblate ellipsoidal shape, cf., [Torza, Cox, and Mason (1971), Fig. 7].

Since the applied electric field is constant, a stationary state is reached at some point and the

resulting shape of the interface is the subject of this section. In order to obtain this result

from the simulation, the numerical time integration is performed until the temporal change of

the velocity falls below a certain threshold and the interface does not change over time. The

conditions for this stationary state are given by

𝜕˜

𝑣

𝜕˜

𝑡=0,𝑛·˜

𝑣(𝑥𝐼) = 0,(7.21)

for all

𝑥𝐼∈𝐼

. The second condition ensures that the interface itself does not change its shape.

However, in the vicinity of the interface tangential motion is still possible, i.e.,𝜏·˜

𝑣(𝑥𝐼)=0.

The jump condition in Eq. (7.10d) reduces to

𝑛·J˜

𝜎(em)K·𝑛=1

We (˜

∇·𝑛) + J˜𝑝K−2

Re 𝑛·q˜𝜇(˜

∇⊗ ˜

𝑣)y·𝑛,(7.22a)

𝑛·J˜

𝜎(em)K·𝜏=1

Re 𝑛·q˜𝜇(˜

𝑣⊗˜

∇+˜

∇⊗ ˜

𝑣)y·𝜏.(7.22b)

Only for the force models (M2) and (M3), a resting droplet with

𝑣≡0

is an admissible

stationary solution because the electromagnetic shear traction is zero. As a result, the mechanical

stress tensor is given by, cf., Sec. 4.3

𝜎=−𝑝1, 𝑝 = ˜𝑝(em) −1

Fr ˜𝜌˜𝑧+𝐶 . (7.23)

Hence, upon noting that for the given materials we have

J˜𝜌K≈

0, the balance of linear momentum

in Eq. (7.10d) reduces to the same equation for both models,

(M2)*:1

2J𝜖r˜

𝐸2

n−𝜖r˜

𝐸2

tK=Weel

We (˜

∇·𝑛) + WeelJ𝐶K.(7.24)

where (M2)

refers to both, (M2) and (M3) as they cannot be distinguished regarding their

Section 7.3. Steady-state deformation and its semi-analytical solution

resulting droplet shape. Therein, the following relations were used:

𝑓(M2)

𝐼·𝜏=˜

𝑓(M3)

𝐼·𝜏= 0 ,J(1 −𝜖r)˜

𝐸2

tK=−J𝜖r˜

𝐸2

tK.(7.25)

Both, the experimental photographs from [Torza, Cox, and Mason (1971)] and numerical tests

indicate an azimuthally symmetric oblate spheroid shape of the interface

𝐼

, with the half axes

𝑎

and

𝑏

. This notion is also supported by the initial force distributions in Fig. 7.3. Therefore,

an analytical solution for the electric field is derived in spheroidal coordinates. The h solution

depends upon the half axes

𝑎

and

𝑏

. The hope is that they can be adjusted such that the

momentum balance in Eq. (7.24) is satisfied.

The problem with spheroidal coordinates is that two versions exist, the oblate and the prolate

ones, which are not directly compatible. Furthermore, two different boundary value problems

arise for an oblate spheroid and a prolate spheroid, because the

𝑧

-axis represents the direction

of the external excitation. Hence, the electric problem must be solved twice. To this end, it

is assumed that the castor oil domain

𝛺+

extends until infinity, i.e., the container walls are

neglected. As a result, the electric field for both spheroids is calculated by means of solving the

following boundary value problem for the shifted electric potential ˜

𝑉of the stray field:

⎧

⎪

⎨

⎪

⎩

Δ˜

𝑉= 0 ,𝑥∈(𝛺−∪𝛺+),

sup{˜

𝑉}<∞,𝑥∈(𝛺−∪𝛺+),

𝑛·J𝜖r˜

∇˜

𝑉K=J𝜖rK𝑛·𝑒𝑧,J˜

𝑉K= 0 ,𝑥∈𝐼 ,

lim‖𝑥‖→∞ ˜

𝑉+= 0 ,‖𝑥‖→∞.

(7.26)

where

is the

Laplace

operator. Note that in order to find an analytical solution, the

constraints of regularity and attenuation have to be added to the boundary value problem.

Subsequently, the half axes

𝑎

and

𝑏

are determined such that the remaining equations in

(7.10)

are satisfied. Note that two different types of spheroids are possible, namely prolate and oblate

ones, which are investigated in the subsequent sections.

7.3.1 The electric field of oblate and prolate spheroids

In this section, the

Laplace

equation for the electric potential

𝑉

is solved in both, oblate and

prolate dimensionless spheroidal coordinates, respectively,

(𝜇o, 𝜈o, 𝜙o)∈[0,∞)×[−π

2,π

2]×(−π,π),

(𝜇p, 𝜈p, 𝜙o)∈[0,∞]×[0,π]×[0,2π].(7.27)

For a detailed description of these coordinates, see [Arfken and Weber (1999)]. Both of these

spheroidal coordinates form orthogonal systems that are obtained from the more general

ellipsoidal coordinates by setting two semi-axes equal. A spheroid with the half axes

{𝑎, 𝑎, 𝑏}

the

Cartesian coordinates

{𝑥, 𝑦, 𝑧}

is said to be oblate if

𝑎>𝑏

and prolate if

𝑏>𝑎

. The corresponding

coordinates share the property that coordinate lines of constant

𝜇o

and

𝜇p

correspond to ellipses.

However, lines of constant

𝜈o

form one-sheet half hyperboloids, whereas lines of constant

𝜈p

correspond to two-sheet hyperboloids. Due to their similarities, the derivation of the solution

for the electric field is only shown for the oblate spheroidal coordinates. Subsequently, the key

difference in the derivation of the electric field solution in prolate spheroidal coordinates is

discussed and the resulting solution is shown.

Spheroids are inherently symmetrical with respect to the azimuthal angle

𝜙

, which is known

from cylindrical coordinates. Therefore, it is sufficient to analyze one cross-section, e.g., for

Chapter 7. The oil drop experiment

On the electromagnetic coupling problem 93

𝜙= 0. The oblate spheroid coordinates are related to the cylindrical coordinates via

˜𝜌= ^𝜌(𝜇o, 𝜈o) = 𝐹ocosh(𝜇o) cos(𝜈o),˜𝑧= ^𝑧(𝜇o, 𝜈o) = 𝐹osinh(𝜇o) sin(𝜈o),(7.28)

where

𝐹o

is a constant. Therefore, the associated orthonormal base vectors are readily obtained

in terms of the cylindrical base vectors:

𝑒𝜇o=1

√︁sinh2(𝜇o) + sin2(𝜈o)(︁sinh(𝜇o) cos(𝜈o)𝑒𝜌+ cosh(𝜇o) sin(𝜈o)𝑒𝑧)︁,

𝑒𝜈o=1

√︁sinh2(𝜇o) + sin2(𝜈o)(︁−cosh(𝜇o) sin(𝜈o)𝑒𝜌+ sinh(𝜇o) cos(𝜈o)𝑒𝑧)︁.

(7.29)

Thus, an oblate spheroid with major axis

𝑎

𝑅0˜𝑎

and minor axis

𝑏

𝑅0˜

𝑏

is characterized by

𝜇𝐼

o,𝐹oand its normal vector via

𝐹o=√︁˜𝑎2−˜

𝑏2, 𝜇𝐼

o=1

2ln (︂˜𝑎+˜

𝑏

˜𝑎−˜

𝑏)︂,𝑛=𝑒𝜇o,˜𝑎 > ˜

𝑏 . (7.30)

The solution to the scalar

Laplace

equation in this coordinates is given by the spheroidal

harmonics, which are a generalization of the spherical harmonics found from the solution of

the

Laplace

equation in spherical coordinates. However, the spheroidal harmonics are usually

not expressed in terms of the spheroidal coordinates

{𝜇o, 𝜈o, 𝜙}

themselves, but rather

{𝜁, 𝜉, 𝜙}

which are defined as:

𝜁= sinh(𝜇o), 𝜉 = sin(𝜈o)⇒𝜁∈[0,∞), 𝜉 ∈[−1,1) .(7.31)

Note that the interface is thus identified with

𝜁𝐼= sinh(𝜇𝐼

o) = ˜

𝑏

√︀˜𝑎−˜

𝑏√︀˜𝑎+˜

𝑏.(7.32)

In particular, the general spheroidal harmonics reduce drastically in complexity, if azimuthal

symmetry is assumed. Hence, the

Laplace

equation reduces to, see [Arfken and Weber (1999)]

Δ˜

𝑉o=1

𝐹2

o(𝜁2+𝜉2)[︂𝜕

𝜕𝜁 (︂(1 + 𝜁2)𝜕˜

𝑉o

𝜕𝜁 )︂+𝜕

𝜕𝜉 (︂(1 −𝜉2)𝜕˜

𝑉o

𝜕𝜉 )︂]︂ !

= 0 .(7.33)

Upon assuming an ansatz of the form

𝑉o

(

𝜁, 𝜉

) =

𝑍

(

𝜁

)

𝑋

(

𝜉

), two ordinary

Legendre

differential

equations arise. Their solutions are given by the linear combinations of the

Legendre

poly-

nomials of the first and second kind,

𝑃𝑛

and

𝑄𝑛

, respectively. The general solution in both

respective subdomains thus reads

𝑉±

o=∞

∑︁

𝑛=0 [︀𝐴±

𝑛𝑃𝑛(i𝜁) + 𝐵±

𝑛𝑄𝑛(i𝜁)]︀[︀𝐶±

𝑛𝑃𝑛(𝜉) + 𝐷±

𝑛𝑄𝑛(𝜉)]︀,(7.34)

where two parameters could be factored out, such that three types of parameters per value of

𝑛

for each subdomain remain, which results in six types of parameters that must be determined.

However, the boundary value problem in Eq.

(7.26)

does only contain five conditions, which allow

for the determination of five of the six types of coefficients. Therefore, additional constraints

must be imposed on the potential

𝑉o

. To this end, the following normalization condition is used

𝑉−

o(𝜁= 0, 𝜉)!

= 0 ∀𝜉∈[−1,1) .(7.35)

Section 7.3. Steady-state deformation and its semi-analytical solution

In Eq.

(7.34)

, the

Legendre

polynomials of the second kind become singular at

𝜉

−

Hence, regularity requires all coefficients

𝐷±

𝑛

to vanish and the potentials are conveniently

expressed as

𝑉±

o=∞

∑︁

𝑛=0 [︀𝐴±

𝑛𝑃𝑛(i𝜁) + 𝐵±

𝑛𝑄𝑛(i𝜁)]︀𝑃𝑛(𝜉).(7.36)

Note that all

Legendre

polynomials are orthogonal and therefore linear independent. As a

result, any condition for the potential itself or its derivative, that must be true for all

𝜉∈

[

−

naturally results in infinitely many restrictions on the coefficients. For example, it follows from

the attenuation condition

lim

𝜁→∞

𝑉+

o=∞

∑︁

𝑛=0

𝑃𝑛(𝜉) lim

𝜁→∞[𝐴+

𝑛𝑃𝑛(i𝜁) + 𝐵+

𝑛𝑄𝑛(i𝜁)] !

