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Bonart, H., Kahle, C., & Repke, J.-U. (2019). Comparison of energy stable simulation of moving contact
line problems using a thermodynamically consistent Cahn–Hilliard Navier–Stokes model. Journal of
Computational Physics, 399, 108959. https://doi.org/10.1016/j.jcp.2019.108959
Henning Bonart, Christian Kahle, Jens-Uwe Repke
Comparison of energy stable simulation of
moving contact line problems using a
thermodynamically consistent Cahn–
Hilliard Navier–Stokes model
Accepted manuscript (Postprint) Journal article |

Comparison of Energy Stable Sim ulation of Mo ving
Con tact Line Problems using a Thermo dynamically
Consisten t Cahn–Hilliard Na vier–Stok es Mo del ?
Henning Bonart a, ∗ , Christian Kahle b , Jens-Uw e Repk e a
a Pr o c ess Dynamics and Op er ations Gr oup, T e chnische Universität Berlin,
10623 Berlin, Germany
b Center for Mathematic al Scienc es, T e chnische Universität München,
85748 Gar ching b ei München, Germany
Abstract
Liquid droplets sliding along solid surfaces are a frequen tly observed phenomenon
in nature, e.g., raindrops on a leaf, and in ev eryda y situations, e.g., drops of
w ater in a drinking glass. T o mo del this situation, w e use a phase field ap-
proac h. The bulk mo del is giv en by the thermodynamically consistent Cahn–
Hilliard Na vier–Stok es mo del from [Ab els et al., Math. Mo d. Meth. Appl. Sc.,
22(3), 2012]. T o mo del the con tact line dynamics w e apply the generalized
Na vier b oundary condition for the fluid and the dynamically adv ected b oundary
con tact angle condition for the phase field as derived in [Qian et al., J. Fluid Mec h.,
564, 2006]. In recen t y ears sev eral sc hemes w ere prop osed to solv e this mo del
n umerically . While they widely differ in terms of complexity , they all fulfill cer-
tain basic prop erties when it comes to thermo dynamic consistency . Ho w ev er, an
accurate comparison of the influence of the sc hemes on the moving con tact line
is rarely found. Therefore, w e thoughtfully compare the qualit y of the numeri-
cal results obtained with three differen t schemes and t wo differen t bulk energy
p oten tials. Esp ecially , w e discuss the influence of the different sc hemes on the
apparen t con tact angles of a sliding droplet.
Keywor ds: Multiphase flo ws, Drop phenomena, Con tact line dynamics, Phase
field mo deling
2000 MSC: 35Q30, 35Q35, 76D05, 76M10, 76T99
? The first and third author ac knowledge the North-German Supercomputing Alliance
(HLRN) for pro viding HPC resources that hav e con tributed to the researc h results rep orted
in this pap er and thank the German Researc h F oundation (DF G) for the financial supp ort
within the pro ject RE 1705/16-1. The second author gratefully ac knowledges the support by
the German Researc h F oundation (DF G) through the International Researc h T raining Group
IGDK 1754 “Optimization and Numerical Analysis for P artial Differential Equations with
Nonsmo oth Structures".
∗ Corresp onding author
Email addr esses: [email protected] (Henning Bonart ),
[email protected] (Christian Kahle), [email protected] (Jens-Uw e
Repk e)
Pr eprint submitte d to Journal of Computational Physics Septemb er 5, 2019

1. In tro duction
Liquid droplets sliding along solid surfaces are a frequen tly observ ed phe-
nomenon in nature, e.g., raindrops on a leaf, and in ev eryda y situations, e.g.,
drops of w ater in a drinking glass. F urthermore, sliding droplets (and conse-
quen tly the suppression of those) are crucial in man y industrial applications
suc h as coating or pain ting and separation or reaction pro cesses in volving m ul-
tiple phases and thin liquid films. The p osition where the in terface b et w een the
sliding droplet and the surrounding fluid in tersects the solid surface is the mo v-
ing con tact line (or contact point if a t wo dimensional problem is observ ed). F or
details ab out liquids on surfaces and mo ving con tact lines see the reviews [1, 2]
and the references therein. In a con tin uum approac h, applying the common
no-slip b oundary condition at the solid surface close to the con tact line leads
to a non-ph ysical, logarithmically diverging energy dissipation. One p ossibilit y
to circum v en t this difficulty is the coupling of the incompressible Na vier–Stokes
equations with the Cahn–Hilliard equation [3]. This phase field metho d mo dels
the in terface b et w een the fluids with a diffuse in terface of p ositiv e thic kness and
describ es the distribution of the differen t fluids b y a smo oth indicator function.
Esp ecially , the Cahn–Hilliard equation allo ws the con tact line to mov e naturally
on the solid surface due to a diffusiv e flux across the interface, whic h is driv en
b y the gradien t of the c hemical p oten tial. F urthermore, the phase fi eld metho d
is able to calculate top ological c hanges lik e breakup of droplets or merging in-
terfaces [4]. F or example in exp eriments b y [5, 6], it is found that during the
rapid spreading of a droplet the con tact angle can differ from the equilibrium
angle giv en b y Y oung’s equation. T o allo w for nonequilibrium con tact angles,
[3] prop oses a relaxation of the static con tact angle b oundary condition, see
Section 1.1, and [7] extends this approac h to include the slip at the con tact line
stemming from the uncomp ensated Y oung stress.
In [8] a thermo dynamically consisten t Cahn–Hilliard Na vier–Stok es phase
field mo del is prop osed to describ e the dynamics of the t wo phases in the bulk
domain. It is v alid also for differen t densities of the inv olved fluids, but sp ecific
con tact line dynamics are not included. Recen tly , several n umerical schemes
for solving this system ha ve been prop osed, see for example, [9–13]. All these
sc hemes are thermo dynamically consisten t in the sense, that they mimic the
energy la w from [8] in the time discrete or ev en in the fully discrete setting.
They range from fully coupled and nonlinear to decoupled and linear, where
decoupled means, that the Na vier–Stokes and the Cahn–Hilliard equations are
solv ed sequen tially .
These sc hemes are extended to the Cahn–Hilliard Navier–Stok es system with
mo ving con tact lines in v arious pap ers. Here the concepts from the aforemen-
tioned pap ers for the discretization of the bulk equations are straightforw ardly
applied. F or the case of equal densities, sc hemes are prop osed, e.g., in [14–17]
and for the case of differen t densities in [11, 18]. The mo del from [8] contains an
2