= 0 ,(7.37)

that all coefficients must attenuate separately, which determines the relation between

𝐴+

𝑛

and

𝐴−

𝑛. It can be shown that

lim

𝜁→∞[𝐴+

𝑛𝑃𝑛(i𝜁) + 𝐵+

𝑛𝑄𝑛(i𝜁)] !

= 0 ∀𝑛⇒𝐵+

𝑛=2

πi𝐴+

𝑛.(7.38)

Similarly, the normalization condition in Eq.

(7.35)

can be used to obtain a relation between

𝐴−

𝑛and 𝐴+

𝑛, because it follows from the orthogonality of the Legendre polynomials that

𝐴−

𝑛𝑃𝑛(0) + 𝐵−

𝑛𝑄𝑛(0) !

= 0 ∀𝑛 . (7.39)

Noting that the polynomials of the first and second kind vanish at zero for odd and even indices,

respectively,

𝑃2𝑛+1(0) = 0 , 𝑃2𝑛(0) = 0 , 𝑄2𝑛(0) = 0 , 𝑄2𝑛+1(0) = 0 ,(7.40)

it immediately follows that

𝐴−

2𝑛

= 0 and

𝐵−

2𝑛+1

= 0, which means that they appear mutually

exclusive in the solution for the potential. Finally, the transition conditions for the gradient of

the potential,

∇˜

𝑉o=1

𝐹o√︃1 + 𝜁2

𝜁2+𝜉2

𝜕˜

𝑉o

𝜕𝜁 𝑒𝜇o+1

𝐹o√︃1−𝜉2

𝜁2+𝜉2

𝜕˜

𝑉o

𝜕𝜉 𝑒𝜈o.(7.41)

result in

0 = s𝜕˜

𝑉o

𝜕𝜉 {, 𝐹o𝜉J𝜖rK=s𝜖r

𝜕˜

𝑉o

𝜕𝜁 {,(7.42)

which after inserting the functions for the potential for both domains yields

0 = ∞

∑︁

𝑛=0 [︂𝐴+

𝑛{︀𝑃𝑛(i𝜁𝐼) + 2

πi𝑄𝑛(i𝜁𝐼)}︀−𝐴−

𝑛𝑃𝑛(i𝜁𝐼)−𝐵−

𝑛𝑄𝑛(i𝜁𝐼)]︂𝜕𝑃𝑛(𝜉)

𝜕𝜉 ,

0 = 𝐹oJ𝜖rK𝑃1(𝜉) + ∞

∑︁

𝑛=0 [︂(1 + 𝑛)

(1 + 𝜁2

𝐼)]︂{︂𝜖+

r𝐴+

𝑛(︂𝜁𝐼𝑃𝑛(i𝜁𝐼)+i𝑃𝑛+1(i𝜁𝐼) + 2i

π𝜁𝐼𝑄𝑛(i𝜁𝐼)−2

π𝑄𝑛+1(i𝜁𝐼))︂

+𝜖−

r𝐵−

𝑛𝜁𝐼𝑄𝑛(i𝜁)+i𝜖−

r𝐵−

𝑛𝑄𝑛+1(i𝜁)−𝜖−

r𝐴−

𝑛𝜁𝐼𝑃𝑛(i𝜁𝐼)−i𝜖−

r𝐴−

𝑛𝑃𝑛+1(i𝜁𝐼)}︂𝑃𝑛(𝜉).

Note that not only the

Legendre

polynomials but also their derivatives are orthogonal. Hence,

by using

𝐴−

2𝑛

= 0 and

𝐵−

2𝑛+1

= 0 it follows that most of the coefficients vanish. In fact, after

Chapter 7. The oil drop experiment

On the electromagnetic coupling problem 95

some algebra, the solutions for the potentials reduce to

𝑉+=𝐾+

o𝐹o[︀1−π

2𝜁+𝜁arctan(𝜁)]︀𝜉 ,

𝑉−=𝐾−

o𝐹o𝜁𝜉 , (7.43)

with the parameters

𝐾+

o=𝜁𝐼𝐾o, 𝐾−

o= [1 −π

2𝜁𝐼+𝜁𝐼arctan(𝜁𝐼)]𝐾o,

𝐾o=(1 + 𝜁2

𝐼)

(1 + 𝜁2

𝐼)[1 −π

2𝜁𝐼+𝜁𝐼arctan(𝜁𝐼)] −𝜖+

J𝜖rK

.(7.44)

Note that in the limiting case of a spherical droplet, with

˜𝑎→˜

𝑏

, the parameters reduce to, e.g.,

lim

˜𝑎→˜

𝑏

𝑉2=−J𝜖rK

J𝜖rK−3𝜖+

,(7.45)

which represents exactly the solution for the electric potential of a sphere, see [Kovetz (2000)].

The electric field itself is obtained as:

𝐸+

o=𝑒𝑧−𝜁𝐼𝐾o

√︀𝜁2+𝜉2(︂√︁1 + 𝜁2𝜉[︂𝜁

1 + 𝜁2−π

2+ arctan(𝜁)]︂𝑒𝜇o

+√︁1−𝜉2[︂1−π

2𝜁+𝜁arctan(𝜁)]︂𝑒𝜈o)︂

𝐸−

o=𝑒𝑧−𝐾o[1 −π

2𝜁𝐼+𝜁𝐼arctan(𝜁𝐼)]

√︀𝜁2+𝜉2(︂√︁1 + 𝜁2𝜉𝑒𝜇o+√︁1−𝜉2𝜁𝑒𝜈o)︂,

(7.46)

where 𝑒𝑧may be represented in terms of the spheroidal base vectors via

𝑒𝑧=√︀𝜁2+𝜉2

(1 + 𝜁2)𝜉2+𝜁2(1 −𝜉2)(︁𝜉√︁1 + 𝜁2𝑒𝜇o+𝜁√︁1−𝜉2𝑒𝜈o)︁.(7.47)

For visualization purposes it is sometimes preferable to express the field vectors in terms of

cylindrical coordinates:

𝐸+

o=𝑒𝑧−𝜁𝐼𝐾o(︂−𝜁𝜉

(𝜁2+𝜉2)√︃1−𝜉2

1 + 𝜁2𝑒𝜌+[︂𝜁

(𝜁2+𝜉2)−π

2+ arctan(𝜁)]︂𝑒𝑧)︂,

𝐸−

o=(︂1−(1 + 𝜁2

𝐼)[1 −π

2𝜁𝐼+𝜁𝐼arctan(𝜁𝐼)]

(1 + 𝜁2

𝐼)[1 −π

2𝜁𝐼+𝜁𝐼arctan(𝜁𝐼)] −𝜖+

J𝜖rK)︂𝑒𝑧,

(7.48)

with the inverse coordinate transformation expressed in terms of 𝐹o=√︀˜𝑎+˜

𝑏√︀˜𝑎−˜

𝑏,

𝜁=1

√2𝐹o√︂˜𝜌2+ ˜𝑧2−𝐹2

o+√︁4𝐹2

o˜𝑧2+ (˜𝜌2+ ˜𝑧2−𝐹2

o)2,

𝜉=√2 ˜𝑧

√︁˜𝜌2+ ˜𝑧2−𝐹2

o+√︀4𝐹2

o˜𝑧2+ (˜𝜌2+ ˜𝑧2−𝐹2

o)2

(7.49)

From the representation in Eq.

(7.48)2

it can be seen that the electric field inside the spheroid is

constant. In Fig. 7.4 the electric field is depicted.

The electric field of a prolate spheroid is depicted in Fig. 7.4b. In order to obtain the

corresponding solution, prolate spheroid coordinates are used, which are related to the cylindrical

Section 7.3. Steady-state deformation and its semi-analytical solution

(a) oblate (b) prolate

Fig. 7.4:

Stream plots of the electric stray field,

𝐸−𝑒𝑧

−˜

∇˜

𝑉

, of the cross sections of an oblate and a

prolate spheroid. Both plots use the same color scaling. The oblate spheroid’s two major axes ar given

˜𝑎

and the minor axis

𝑏

/˜𝑎2

. For the prolate one, the major axis is also

˜𝑎

, but the minor axis

is put to ˜

𝑏=1

/√˜𝑎, such that both spheroids have the same volume.

coordinates via

˜𝜌=𝐹psinh(𝜇p) sin(𝜈p) = 𝐹p√︀𝜎2−1√︀1−𝜏2,

˜𝑧=𝐹pcosh(𝜇p) cos(𝜈p) = 𝐹p𝜎𝜏 , (7.50)

where the modified coordinates

𝜎

cosh

(

𝜇p

)and

𝜏

cos

(

𝜈p

)are introduced. Upon comparing

these expressions with the ones in Eq.

(7.28)

it becomes clear that both versions of the spheroidal

coordinates are related via

𝜈o=𝜈p−π

2, 𝜇o=𝜇p+ iπ

2, 𝐹o=−i𝐹p,(7.51)

which results in

𝜁

= i

𝜎

and

𝜉

−𝜏

. However, after this transformation is applied, the

𝑧

-axis is

flipped such that

𝑧→ −𝑧

and thus

𝑒𝑧→ −𝑒𝑧

must be replaced manually. Hence, the general

solution for the electric potential of a prolate spheroid is obtained by means of applying the

transformations in Eq.

(7.51)

to the oblate solution in Eq.

(7.44)

. However, the coefficients

𝐾±

cannot be obtained from this transformation, because the boundary value problem for a prolate

spheroid is different. Therefore, the transmission conditions have to be evaluated separately.

After some algebraic simplification this results in

𝑉+

p=𝐹p𝐾+

p(︁2−𝜎ln (︀𝜎+1

𝜎−1)︀)︁𝜏 , ˜

𝑉−

p=𝐹p𝐾−

p𝜎𝜏 (7.52)

with the definitions

𝐾+

p=𝜎𝐼𝐾p, 𝐾−

p=(︁2−𝜎𝐼ln(︁𝜎𝐼+ 1

𝜎𝐼−1)︁)︁𝐾p,

𝐾p=𝜎𝐼(𝜎2

𝐼−1)

2(︀𝜎2

𝐼+𝜖−

J𝜖rK)︀−𝜎𝐼(𝜎2

𝐼−1) ln(𝜎𝐼+1

𝜎𝐼−1),(7.53)

Chapter 7. The oil drop experiment

On the electromagnetic coupling problem 97

and the interface description of a prolate spheroid:

𝜎𝐼=˜

𝑏

√︀˜

𝑏2−˜𝑎2, 𝐹p=√︁˜

𝑏2−˜𝑎2,𝑛=𝑒𝜇p.(7.54)

The electric field is thus given by:

𝐸−

p= (1 −𝐾−

p)𝑒𝑧=1−𝐾−

√𝜎2−𝜏2(︀𝜏√︀𝜎2−1𝑒𝜇p−𝜎√︀1−𝜏2𝑒𝜈p)︀,

𝐸+

p=−2𝐾+

𝜏√1−𝜏2

(𝜎2−𝜏2)√𝜎2−1𝑒𝜌+(︂1−2𝐾+

p𝜎

𝜎2−𝜏2+𝐾+

pln(︁𝜎+ 1

𝜎−1)︁)︂𝑒𝑧

√︀(𝜎−𝜏)(𝜎+𝜏)

𝜏

√𝜎2−1(︂𝐾+

p(𝜎2−1) ln(︁𝜎+ 1

𝜎−1)︁+𝜎2−2𝐾+

p𝜎−1)︂𝑒𝜇p

−𝐾+

√1−𝜏2

√︀(𝜎−𝜏)(𝜎+𝜏)(︂𝜎ln(︁𝜎+ 1

𝜎−1)︁−2 + 𝜎)︂𝑒𝜈p.

(7.55)

Therein, the contained prolate quantities are connected to the cylindrical basis via

𝑒𝜇p=1

√𝜎2−𝜏2(︀𝜎√︀1−𝜏2𝑒𝜌+𝜏√︀𝜎2−1𝑒𝑧)︀,

𝑒𝜈p=1

√𝜎2−𝜏2(︀𝜏√︀𝜎2−1𝑒𝜌−𝜎√︀1−𝜏2𝑒𝑧)︀,

𝜎=1

√2𝐹p√︂˜𝜌2+ ˜𝑧2+𝐹2

p+√︁(𝐹2

p−˜𝑧2)2+ 2(𝐹2

p+ ˜𝑧2)˜𝜌2+ ˜𝜌4,

𝜏=√2 ˜𝑧

√︂˜𝜌2+ ˜𝑧2+𝐹2

p+√︁(𝐹2

p−˜𝑧2)2+ 2(𝐹2

p+ ˜𝑧2)˜𝜌2+ ˜𝜌4

(7.56)

7.3.2 Surface force equilibrium

In the steady state, the electromagnetic forces are in equilibrium with the surface tension force

and the pressure difference. The surface tension force is obtained from the curvature. By

definition, the surface normal of a prolate spheroid is given by

𝑛

𝑒𝜇p

and thus its divergence

evaluated at the interface reads

∇·𝑛=𝜎𝐼(2𝜎𝐼−𝜏2−1)

𝐹p√𝜎2−1√︀(𝜎2

𝐼−𝜏2)3, 𝜎𝐼=˜

𝑏

√︀˜

𝑏2−˜𝑎2.(7.57)

Bearing in mind that the normal and tangential components of the electric field solutions in

Eq.

(7.55)

are given by the

𝜇p

and

𝜈p

components, respectively, the linear momentum balance

in (7.24) reduces, for both models respectively, to:

(M2)*:1

2J˜

𝐸2

n+ (1 −𝜖r)˜

𝐸2

tK=Weel

𝜎𝐼(2𝜎𝐼−𝜏2−1)

𝐹p√𝜎2−1√︀(𝜎2

𝐼−𝜏2)3+WeelJ𝐶K,(7.58)

with the auxiliary parameters

𝛼0=𝐾+

p(𝜎2

𝐼−1) ln(︁𝜎𝐼+ 1

𝜎𝐼−1)︁+𝜎2

𝐼−2𝐾+

p𝜎𝐼−1,

𝛼1=𝜖+

𝛼2

𝜎2

𝐼−1−𝜖−

r(1 −𝐾−

p)2(𝜎2

𝐼−1) + J𝜖rK(1 −𝐾−

p)2𝜎2

𝐼,

(7.59)

Section 7.3. Steady-state deformation and its semi-analytical solution

Unfortunately, neither of the Eqs.

(7.58)

has an exact solution for all values of

𝜏∈

[

−

1]. This

means that both models, (M2) and (M3), do not result in perfectly spheroidal shapes, but

something different. However, the equations can be solved approximately, e.g., by means of a

series expansion in terms of

𝜏

around

𝜏

= 0. While respecting the volume conservation imposed

by the incompressibility constraint, ˜

𝑏= ˜𝑎−2, the approximate solution for the half axes is:

(M2)*: ˜𝑎= 0.977 ,˜

𝑏= 1.047 ,J𝐶K=−2.5,(7.60)

where the material parameters from Tab. 7.1–7.2 were used. In Fig. 7.5 the left-hand side and the

right-hand side of Eq.

(7.58)

are plotted, i.e., the effective electromagnetic forces are compared

to the curvature forces. This shows, that solutions in Eq.

(7.60)

only produce marginal errors.

However, upon performing the same analysis for the oblate case

, the electric field solution

from Eq.

(7.46)

must be used. It is interesting to note that the electromagnetic surface force

prediction does not change qualitatively when compared to the prolate case. However, the

curvature force does change its behavior, because the normal vector is given by

𝑛

𝑒𝜇o

and

the curvature reads

∇·𝑛=𝜁𝐼(1 + 2𝜁2

𝐼+𝜉2)

𝐹o√︀(1 + 𝜁2

𝐼)(𝜁2

𝐼+𝜉2)3.(7.61)

For example, at the equator with 𝜗=π

2, the curvature force increases with increasing ˜𝑎in the

oblate case. Since the electromagnetic force solution cannot this curvature force, a prolate shape

results.

0π

23π

1.9

2.1

2.2

prolate

0π

23π

1.7

1.8

1.9

oblate

curvature force 1

We ˜

∇ · neffective electromagnetic force for (M2)∗

Fig. 7.5:

Effective electromagnetic forces and curvature forces for the force models in (M2)

for a prolate

and an oblate spheroid, respectively, plotted against the polar angle. In the prolate case the parameter

solution from Eq. (7.60) is used and in the oblate case ˜𝑎= 1.005 was employed.

2The resulting surface equation is similar to Eq. (7.58).

Chapter 7. The oil drop experiment

On the electromagnetic coupling problem 99

Another result worth noting is the jump of the pressure constant. Without any electromagnetic

force, its value is given by

J𝐶K

−

2, which represents the dimensionless pressure difference

due to the curvature of a sphere. The addition to that value in Eq.

(7.60)

arises from the

electromagnetic forces acting on the interface. However, the whole pressure difference is given

by Eq. (7.23) and is a highly nonlinear function of the azimuthal angle.

7.4 Comparison to experimental data and discussion

In order to compare the numerical results to the experimental results from [Torza, Cox, and

Mason (1971), Fig. 7], the photographs of the deformed droplet could be digitized. The two-

dimensional photographs do not show the interface directly as the curved surface bends the

light. Thus, the obtained values for

˜𝑎

and

𝑏

are only approximations. However,

Torza

et al.

do provide some numerical values representing the deformation by means of the parameter

𝐷:

= (

𝑏−˜𝑎

)

(

𝑏

˜𝑎

). Upon invoking the volume conservation constraint, the dimensionless

half-axes are connected via ˜

𝑏= ˜𝑎−2and it follows that

𝐷=1−˜𝑎3

1 + ˜𝑎3⇒𝑎=3

√︃1−𝐷

1 + 𝐷,(7.62)

which in turn can be verified by comparing the resulting ellipses to the photographs.

From the numerical simulation as well as the experimental data, see [Torza, Cox, and Mason

(1971), Tab. 2, class C16], the following parameters can be extracted:

(M1) : ˜𝑎= 1.03 ,˜

𝑏= 0.943 , 𝐷 =−0.0441 ,(7.63)

(M2)*: ˜𝑎= 0.972 ,˜

𝑏= 1.06 , 𝐷 = 0.0433 ,(7.64)

(EXP) : ˜𝑎= 1.034 ±0.04 ,˜

𝑏= 0.933 ±0.006 , 𝐷 =−0.0515 ±0.005 ,(7.65)

From these values it can be seen, that the prediction by the generalized

Lorentz

model (M1)

does agree well with the experimental data. All the other analyzed models (M2)

yield a prolate

shape of the droplet, which does not agree with experimental observations. This was rigorously

proven by means of the semi-analytical calculation resulting in Eq.

(7.60)

as well as through

numerical calculations. It can be noted that the numerical predictions for (M2)

in Eq.

(7.64)

do in fact closely resemble the semi-analytical solution in Eq. (7.60).

In Fig.

(7.6)

the calculated steady-state interface displacement solutions are shown. In

particular, for each numerical solution, the approximation by means of a spheroid with half-axes

˜𝑎and ˜

𝑏in dashed lines is shown.

0π

23π

0.95

1.05

r/R

(M1)

(M2)∗

experimental

Fig. 7.6: Steady-state interface solutions.

Section 7.4. Comparison to experimental data and discussion

100

For the sake of completeness, the stationary flow field as predicted by the generalized

Lorentz

model is depicted in Fig. 7.7. Due to symmetry, only the first quadrant is shown. It can be

seen that a vortex flow inside the droplet has developed. Interestingly, the maximum velocity

magnitude develops at the interface itself and is located at the polar angle of

𝜗≈π

. Both fluids

are at rest near the poles and the equator.

Fig. 7.7:

Stationary flow field as predicted by the generalized

Lorentz

force model. The arrows indicate

the flow direction and the background color represents the magnitude of the velocity field.

Chapter 7. The oil drop experiment

On the electromagnetic coupling problem 101

8 Summary and conclusions

In this thesis, the electromagnetic coupling was investigated and its impact on measurable

quantities was evaluated. In the first part, the electromagnetic coupling problem was analyzed

theoretically. A rational framework for the evaluation of the electromagnetic coupling problem

was developed and subsequently applied to a variety of electromagnetic coupling models. In

particular, the interplay between a general electromagnetic coupling model and constitutive

theory was investigated. It was found that the electromagnetic force model cannot be subject

to the entropy principle. Rather, it affects the thermodynamic restrictions that follow for

the material functions. Moreover, the choice of the electromagnetic force model impacts the

electromagnetic energy model, which has to be constructed in accordance with the second law

of thermodynamics. From this, it was concluded that the electromagnetic force model must be

investigated first.

The developed rational framework renders force models that employ different formulations

of continuum physics comparable. This facilitated the evaluation of different theoretical

predictions of modern force models. The impact of differences in electromagnetic force models

on the prediction of measurable quantities was analyzed systematically. Therefore, global and

local effects were evaluated and the circumstances under which relevant differences arise were

determined. In particular, it was found that the total electromagnetic force and the total

electromagnetic moment acting upon solid bodies differ between two particular models if the

employed electromagnetic stress tensors are different. Furthermore, the same applies to the

local deformation predictions in quasi-static settings, which was proved using the principle of

virtual work. As a an additional result, the concerns raised in [Reich (2017)], regarding the

ambiguities in electromagnetic force models, were laid to rest by showing that the local motion

as well as the total force computation are unaffected by the reshuffling between electromagnetic

force, momentum and stress.

Using the developed framework, the comparison of different force models was facilitated. The

analyzed models are the ones proposed by

Abraham

Minkowski

Einstein

–

Laub

Erigen

–

Maugin

Pao

–

Hutter

and

Kovetz

. The unitized comparison of these models allowed to

determine suitable experiments in which force model differences can be observed. Consequently,

three experiments have been identified. Two of the experiments were designed and conducted

in this work. These encompass the measurement of forces and moments between cylindrical

permanent magnets. It should be noted that by utilizing ferrofluid, magnetically nonlinear

effects were present. For the third experiment concerning the deformation of oil droplets, the

experimental data provided in [Torza, Cox, and Mason (1971)] was used. In order to evaluate

the considered experiments numerically, a finite element method in curvilinear coordinates was

developed. This allowed the efficient computation of the electromagnetic fields by means of

exploiting the inherent symmetries of the experiments. As a result, the temporal interface

evolution of the droplet in the third experiment could be calculated with minimal computational

costs.

By means of comparing the experimental observations with the theoretical results, several force

models have been found to be not applicable in general. In particular, the models by

Abraham

Minkowksi

and

Einstein

–

Laub

could not predict the electromagnetic moment between

cylindrical permanent magnets correctly. Most noteworthy is the oil drop experiment. Here,

all analyzed force models except for the generalized

Lorentz

model resulted in qualitatively

different droplet shapes when compared to the experiment. Notwithstanding that quasi-static

102

conditions were employed in the experiment, it can be concluded that all analyzed models, except

the generalized

Lorentz

model, are not generally-applicable. This includes the contemporary

force model proposed by

Eringen

and

Maugin

as well as the model by

Kovetz

, see [Eringen

and Maugin (1990)] and [Kovetz (2000)].

8.1 Contribution of the thesis

In the current work, a rational framework for the unitized comparison of electromagnetic

coupling models was developed, which facilitated the analysis of substantial differences between

the models. The framework also aids the design of suitable experiments that test the limitations

of a given coupling model. As a result, several novel insights into the electromagnetic coupling

problem as well as the numerical analysis of electromechanical computations have been obtained:

For the case of simple material-response functions it was shown that the electromagnetic

force model is not subject to the restriction imposed by the second law of thermodynamics.

Rather, the electromagnetic force model influences the conditions arising for the electro-

magnetic energy model. This also demonstrates that the electromagnetic force model

must be investigated first before the whole coupling model is considered.

A classification of the circumstances under which the electromagnetic coupling problem

was provided. In particular, it was shown that the total electromagnetic force, the

total electromagnetic moment and the local deformation of bodies are governed by the

electromagnetic stress tensor if quasi-static conditions can be assumed. Hence, the vast

number of possible electromagnetic force models that need to be analyzed is drastically

reduced, because the number of suitable electromagnetic stress tensors is finite.

An efficient numerical method was developed that employs a finite element analysis in

curvilinear coordinates. The method was used to study electromagnetic phenomena such

as the three-dimensional magnetic field calculation as well as the corresponding force

computation while utilizing two-dimensional computational domains.

Guidelines for the numerical force calculation from electromagnetic fields were developed

based on an error analysis for mathematically equivalent force expressions. It was found

that the force computation via the electromagnetic stress tensor is not always the most

accurate, but can safely be applied in quasi-stationary settings.

The developed numerical method together with a simple interface tracking method allowed

for a computationally efficient analysis of the three-dimensional oil drop experiment.

By means of comparing to this experimental evidence, it was shown that some of the

popular electromagnetic coupling models are not generally applicable. These include

the force models by

Einstein

and

Laub

Abraham

, and

Minkowski

. Furthermore,

they yield incorrect predictions for quite simple initial-boundary value problems as was

demonstrated in the oil drop experiment. Consequently, the remaining models by

Pao

and

Hutter

Eringen

and

Maugin

as well as

Kovetz

were found to be not applicable

in general, which is particularly striking because they are thermodynamically consistent.

It was found that only the generalized

Lorentz

model is applicable to all experiments

analyzed in this thesis. In particular, from the models analyzed in this thesis, it was the

only one that resulted in correct predictions for the oil drop experiment for which the

experimental data provided in [Torza, Cox, and Mason (1971)] was used.

These scientific findings impact technology because in stationary applications the generalized

Lorentz

model can safely be recommended. Furthermore, the developed numerical method can

be applied to a variety of simulation problems, which aid, e.g., the dimensioning of machines.

Chapter 8. Summary and conclusions

On the electromagnetic coupling problem 103

8.2 Outlook

The identification procedure for limitations of electromagnetic coupling models presented in this

thesis provided conclusive evidence that many of the popular coupling models are not generally

applicable. However, that does not guarantee that the generalized

Lorentz

model is applicable

under all circumstances, for example when the accelerated rigid body motion of magnets is

concerned. Possible lines of future work can be summarized as follows:

•

Dynamical experiments must be investigated. These include both, analyzing the dynamic

motion of matter under the influence of electromagnetic fields and studying the impact of

non-stationary electromagnetic fields on the electromagnetic coupling. In particular, the

incorporation of the results from special relativity needs to be investigated.

•

Local deformations due to electromagnetic forces must be analyzed further. In this thesis,

only quasi-electrostatic conditions were investigated. The analysis of the deformation

of ferrofluid droplets by means of the methods devised in this thesis will probably be

fruitful as well. This is particularly interesting because the electromagnetic stress tensors

Eringen

and

Maugin

as well as

Kovetz

do not employ the same structure regarding

the magnetic contributions when compared to the electric ones.

•

The electromagnetic energy model must be considered as it may impact the constitutive

theory. To this end, general representation theorems could be constructed and subsequently

exploited similarly to the general electromagnetic stress tensor representation.

•

In this thesis, only simple material-response functions were analyzed. For example, how do

the statements regarding the total electromagnetic force change if more complex material

behavior is analyzed? In particular, the impact on more sophisticated theories such as the

higher gradient theory or the micro-morphic theory need to be addressed.

Section 8.2. Outlook

On the electromagnetic coupling problem XV

List of Figures

2.1 Depiction of material and non-material control domains . . . . . . . . . . . . . 6

2.2 Schematic depictions of control domains . . . . . . . . . . . . . . . . . . . . . . 7

5.1 Coordinate space transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.2 Azimuthal symmetry exploitation . . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.3 Magnetic field error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.4 Magnetic force error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.5 Mesh structuring rule in corner regions . . . . . . . . . . . . . . . . . . . . . . . 58

5.6 Self-force error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.7 Relative force error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.8 Immiscible fluids in a spherical container . . . . . . . . . . . . . . . . . . . . . . 60

5.9 Numerical interface interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.10Snaptestinitialstate ................................ 66

5.11 The interface of a droplet in the snap test for different time steps . . . . . . . . 66

6.1 Ferrofluid container assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

6.2 Photographs of the first attempt of the ferrofluid assembly . . . . . . . . . . . . 68

6.3 Photographs of the second assembly attempt . . . . . . . . . . . . . . . . . . . 69

6.4 Force measurement over the course of the experiment. . . . . . . . . . . . . . . 70

6.5 Schematic of the global force experiment between permanent magnets . . . . . 70

6.6 Force measurement of the magnet immersed in ferrofluid . . . . . . . . . . . . . 73

6.7 Schematics of magnets and electrets in external electromagnetic fields . . . . . 74

6.8 Schematic of the turntable assembly . . . . . . . . . . . . . . . . . . . . . . . . 79

6.9 Photographs of the turntable assembly . . . . . . . . . . . . . . . . . . . . . . . 79

6.10 Schematic of the turntable experiment . . . . . . . . . . . . . . . . . . . . . . . 80

6.11 Theoretical moment parameter study . . . . . . . . . . . . . . . . . . . . . . . . 81

6.12 Machine force parameter analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.13 Electromagnetic moment measurement . . . . . . . . . . . . . . . . . . . . . . . 84

7.1 Schematic depiction of the oil-drop experiment . . . . . . . . . . . . . . . . . . 86

7.2 Dimensional reduction of the oil-drop experiment . . . . . . . . . . . . . . . . . 86

7.3 Effective electromagnetic surface force . . . . . . . . . . . . . . . . . . . . . . . 90

7.4 Electric field of spheroidal dielectrics . . . . . . . . . . . . . . . . . . . . . . . . 96

7.5 Effective electromagnetic forces and curvature forces . . . . . . . . . . . . . . . 98

7.6 Steady-state interface solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7.7 Stationaryflowfield ................................. 100

A.1 Magnetic flux measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

B.1 Schematic depiction of two observers surveying the same point space . . . . . . v

C.1 Special control domain for the localization on a singular line . . . . . . . . . . . xiv

C.2 Control domain for a surface theorem. . . . . . . . . . . . . . . . . . . . . . . . xv

List of Figures

On the electromagnetic coupling problem XVII

List of Tables

4.1 Summary of the stress tensor coefficients resulting from different force models . 42

5.1 Finite element function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

6.1 Selected properties of ferrofluid . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6.2 Selected properties of permanent magnets . . . . . . . . . . . . . . . . . . . . . 83

7.1 Reference parameters for the oil drop experiment . . . . . . . . . . . . . . . . . 88

7.2 Dimensionless numbers for the two-phase flow problem . . . . . . . . . . . . . 88

List of Tables

On the electromagnetic coupling problem XIX

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Bibliography

On the electromagnetic coupling problem i

A Magnetic field measurements

In Tab. 6.2 several parameters of the used permanent magnets are listed. In particular the

remanence values are of signal importance. Therefore, the corresponding values provided by

the data sheets are measured independently. To this end, a magnetometer is placed at various

distances along the

𝑧

-axis as shown in Fig. A.1. The remanence

𝜇0𝑀0

occurring in the analytical

𝑑

𝑧

magnetometer

0 5 10 15 20 25

200

400

600

d(mm)

kBk(mT)

magnet 1

magnet 2

analytical

Fig. A.1: Magnetic flux measurement

solution is then found such that the resulting

𝐵𝑧

magnitude fits the measured values. As it

turns out, a remanence of

1.3 T

results in an almost perfect fit. This value is

0.1 T

lower than

the value provided in the data sheet.

On the electromagnetic coupling problem iii

B Observer transformations

In this section, a framework for the concepts of observers, their transformations and the infamous

objective quantities is discussed. To this end, the rather philosophical concepts are introduced

and discussed in detail. Subsequently, an appropriate notation is developed, that mathematically

formalizes the abstract philosophical concepts. The ideas presented in the following are not new

and likewise is their modern representation not new. However, in the literature, mathematical

simplicity is often preferred at the cost of logical rigor. For example, the terms “frame of

reference” and “coordinate systems” are often used synonymously, see the discussion in [Ivanova,

Vilchevskaya, and Müller (2016); Ivanova, Vilchevskaya, and Müller (2017)]. These concepts

represent an integral part of the principles of constitutive theory. In particular, the material of

frame indifference states that constitutive functions should be independent of the frame. There

are other principles, which are closely related and, if properly combined, are equivalent to the

material of frame indifference principle, see [Svendsen and Bertram (1999)].

A comprehensive guide on how the principle is to be employed can be found in [Müller (1985)].

However, if these applications are compared with, e.g., [Svendsen and Bertram (1999)], there

appears to be a contradiction right from the beginning. In both references, the

Euclid

ean

transformation between two observers is introduced differently,

Müller :𝑥′

𝑖=𝑄𝑖𝑗(𝑡)𝑥𝑗+𝑐𝑖(𝑡),(B.1a)

Bertram :𝑥′=𝑄(𝑡)·𝑥+𝑐(𝑡),(B.1b)

where

𝑥

𝑥′

are position vectors and

𝑥𝑖

𝑥′

𝑖

their respective components. Similarly,

𝑐

is the

connection vector between the two origins

𝑂

and

𝑂′

and

𝑐𝑖

are its components. The tensor

𝑄

represents the rotation between both observers and is thus orthogonal.

𝑄𝑖𝑗

is an orthogonal

rotation matrix. Assuming that the observer in

𝑂

uses

{𝑒𝑖}

and the other observer located in

𝑂′uses {𝑒′

𝑖}it follows from Eq. (B.1a)

𝑥′=𝑥+𝑐(𝑡), 𝑄𝑖𝑗(𝑡) = 𝑒′

𝑖·𝑒𝑗.(B.2)

This vector equation seems to be contradicting the transformation proposed in Eq.

(B.1b)

This is only apparently so, as the concepts of a coordinate transformation and an observer

transformation are combined into one operation, but in a different order. At the first glance,

however, Eq.

(B.1a)

looks only like a coordinate transformation and Eq.

(B.1b)

looks only like

an observer transformation. In [Ivanova, Vilchevskaya, and Müller (2017)] both, the precise

definition of an observer and the concept of an observer image are explained in detail to clarify

the situation. The ideas presented by the authors constitute the basis for the extended framework

introduced in the following.

An observer is an entity, which is capable of measuring time and surveying space. Thus, an

observer can measure distances and track motion. Mathematically, an observer is represented

by a frame and a measuring system. The frame of the observer

is defined by a fixed origin

𝑂

as well as a non-collinear vector triad,

frameO={𝑂, 𝑒1,𝑒2,𝑒3}.(B.3)

Without loss of generality, it can be assumed that

𝑒𝑖·𝑒𝑗

𝛿𝑖𝑗

. The observer’s measuring system

consists of a clock to measure time,

𝑡

, and a characteristic coordinate system

𝑥𝑖

. Hence, an

observer may be represented via

O:= frameO∪{𝑡, 𝑥𝑖

O}={𝑂, 𝑒1,𝑒2,𝑒3, 𝑡, 𝑥1

O, 𝑥2

O, 𝑥3

O}.(B.4)

Furthermore, an observer may introduce a position vector via

𝑥O

𝑥𝑖

O𝑒𝑖

. It is important to

note that

𝑥𝑖

and

𝑒𝑖

are quantities defined by the observer

. Consequently, with respect to his

clock, these quantities are fixed and have no temporal change. Most importantly, however, is

that they cannot be accessed directly by another observer. Metaphorically speaking, the vectors

𝑒𝑖

represent the “sticks” of the observer. Another observer cannot access these sticks, but can

only track their motion, i.e., take “images” of them.

The concept of the image operation

introduced in [Zhilin (2012)], aids to separate the

mathematical coordinate transformation from the actual change of observer. Consider two

observers, respectively,

A={𝐴, 𝑎𝑖, 𝑡, 𝑥𝑖

A},B={𝐵, 𝑏𝑖, 𝑡, 𝑥𝑖

B}.(B.5)

Since the observer

can neither access the coordinates

𝑥𝑖

nor the directors

𝑏𝑖

directly, he can

only determine an image of the position vector

𝑥B

using his own instruments of measurement:

imA{𝑥B}

. Therein,

𝜓B

is the measurement of some quantity as performed by

and the image

operator

imA{𝜓B}

represents the measurement of this measurement as performed by the observer

. Mathematically, the image operator returns the measurement of another observer and can

thus be understood as a mapping between observers,

imA{𝜓B}:B↦→ A.(B.6)

The image operation itself cannot be specified further. Instead, its action upon particular

quantities can be studied. For instance, the image operation of one observer applied to his own

measurements has the same effect as the identity operator, i.e.,

imA{𝑎𝑖}=𝑎𝑖,imB{𝑏𝑖}=𝑏𝑖.(B.7)

However, the images of the basis vectors of another observer are given by

imA{𝑏𝑖}=𝑄BA

(𝑡)·𝑎𝑖,imB{𝑎𝑖}=𝑄AB

(𝑡)·𝑏𝑖.(B.8)

where

𝑄BA

(

𝑡

)and

𝑄AB

(

𝑡

)are time-dependent transformation tensors as measured from their

respective observers. The image operator offers a resolution to the apparent paradox: How can

the connection between two time-independent sets of vectors

{𝑎𝑖}

and

{𝑏𝑖}

be time-dependent?

Of course, it is not. Rather, their images are time-dependent. Without loss of generality the

transformation tensors

𝑄BA

and

𝑄AB

can be assumed to be orthogonal, i.e.,

𝑄BA

(

𝑡

)

·𝑄T

(

𝑡

) =

where

is the unit tensor, see the comments in [Ivanova, Vilchevskaya, and Müller (2017)].

Hence, they are referred to as rotation tensors.

It seems natural to interpret

𝑄BA

as the mapping between the directors of

and those of

. However, this is inappropriate for two reasons. Firstly, the rotation tensor maps

𝑎𝑖

onto

the images

imA{𝑏𝑖}

rather than onto the basis vectors

𝑏𝑖

themselves. Secondly, this mapping is

completely performed in the frame of

and hence with the tools of the observer

. Therefore,

one has the following natural representations:

𝑄BA=𝑄𝑖𝑗

BA𝑎𝑖⊗𝑎𝑗,𝑄AB=𝑄𝑖𝑗

AB𝑏𝑖⊗𝑏𝑗.(B.9)

There is no representation of

𝑄BA

in the basis

{𝑏𝑖}

even though one could construct such

1A similar notion is introduced utilizing “monitors” in [Bertram (2012)]

Chapter B. Observer transformations

On the electromagnetic coupling problem v

a representation for the

-image of

𝑄BA

, which is not a meaningful thing to do. Note that

Eq.

(B.8)

does not represent a definition of the image operation. Rather, it defines the action

upon basis vectors. Measurements, as performed by observers, are subjective in the sense that

two measurements of two observers of the same object may differ. That is to say, the observer’s

measurement may not always capture the essence of the object. A physical theory is therefore to

be built upon objects for which two observers can agree upon their measurements. To formalize

this idea, another operation is introduced. Let the field quantity

𝜓

be an object, which means

that its existence is not tied to a measurement. Then, the measurement or observation of

𝜓

performed by Amay formally be denoted by

𝜓A= obsA(𝜓).(B.10)

Subsequently, an objectively measurable quantity 𝜓satisfies

imB{obsA(𝜓)}=𝑄AB*obsB(𝜓),(B.11)

for any two observers Aand B, where “*” is the Rayleigh product defined via

𝐴*(𝑐1⊗𝑐2⊗. . . ⊗𝑐𝑛):=𝐴·𝑐1⊗𝐴·𝑐2⊗. . . ⊗𝐴·𝑐𝑛.(B.12)

That is to say, the measurements of the observers differ only by rotation. Since Eq.

(B.11)

must

hold for any two observers, the converse is also true

imA{obsB(𝜓)}=𝑄BA*obsA(𝜓).(B.13)

In the literature, such a quantity is often referred to as objective itself. In the current framework,

such a statement is redundant as

𝜓

was assumed to be an object as an initial assumption.

Rather, as introduced above, in the following these quantities are referred to as objectively

measurable. The operations in Eq.

(B.11)

are only defined abstractly. Following the arguments

above, one could refer to Eq.

(B.11)

as a measurement comparison. How this comparison is to

be performed mathematically, depends upon the quantity under consideration. Note that the

basis vectors in Eq.

(B.8)

are by definition objectively measurable. Before the measurements

of particular physical quantities can be compared, the comparison of the measuring tools of

the observers is to be investigated first. To this end, the so-called observer transformation

is introduced. Consider the situation depicted in Fig. B.1. Therein, the same point space is

surveyed by two different observers

and

from Eq.

(B.5)

in their respective frames. From

𝐴

𝐵

𝑃

𝐴

𝐵

𝑃

frame of Aframe of B

𝑐A

𝑑A

𝑥A

𝑐B

𝑑B

𝑥B

𝑏1

𝑏2

𝑎1

𝑎2

Fig. B.1: Schematic depiction of two observers surveying the same point space.

simple vector algebra in each observer’s frame, one has for the position vectors to the observation

point 𝑃

𝑥A(𝑡) = 𝑑A(𝑡) + 𝑐A(𝑡),𝑥B(𝑡) = 𝑑B(𝑡) + 𝑐B(𝑡).(B.14)

Therein, the indices

and

denote the measurements with respect to the corresponding

observers according to Eq.

(B.10)

. Each measurement may be represented only with respect to

the basis of its respective observer:

𝑥A(𝑡) = 𝑥𝑖

A(𝑡)𝑎𝑖,𝑐A(𝑡) = 𝑐𝑖

A(𝑡)𝑎𝑖,𝑑A(𝑡) = 𝑑𝑖

A(𝑡)𝑎𝑖,

𝑥B(𝑡) = 𝑥𝑖

B(𝑡)𝑏𝑖,𝑐B(𝑡) = 𝑐𝑖

B(𝑡)𝑏𝑖,𝑑B(𝑡) = 𝑑𝑖

B(𝑡)𝑏𝑖.(B.15)

Hence, the following component equations could be derived:

𝑥𝑖

A(𝑡) = 𝑑𝑖

A(𝑡) + 𝑐𝑖

A(𝑡), 𝑥𝑖

B(𝑡) = 𝑑𝑖

B(𝑡) + 𝑐𝑖

B(𝑡).(B.16)

However, these equations are rather unhelpful as can be seen shortly. To connect both observa-

tions, one has to identify, e.g., that

𝑑A

imA{𝑥B}

. Bearing in mind that the position vector

𝑥

to the point

𝑃

is constructed specifically by each observer, it is only natural that

𝑥

is not an

object and thus is not an objectively measurable quantity in the sense of Eq.(B.11),

imB{obsA(𝑥)}=𝑑B=𝑄AB·obsB(𝑥) = 𝑄AB·𝑥B.(B.17)

Instead, the connection between the position vectors from Eq. (B.14) can be rewritten:

𝑥A(𝑡) = imA{𝑥B}(𝑡) + 𝑐A(𝑡),𝑥B(𝑡) = imB{𝑥A}(𝑡) + 𝑐B(𝑡).(B.18)

In the context of classical physics, it is assumed that the images of lengths corresponding to one

observer are identical to the lengths measured by the said observer, i.e., lengths are observer

invariant. Therefore, the images of position vectors are given by:

imA{𝑥B}(𝑡) = 𝑥𝑖

B(𝑡)𝑄BA(𝑡)·𝑎𝑖=𝑄𝑖𝑗

BA(𝑡)𝑥𝑗

B(𝑡)𝑎𝑖,

imB{𝑥A}(𝑡) = 𝑥𝑖

A(𝑡)𝑄AB(𝑡)·𝑏𝑖=𝑄𝑖𝑗

AB(𝑡)𝑥𝑗

A(𝑡)𝑏𝑖,(B.19)

From the first equation, it follows that

𝑥𝑖

A(𝑡) = 𝑄𝑖𝑗

BA(𝑡)𝑥𝑗

B(𝑡) + 𝑐𝑖

A(𝑡) = 𝑄𝑖𝑗

BA(𝑡)(︀𝑥𝑗

B(𝑡)−𝑐𝑗

B(𝑡))︀.(B.20)

This matrix equation resembles the

Euclid

ean transformation from Eq.

(B.1a)

as introduced

in, e.g., [Müller (1985)]. However, upon defining a mixed quantity 𝑥*A

Bone has

𝑥A(𝑡) = 𝑄BA(𝑡)·𝑥*A

B(𝑡) + 𝑐A(𝑡),𝑥*A

B:=𝑥𝑖

B(𝑡)𝑎𝑖,(B.21)

which in turn is of the form of Eq.

(B.1b)

. Hence, if interpreted appropriately, both representa-

tions of the

Euclid

ean transformation are equivalent in their essence, albeit being completely

different mathematical statements. They are different representations of the transformation

in Eq.

(B.18)

. The latter utilizes the image operator, which offers a clear interpretation, but

is inconvenient for calculations. In contrast, Eq.

(B.21)

is a tensor equation, which employs

the direct tensor notation and is thus easier to use. However, only when derived from the

proper

Euclid

ean transformation in Eq.

(B.18)

its interpretation is readily accessible. This

interpretation is most important when time derivatives are considered. Any two observers

with identical clocks may perceive temporal changes differently. The velocities of the same

observational point 𝑃as perceived by both, Aand B, respectively, are given by

𝑣A(𝑡):=d𝑥A(𝑡)

d𝑡=d𝑥𝑖

A(𝑡)

d𝑡𝑎𝑖,

𝑣B(𝑡):=d𝑥B(𝑡)

d𝑡=d𝑥𝑖

B(𝑡)

d𝑡𝑏𝑖.

(B.22)

Chapter B. Observer transformations

On the electromagnetic coupling problem vii

Hence, from Eq. (B.21)

𝑣A(𝑡) = 𝛺BA(𝑡)·𝑄BA(𝑡)·𝑥*A

B(𝑡) + 𝑄BA(𝑡)·𝑣*A

B(𝑡) + d𝑐A(𝑡)

d𝑡

=𝜔BA(𝑡)×(︀𝑥A(𝑡)−𝑐A(𝑡))︀+𝑄BA(𝑡)·𝑣*A

B(𝑡) + d𝑐A(𝑡)

d𝑡.

(B.23)

with

𝛺BA=d𝑄BA

d𝑡·𝑄T

BA,𝑣*A

B(𝑡) = d𝑥𝑖

B(𝑡)

d𝑡𝑎𝑖(B.24)

and the angular velocity

𝜔BA

defined as the axial vector to the angular velocity tensor

𝛺BA

such that for any vector 𝑦=0one has

𝛺BA(𝑡)·𝑦!

=𝜔BA(𝑡)×𝑦⇔𝜔BA(𝑡) = −1

⟨3⟩

𝜖··𝛺BA(𝑡),(B.25)

where

⟨3⟩

𝜖

is the

Levi

Civita

tensor. Its components are always given by the permutation symbol

𝜀𝑖𝑗𝑘. However, each observer uses their own basis, such that

imA{⟨3⟩

𝜖B}= imB{𝜀𝑖𝑗𝑘𝑏𝑖⊗𝑏𝑗⊗𝑏𝑘}=𝜀𝑖𝑗𝑘𝑄BA*(𝑎𝑖⊗𝑎𝑗⊗𝑎𝑘) = 𝑄BA*⟨3⟩

𝜖A,(B.26)

with the Rayleigh product

𝐴*(𝑐1⊗𝑐2⊗. . . ⊗𝑐𝑛):=𝐴·𝑐1⊗𝐴·𝑐2⊗. . . ⊗𝐴·𝑐𝑛.(B.27)

Hence, according to Eq.

(B.11)

the

Levi

Civita

tensor is an objectively measurable quantity.

Note that in the literature, ⟨3⟩

𝜖is referred to as “tensor density,” because of

imA{⟨3⟩

𝜖B}=𝑄BA*⟨3⟩

𝜖A= det(𝑄BA)⟨3⟩

𝜖A.(B.28)

On the electromagnetic coupling problem ix

C Mathematical identities and coordinate

transformations

In this chapter, several mathematical identities are discussed. None of them are a novelty and

they can be found in, e.g., [Eringen and Maugin (1990)].

C.1 Integral theorems and general balance laws

Consider the following general volumetric balance

d𝑡ˆ

𝑊s

𝜓d𝑊+d

d𝑡ˆ

𝐼

𝜓𝐼d𝐴=−ˆ

𝛤

𝑛·(︀𝜙+ (𝑣s−𝑤)⊗𝜓)︀d𝐴+ˆ

𝑊s

(𝑝+𝑠) d𝑊

−˛

𝜕𝐼

𝜈·𝜙

𝐼dℓ+ˆ

𝐼

(𝑝𝐼+𝑠𝐼) d𝐴 .

(C.1)

Neither

𝑊s

nor

𝐼

need to be material in general, i.e., they move with their independent velocities

𝑤

and

𝑤𝐼

, respectively. However, the surface terms

𝜓𝐼

𝜙

𝐼

and

𝑝

𝐼

as well as

𝑠

𝐼

are restricted to

the surface. Therefore, the supply to the surface does only contain the non-convective part,

𝜙

𝐼

However, since there could be a relative motion on the surface, the notion of another velocity

field,

𝑣𝐼

, is required. The field

𝑣𝐼

represents the velocity of a point on the surface, that is

restricted to the surface. It can be referred to as pseudo-material surface velocity as it is a

function of a surface point 𝑥𝐼corresponding to a particular initial position 𝑋𝐼,

𝑥𝐼=𝜒𝐼(𝑋𝐼, 𝑡).(C.2)

In turn, the reference surface point

𝑋𝐼

may be represented in terms of the intrinsic surface

coordinates

𝑈1

and

𝑈2

, for which yet another mapping onto the intrinsic surface coordinates

𝑢1

and 𝑢2in the current configuration exists,

𝑢𝛼= ^𝑢𝛼(𝑈1, 𝑈2, 𝑡), 𝛼 ∈ {1,2},(C.3)

with Greek indices always ranging from one to two. Analogously to the material velocity one

has

𝑣𝐼:=𝜕𝜒𝐼

𝜕𝑡 𝑈𝛼

=𝑤𝐼+𝑢𝐼,(C.4)

where the motion of the surface and the relative velocity are given by, respectively,

𝑤𝐼=𝜕𝜒𝐼

𝜕𝑡 𝑢𝛼

,𝑢𝐼=𝜕^𝑢𝛼

𝜕𝑡 𝑈𝛽

𝜕𝜒𝐼

𝜕𝑢𝛼.(C.5)

The relative velocity

𝑢𝐼

represents the material velocity of points with respect to the surface

itself. If there is no relative motion on the surface,

𝑢𝐼≡0

, and the pseudo-particle velocity and

the surface velocity coincide, 𝑣𝐼=𝑤𝐼.

Finally, consider the general surface flux balance

d𝑡ˆ

𝑆−∪𝑆+

𝑛·𝛾d𝐴=−ˆ

ℓ−∪ℓ+

𝜏·(︀𝜙𝑆+𝛾×[𝑣s−𝑤])︀dℓ+ˆ

𝑆−∪𝑆+

𝑛·(𝑝𝑆+𝑠𝑆) d𝐴+ˆ

ℓ𝐼

𝜈𝐼·𝑠𝑆

𝐼dℓ . (C.6)

C.1.1 Regular integral theorems

Depending on the type of the integration domain the occurring velocities in the integral theorems

may differ. In order to obtain the most general formulæ non-material domains are considered.

The velocity of a non-material point is given by

𝑤

. However, all the following theorems hold

true for material domains by means of the identification

𝑤

𝑣s

and by replacing

∇s→ ∇m

The following theorems are only applicable to domains in which the kernels are continuous and

differentiable.

Theorem of GAUSS For a tensor 𝐴of an arbitrary rank greater or equal than one,

𝑊s

∇s⊕𝐴d𝑊=˛

𝜕𝑊s

𝑛⊕𝐴d𝐴(C.7)

for a product ⊕ ∈ {·,×,⊗}.

Theorem of STOKES sometimes also referred to as theorem of Kelvin–Stokes, reads

𝑆

𝑛·(∇s×𝐴) d𝐴=˛

𝜕𝑆

𝜏·𝐴dℓ . (C.8)

Volumetric transport theorem also referred to as Reynolds transport theorem

d𝑡ˆ

𝑊s

𝐴d𝑊=ˆ

𝑊s

d𝐴

d𝑡d𝑊+ˆ

𝑊s

𝐴(∇s·𝑤) d𝑊 . (C.9)

Naturally, the kernel is a spatial function, i.e.,

𝐴

(

𝑥s, 𝑡

)such that its total time derivative

is given by d𝐴

d𝑡=𝜕𝐴

𝜕𝑡 +𝑤·(∇s⊗𝐴).(C.10)

Therefore, using Eq. (C.7) the transport theorem may be rewritten

d𝑡ˆ

𝑊s

𝐴d𝑊=ˆ

𝑊s(︂𝜕𝐴

𝜕𝑡 +∇s·[𝑤⊗𝐴])︂d𝑊=ˆ

𝑊s

𝜕𝐴

𝜕𝑡 d𝑊+˛

𝜕𝑊s

(𝑛·𝑤)𝐴d𝐴 . (C.11)

Surface flux theorem for a vectorial flux 𝑎through a surface 𝑆

d𝑡ˆ

𝑆

𝑛·𝑎d𝐴=ˆ

𝑆

𝑛·(︂𝜕𝑎

𝜕𝑡 +∇s·[𝑤⊗𝑎]−𝑎·(∇s⊗𝑤))︂d𝐴(C.12)

By noting the identity

∇s×(𝑎×𝑤)=(∇s·𝑤)𝑎+𝑤·(∇s⊗𝑎)−(∇s·𝑎)𝑤−𝑎·(∇s⊗𝑤)(C.13)

Chapter C. Mathematical identities and coordinate transformations

On the electromagnetic coupling problem xi

and subsequent application of

Stokes

’ theorem, the surface transport theorem can be denoted

compactly,

d𝑡ˆ

𝑆

𝑛·𝑎d𝐴=ˆ

𝑆

𝑛·[︂𝜕𝑎

𝜕𝑡 + (∇s·𝑎)𝑤]︂d𝐴+˛

𝜕𝑆

𝜏·(𝑎×𝑤) dℓ . (C.14)

C.1.2 Integral theorems with discontinuities

Consider an integration domain

𝑊s

that is cut by an interface

𝐼

such that two regular subdomains

𝑊+

and

𝑊−

result. At the interface

𝐼

the kernel functions may be discontinuous. Hence, the

integral theorems from Sec. C.1.1 can only be applied to the subdomains

𝑊±

. In order to

obtain the generalized versions, which are valid for discontinuous kernels, the regular integral

theorems are applied to both regular subdomains and the resulting expressions are subsequently

added. Since the boundary of each subdomain contains the interface, it is advisable to use the

decomposition

𝜕𝑊±

𝛤±

s∪𝐼

, where the boundary of the total region is given by

𝜕𝑊s

𝛤+

s∪𝛤−

By adding the integral theorems applied to both subdomains, the interface is counted twice.

However, the respective outward normal at the interface differs by the sign. Conventionally, the

normal to the interface

𝑒

is defined to point towards the

𝑊+

domain such that

𝑛±

∓𝑒

. At

the interface, the kernel values may differ depending on the side from which the limit is taken.

This gives rise to the jump operator defined as

J𝑎K:=𝑎+−𝑎−,𝑎±:= lim

𝑥±→𝑦𝑎(𝑥±),𝑥±∈𝑊±

s,𝑦∈𝐼 . (C.15)

Generalized theorem of GAUSS

For a tensor

𝐴

of an arbitrary rank greater or equal than one,

𝑊+

s∪𝑊−

∇s⊕𝐴d𝑊=ˆ

𝛤+

s∪𝛤−

𝑛⊕𝐴d𝐴−ˆ

𝐼

𝑒⊕J𝐴Kd𝐴(C.16)

for a product ⊕ ∈ {·,×,⊗}.

Generalized theorem of STOKES

For a tensor

𝐴

of an arbitrary rank greater or equal than

one,

𝑆+

s∪𝑆−

𝑛·(∇s×𝐴) d𝐴=ˆ

ℓ+

s∪ℓ−

𝜏·𝐴dℓ−ˆ

ℓ𝐼

𝜏𝐼·J𝐴Kdℓ , (C.17)

where it was tacitly assumed that 𝜏𝐼:=𝜏−such that 𝜏+=−𝜏𝐼.

Generalized transport theorem

d𝑡ˆ

𝑊+

s∪𝑊−

𝐴d𝑊=ˆ

𝑊+

s∪𝑊−

(︂𝜕𝐴

𝜕𝑡 +∇s·(𝑤⊗𝐴))︂d𝑊+ˆ

𝐼

𝑒·J(𝑤−𝑤𝐼)⊗𝐴Kd𝐴 . (C.18)

Generalized surface flux theorem

d𝑡ˆ

𝑆+∪𝑆−

𝑛·𝑎d𝐴=ˆ

𝑆+∪𝑆−

𝑛·[︂𝜕𝑎

𝜕𝑡 + (∇s·𝑎)𝑤]︂d𝐴+˛

ℓ+∪ℓ−

𝜏·(𝑎×𝑤) dℓ−ˆ

ℓ𝐼

𝜏𝐼·(J𝑎K×𝑤𝐼) dℓ . (C.19)

Surface transport theorem

In contrast to the surface flux theorem, the surface transport

theorem aims at describing singular surfaces

𝐼

and the temporal change of some additive quantity

Section C.1. Integral theorems and general balance laws

xii

that is bound to the surface itself. Consider a surface density

𝑎𝐼

that is restricted to the surface

𝐼

such that

𝑎𝐼

𝑎E

𝐼(︀𝑢𝛼

(

𝑈1, 𝑈2, 𝑡

)

, 𝑡)︀

in a pseudo

Lagrange

an manner. Due to the surface

restriction and the subsequent intrinsic coordinate dependence, one has

d𝑎𝐼

d𝑡=𝜕^

𝑎E

𝐼

𝜕𝑡 𝑢𝛼

+ (𝑣𝐼−𝑤𝐼)·(∇𝐼⊗^

𝑎E

𝐼),∇𝐼=𝜕

𝜕𝑥

𝐼

𝐼·∇ .(C.20)

Therein,

𝐼

1−𝑒⊗𝑒

is the surface projector. Observing that for the undirected surface

element one has, see [Dziubek (2011)],

d𝑡(d𝐴)=(∇𝐼·𝑣𝐼) d𝐴 , ∇𝐼·𝑣𝐼=1

𝐼··(∇⊗𝑣𝐼),(C.21)

with the intrinsic surface velocity 𝑣𝐼, the surface transport theorem reads

d𝑡ˆ

𝐼

𝑎𝐼d𝐴=ˆ

𝐼(︂𝜕𝑎𝐼

𝜕𝑡 𝑢𝛼

+ (𝑣𝐼−𝑤𝐼)·(∇𝐼⊗𝑎𝐼) + 𝑎𝐼(∇𝐼·𝑣𝐼))︂d𝐴

=ˆ

𝐼(︂𝜕𝑎𝐼

𝜕𝑡 𝑢𝛼

+∇𝐼·([𝑣𝐼−𝑤𝐼]⊗𝑎𝐼) + 𝑎𝐼(∇𝐼·𝑤𝐼))︂d𝐴 .

(C.22)

Surface divergence theorem

𝜕𝐼

𝜈·𝑎dℓ=ˆ

𝐼

(∇𝐼·𝑎−[∇·𝑒]𝑒·𝑎) d𝐴 . (C.23)

After applying the product rule,

[∇·𝑒]𝑒·𝑎=∇𝐼·(︀𝑒⊗𝑒·𝑎)︀−𝑒·(∇𝐼⊗𝑒)·𝑎−(𝑒⊗𝑒)··(∇𝐼⊗𝑎)(C.24)

it can be observed that for any differentiable field, 𝑏one has

𝑒·(∇𝐼⊗𝑏) = 𝑒·(∇⊗𝑏−𝑒⊗𝑒·[∇⊗𝑏]) = 0,(C.25)

and thus the last two terms in Eq.

(C.24)

vanish. Hence, another useful representation of the

surface divergence theorem is obtained

𝜕𝐼

𝜈·𝑎dℓ=ˆ

𝐼

∇𝐼·(1𝐼·𝑎) d𝐴 . (C.26)

Chapter C. Mathematical identities and coordinate transformations

On the electromagnetic coupling problem xiii

C.1.3 Localization of global balance laws

With the integral theorems from the previous section, the balances in Eqs.

(C.1)

–

(C.6)

can be

localized. This is done in two steps. First, all integrals are transformed such that only regular

and singular integrals of the appropriate type remain. For the volumetric balance, this means

that only volumetric integrals and integrals with respect to

𝐼

remain. For the surface balance,

this means two generate surface integrals and line integrals with respect to

ℓ

𝐼

, respectively.

Hence, the balances reduce to

0=ˆ

𝑊s(︂𝜕𝜓

𝜕𝑡 +∇s·(𝑣s⊗𝜓+𝜙)−𝑝−𝑠)︂d𝑊+(C.27a)

+ˆ

𝐼(︂𝜕𝜓𝐼

𝜕𝑡 𝑢𝛼

+ (𝑣𝐼−𝑤𝐼)·(∇𝐼⊗𝜓𝐼) + 𝜓𝐼(∇𝐼·𝑣𝐼) + ∇𝐼·(︀1

𝐼·𝜙

𝐼)︀)︂d𝐴

+ˆ

𝐼(︂𝑒·J(𝑣s−𝑤𝐼)⊗𝜓+𝜙K−𝑝𝐼−𝑠𝐼)︀)︂d𝐴 .

The second step involves the actual localization as outlined in [Rickert and Müller (2020)]. The

balance in Eq. (C.1) reduces to

𝜕𝜓

𝜕𝑡 +∇s·(𝑣s⊗𝜓+𝜙) = 𝑝+𝑠(C.28)

in regular points 𝑥∈(𝑊+

s∪𝑊−

s)and in singular points 𝑥∈𝐼to

𝜕𝜓𝐼

𝜕𝑡 𝑢𝛼

+(𝑣𝐼−𝑤𝐼)·(∇𝐼⊗𝜓𝐼)+𝜓𝐼(∇𝐼·𝑣𝐼)+∇𝐼·(︀1

𝐼·𝜙

𝐼)︀=−𝑒·J(𝑣s−𝑤𝐼)⊗𝜓+𝜙K+𝑝𝐼+𝑠𝐼.(C.29)

By means of the localized mass balance,

𝜕𝜌s

𝜕𝑡 +∇s·(𝜌s𝑣s)=0 (C.30)

the representation via a mass-specific quantity, i.e.,

𝜓

𝜌s¯

𝜓

, yields the following simplification

of the local regular balance law

𝜌s(︂𝜕¯

𝜓

𝜕𝑡 +𝑣s·(∇s⊗¯

𝜓))︂=−∇s·𝜙+𝑝+𝑠.(C.31)

Similarly, the surface balance in Eq. (C.6) can be reformulated to read

0 = ˆ

𝛶+∪𝛶−

𝑛·[︂𝜕𝛾

𝜕𝑡 + (∇s·𝑎)𝑤−𝑝𝑆−𝑠𝑆]︂d𝐴+

+ˆ

𝐿+∪𝐿−

𝜏·(𝜙𝑆+𝛾×𝑣s) dℓ−ˆ

ℓ𝐼[︀𝜏𝐼·(J𝛾K×𝑤𝐼) + 𝜈𝐼·𝑠𝑆

𝐼]︀dℓ .

(C.32)

Therein, the domains are labeled according to Fig. C.1. Using this specialized control domain,

the balance can be localized to points on the singular line. In the limit, the noose contains no

area and the integral with respect to

𝛶±

vanishes. However, the boundary parts

𝐿±

remain

Section C.1. Integral theorems and general balance laws

xiv

Fig. C.1:

Special control domain for the localization on a singular line as presented in [Rickert and

Müller (2020)], reproduced with permission from Springer Nature.

and reduce to ℓ𝐼, where 𝜏=±𝜏𝐼, such that

𝐿+∪𝐿−

𝜏·(𝛾×𝑣s) dℓ→ˆ

ℓ𝐼

𝜏𝐼·𝛾×𝑣sdℓ . (C.33)

Thus, only two types of integrals in Eq.

(C.32)

remain, i.e., with respect to

𝛶±

and

ℓ𝐼

. The

former vanish and the localization onto the singular line results in

𝜏𝐼·𝜙𝑆+𝛾×(𝑣s−𝑤𝐼)=𝜈𝐼·𝑠𝑆

𝐼.(C.34)

Upon noting that 𝜈𝐼=𝑒𝐼×𝜏𝐼it follows that

𝑒𝐼×𝜙𝑆+𝛾×(𝑣s−𝑤𝐼)=𝑠𝑆

𝐼.(C.35)

Chapter C. Mathematical identities and coordinate transformations

On the electromagnetic coupling problem xv

C.2 An extension of CAUCHY’s tetrahedron argument to surfaces

The famous tetrahedron argument by C

AUCHY

allows for the deduction of the dependence of the

non-convective surface supply term in a balance equation on the normal vector of a considered

plane. Consider a global volumetric balance of an additive quantity

𝜓

over a not necessarily

material but otherwise arbitrary regular volume 𝛺of the form

d𝑡ˆ

𝛺

𝜓d𝑉=˛

𝜕𝛺

(𝜙−𝜓[𝑣−𝑤]·𝑛) d𝐴+ˆ

𝛺

(𝑝+𝑠) d𝑉 , (C.36)

where

𝜙

is the non-convective surface supply and

𝑝

and

𝑠

are the volumetric production density

and volumetric supply density, respectively. Since the domain

𝛺

is arbitrary in its extent, the

balance law is also valid for all subdomains. By assuming that

𝑓

depends parametrically upon

the outward normal vector

𝑛

of the boundary

𝜕𝛺

𝜙

^𝜙

(

𝑥, 𝑡

;

𝑛

), it can be shown that the

dependence is linear, see for example [Flügge (1972)]:

^𝜙(𝑥, 𝑡;𝑛) = −^

𝛷(𝑥, 𝑡)·𝑛,(C.37)

where the minus sign is conventional and

𝛷

is referred to as flux tensor. The proof of this

linear relation is known as

Cauchy

’s tetrahedron argument. The representation in Eq.

(C.37)

is required for the localization of the global balance law. It is important to note that

^𝜙

is not a

priori a linear function of

𝑛

. Rather it is a consequence of the balance law itself and the given

assumptions.

Note that one step in the derivation of Eq.

(C.37)

is the pseudo localization by means of

the mean value theorem applied to a small regular domain. Hence, any contributions from

singular surfaces as in the general balance law in Eq.

(C.1)

would vanish and have no effect on

the obtained linear relation. A less well-known, but similar argument can be used to obtain a

representation for the non-convective flux

𝑓

in a general surface flux balance. For a non-material

𝑒

𝜏

𝑒

𝑇

𝑎1

𝑎2

𝑎3

ℓ2

ℓ1

ℓ3

𝑆

𝜕𝑆

𝑏

Fig. C.2:

Depiction of a surface

𝑆

with normal vector

𝑒

. The boundary line

𝜕𝑆

has the tangential vector

𝜏

and the binormal vector

𝑏

𝜏×𝑒

. From the surface,

𝑆

a small triangular patch

𝑇

is cut free. The

boundary

𝜕𝑇

consists of three straight line segments

ℓ1

ℓ2

and

ℓ3

with the tangent vectors

𝑎1

𝑎2

and

𝑎3, respectively.

arbitrary regular surface, 𝑆the balance reads

d𝑡ˆ

𝑆

𝛾·𝑒d𝐴=˛

𝜕𝑆 (︀𝑓−[𝛾×(𝑣−𝑤)] ·𝜏)︀dℓ+ˆ

𝑆

(𝑝+𝑠)·𝑒d𝐴 , (C.38)

where

𝑒

is the normal vector to the surface as depicted in Fig. C.2. The quantity

𝛾

is the

surface flux density and

𝑓

is the non-convective boundary flux. Again, possible contributions

Section C.2. An extension of CAUCHY’s tetrahedron argument to surfaces

xvi

from singular lines can be neglected for the following analysis without loss of generality. The

additional terms

𝑝

and

𝑠

represent the production and supply terms. The same symbols as in

Eq.

(C.36)

were used even though they are not the same. Since

𝑆

is an arbitrary surface, the

balance must also be valid for the small triangular patch

𝑇

from Fig. C.2. Note that since the

boundary line 𝜕𝑇 is closed, it follows by means of Stokes’ theorem that

𝜕𝑇

𝜏dℓ=˛

𝑇

𝑒·(∇s×1) d𝐴=0⇒

∑︁

𝛼=1 ˆ

ℓ𝛼

𝜏dℓ=0,(C.39)

where on the

𝛼

-th line segment

ℓ𝛼

the tangent vector to the curve is given by

𝜏

𝑎𝛼

. Therefore,

the integral along the closed path yields

0=˛

𝜕𝑆

𝜏dℓ=𝑎1Δℓ1+𝑎2Δℓ2+𝑎3Δℓ3⇒𝑎3=−Δℓ1

Δℓ3

𝑎1−Δℓ2

Δℓ3

𝑎2.(C.40)

By means of the surface transport theorem from Eq.

(C.14)

the left-hand side of the surface

balance in Eq. (C.38) is modified such that the balance can be written as

𝑇

𝑔d𝐴−˛

𝜕𝑇 (︀𝑓−[𝛾×𝑣]·𝜏)︀dℓ= 0 , 𝑔 =[︂𝜕𝛾

𝜕𝑡 + (∇s·𝛾)𝑤−𝑝−𝑠]︂·𝑒.(C.41)

It is now assumed that the non-convective boundary flux

𝑓

depends parametrically upon the

tangent vector,

𝑓

(

𝑥, 𝑡

;

𝜏

). Bearing in mind that (

𝑎×𝑏

)

·𝑐

= (

𝑎⊗𝑏

)

··⟨3⟩

𝜖·𝑐

and by applying

the mean value theorem to each integral one obtains

¯𝑔(𝑥, 𝑡)1

2Δℓ1Δℓ2−

∑︁

𝛼=1

𝑓(𝑥, 𝑡;𝑎𝛼)Δℓ𝛼+ (𝛾⊗𝑣)··⟨3⟩

𝜖·˛

𝜕𝑇

𝜏dℓ= 0 ,(C.42)

where quantities with a bar are averaged. The last term vanishes since the boundary of the

surface is closed. Furthermore, in the limiting cases Δ

ℓ1→

0and Δ

ℓ2→

0the surface area of

𝑇shrinks faster than the lengths themselves. Hence, the first term with ¯𝑔can be neglected as

well, which results in ∑︀3

𝛼=1 ¯

𝑓(𝑥, 𝑡;𝑎𝛼)Δℓ𝛼= 0. Together with Eq. C.40 it follows that

𝑓(︂𝑥, 𝑡;−Δℓ1

Δℓ3

𝑎1−Δℓ2

Δℓ3

𝑎2)︂=−Δℓ1

Δℓ3

𝑓(𝑥, 𝑡;𝑎1)−Δℓ2

Δℓ3

𝑓(𝑥, 𝑡;𝑎2).(C.43)

This encapsulates precisely the conditions for linearity, namely additivity, and homogeneity

of degree one. Hence, the function

𝑓

(

𝑥, 𝑡

;

𝜏

)is linear in its third argument

𝜏

. In the limit

ℓ𝑖→

0the mean value

𝑓

and the local function value

𝑓

both approach the same value.

Thus,

𝑓

(

𝑥, 𝑡

;

𝜏

)is a linear function in its third argument as well and there exists a vector

𝑓=^

𝑓(𝑥, 𝑡)such that ^

𝑓(𝑥, 𝑡;𝜏) = ^

𝑓(𝑥, 𝑡)·𝜏.(C.44)

This concludes the proof of the linear dependence of the boundary flux on the tangent vector.

C.3 Cylindrical coordinate transformations

Two cylindrical coordinate systems as in Fig. 6.10 are considered. The dashed system

{𝜌′, 𝜙′, 𝑧′}

is translated by

𝐷𝑒0

𝑧

and rotated by an angle

𝛼

around the shared direction

𝑒0

𝑥

𝑒′

𝑥

with

respect to the reference system

{𝜌0, 𝜙0, 𝑧′}

with the global Cartesian basis

{𝑒0

𝑥,𝑒0

𝑦,𝑒0

𝑧}

. The

Chapter C. Mathematical identities and coordinate transformations

On the electromagnetic coupling problem xvii

rotated cylindrical base vector are readily expressed with respect to the global Cartesian basis:

𝑒′

𝜌= cos(𝜙′)𝑒0

𝑥+ sin(𝜙′)[︀cos(𝛼)𝑒0

𝑦−sin(𝛼)𝑒0

𝑧]︀,

𝑒′

𝜙=−sin(𝜙′)𝑒0

𝑥+ cos(𝜙′)[︀cos(𝛼)𝑒0

𝑦−sin(𝛼)𝑒0

𝑧]︀,

𝑒′

𝑧= sin(𝛼)𝑒0

𝑦+ cos(𝛼)𝑒0

𝑧.

(C.45)

From the connection of the position vectors,

𝑥0=𝑥′+𝐷𝑒0

𝑧⇒𝜌0𝑒0

𝜌+𝑧0𝑒0

𝑧=𝜌′𝑒′

𝜌+𝑧′𝑒′

𝑧+𝐷𝑒0

𝑧,(C.46)

it follows via projection that

𝜌0cos(𝜙0) = 𝜌′cos(𝜙′),

𝜌0sin(𝜙0) = 𝜌′sin(𝜙′) cos(𝛼) + 𝑧′sin(𝛼),

𝑧0=−𝜌′sin(𝜙′) sin(𝛼) + 𝑧′cos(𝛼) + 𝐷 ,

(C.47)

which then results in Eqs.

(C.49)

. Furthermore, the corresponding cylindrical base vectors are

connected via:

𝑒0

𝜉= cos(𝜙0)𝑒0

𝑥+ sin(𝜙0)𝑒0

𝑦

= cos(𝜙0)𝑒′

𝑥+ sin(𝜙0) cos(𝛼)𝑒′

𝑦+ sin(𝜙0) sin(𝛼)𝑒′

𝑧

=[︀cos(𝜙0) cos(𝜙′) + sin(𝜙0) cos(𝛼) sin(𝜙′)]︀𝑒′

𝜉+

+[︀sin(𝜙0) cos(𝛼) cos(𝜙′)−cos(𝜙0) sin(𝜙′)]︀𝑒′

𝜙+ sin(𝜙0) sin(𝛼)𝑒′

𝑧,

𝑒0

𝜙=−sin(𝜙0)𝑒0

𝑥+ cos(𝜙0)𝑒0

𝑦

=−sin(𝜙0)𝑒′

𝑥+ cos(𝜙0) cos(𝛼)𝑒′

𝑦+ cos(𝜙0) sin(𝛼)𝑒′

𝑧

=−sin(𝜙0)(︀cos(𝜙′)𝑒′

𝜉−sin(𝜙′)𝑒′

𝜙)︀+ cos(𝜙0) cos(𝛼)(︀sin(𝜙′)𝑒′

𝜉+ cos(𝜙′)𝑒′

𝜙)︀+ cos(𝜙0) sin(𝛼)𝑒′

𝑧

=[︀cos(𝜙0) cos(𝛼) sin(𝜙′)−sin(𝜙0) cos(𝜙′)]︀𝑒′

𝜉+

+[︀sin(𝜙0) sin(𝜙′) + cos(𝜙0) cos(𝛼) cos(𝜙′)]︀𝑒′

𝜙+ cos(𝜙0) sin(𝛼)𝑒′

𝑧,

𝑒0

𝑧=−sin(𝛼)𝑒′

𝑦+ cos(𝛼)𝑒′

𝑧

=−sin(𝛼) sin(𝜙′)𝑒′

𝜉−sin(𝛼) cos(𝜙′)𝑒′

𝜙+ cos(𝛼)𝑒′

𝑧.

(C.48)

𝜉0=√︁(𝜉′)2[︀cos2(𝜙′) + sin2(𝜙′) cos2(𝛼)]︀+ 2𝜉′𝑧′sin(𝜙′) sin(𝛼) cos(𝛼)+(𝑧′)2sin2(𝛼),

tan(𝜙0) = 𝜉′sin(𝜙′) cos(𝛼) + 𝑧′sin(𝛼)

𝜉′cos(𝜙′),

𝑧0=−𝜉′sin(𝜙′) sin(𝛼) + 𝑧′cos(𝛼) + 𝐷 ,

(C.49)

as well the basis vector representations:

𝑒0

𝜉=[︀cos(𝜙0) cos(𝜙′) + sin(𝜙0) cos(𝛼) sin(𝜙′)]︀𝑒′

𝜉+

+[︀sin(𝜙0) cos(𝛼) cos(𝜙′)−cos(𝜙0) sin(𝜙′)]︀𝑒′

𝜙+ sin(𝜙0) sin(𝛼)𝑒′

𝑧,

𝑒0

𝜙=[︀cos(𝜙0) cos(𝛼) sin(𝜙′)−sin(𝜙0) cos(𝜙′)]︀𝑒′

𝜉+

+[︀sin(𝜙0) sin(𝜙′) + cos(𝜙0) cos(𝛼) cos(𝜙′)]︀𝑒′

𝜙+ cos(𝜙0) sin(𝛼)𝑒′

𝑧,

𝑒0

𝑧=−sin(𝛼) sin(𝜙′)𝑒′

𝜉−sin(𝛼) cos(𝜙′)𝑒′

𝜙+ cos(𝛼)𝑒′

𝑧,

(C.50)

are derived.

Section C.3. Cylindrical coordinate transformations

xviii

To this end, the position vectors, as well as base vectors, have to be transformed from the

𝒪-system to dashed one according to

𝑥0=𝑥′+𝐷𝑒0

𝑧,𝑒0

𝑥=𝑒′

𝑥,𝑒0

𝑦= cos(𝛼)𝑒′

𝑦+ sin(𝛼)𝑒′

𝑧,𝑒0

𝑧=−sin(𝛼)𝑒′

𝑦+ cos(𝛼)𝑒′

𝑧.

In order to obtain the

𝑒0

𝜉= cos(𝜙0)𝑒0

𝑥+ sin(𝜙0)𝑒0

𝑦= cos(𝜙0)𝑒′

𝑥+ sin(𝜙0) cos(𝛼)𝑒′

𝑦+ sin(𝜙0) sin(𝛼)𝑒′

𝑧,(C.51)

Chapter C. Mathematical identities and coordinate transformations