additional flux term in the momen tum equation, that renders the mo del ther-
mo dynamically consisten t. This term is often neglected, see e.g., [19]. F or the
resulting mo del sev eral discretization sc hemes are prop osed and w e refer to the
references in [18] for details. In all these simulations in v olving mo ving contact
lines a p olynomial bulk energy p otential is applied. In con trast, w e include a
double obstacle p oten tial, whic h is subsequently relaxed, see [20]. In [9, 21] in a
n umerical b enc hmark setting the results with this kind of energy are t ypically
closer to sharp in terface numeric than with the polynomial p oten tial.
T o prepare future researc h on the passive con trol of droplets sliding on struc-
tured or c hemically patterned surfaces, we extend the w ork of [9] in this pap er
to the case of mo ving con tact line dynamics and compare the n umerical re-
sults with the corresp onding decoupled scheme from, e.g., [11] and a fully linear
sc heme, that b oth uses decoupling and stabilization as in [16]. W e test b oth
the p olynomially and the relaxed double-obstacle bulk energy p otential, so that
in total w e compare six differen t com binations of bulk energy p otentials and
solution sc hemes.
The remainder of the pap er is organized as follo ws. In the second part of
the in tro duction, Section 1.1, w e introduce the con tin uous mo del as w ell as the
bulk energy p oten tials and the con tact line energies. Afterw ards, w e deriv e a
w eak form ulation in Section 2 and the n umerical sc hemes in Section 3. In Sec-
tion 4.1 w e compare the differen t com binations at first in the bulk without an y
con tact line. Finally , w e compare sim ulation results of sliding droplets on in-
clined surfaces to in v estigate the accuracy and efficiency of the linearization and
decoupling strategies as w ell as the bulk energy p oten tials for mo ving contact
line problems in Section 4.2. W e conclude our w ork in Section 5.
1.1. Mo del
In the fluid domain w e consider the thermo dynamically consisten t mo del for
the sim ulation of t w o-phase flo w presen ted in [8], in the v ariant for nonlinear
densit y functions prop osed in [22, Eq. 1.10]. T o mo del the contact line dynamics
w e use generalized Na vier b oundary conditions for the v elo cit y field together
with dynamically adv ected b oundary conditions for the phase field as prop osed
in [7].
In strong form the mo del reads as follo ws. Let Ω ⊂ R d with d ∈ { 2 , 3 } denote
an op en, p olygonally/p olyhedrally b ounded Lipsc hitz domain and I = (0 , T ]
with 0 < T < ∞ denote a time in terv al. The outer unit normal on ∂ Ω is ν Ω .
A t time t ∈ I the primal v ariables are giv en b y the v elo cit y field v , the pressure
field p , the phase field ϕ and the c hemical p oten tial µ . They satisfy the follo wing
3

system of equations
ρ∂ t v + (( ρv + J ) · ∇ ) v + R v
2 − div (2 η D v ) + ∇ p = − ϕ ∇ µ + ρg in Ω , (1)
− div ( v )=0 in Ω , (2)
∂ t ϕ + v · ∇ ϕ − b ∆ µ = 0 in Ω , (3)
− σ  ∆ ϕ + σ
 W 0 ( ϕ ) = µ in Ω , (4)
v · ν Ω = 0 on ∂ Ω , (5)
[2 η D v ν Ω + l ( ϕ ) v tan − L ( ϕ ) ∇ ϕ ] × ν Ω = 0 on ∂ Ω , (6)
r B + L ( ϕ )=0 on ∂ Ω , (7)
∇ µ · ν Ω = 0 on ∂ Ω , (8)
where w e abbreviate J := − b ∂ ρ
∂ ϕ ∇ µ , R := − b ∇ ∂ ρ
∂ ϕ · ∇ µ , B := ∂ t ϕ + v · ∇ ϕ ,
L := σ  ∇ ϕ · ν Ω + γ 0 ( ϕ ) . The gra vitational acceleration is denoted b y g and
w e abbreviate 2 D v := ∇ v + ( ∇ v ) t . The function W ( ϕ ) denotes a dimension-
less p oten tial of double-w ell t yp e, with t wo strict minima at ± 1 . W e refer to
Remark 2 for a discussion of p ossible c hoices for W . W e form ulate (1) with
a shifted pressure v ariable p = p phy s − µϕ , where p phy s denotes the ph ysical
pressure.
The con tact line energy is denoted b y γ , see Remark 3. The strictly p ositiv e,
constan t parameters for the equations in Ω are given b y the mobility b > 0 , the
scaled surface tension σ , see Remark 2, and the in terfacial thic kness parameter
 . The constant mobilit y is used for simplicit y but the following is also v alid for
mobilities that dep end on ϕ .
The (nonlinear) densit y function is denoted b y ρ ≡ ρ ( ϕ ) > 0 and satisfies
ρ ( − 1) = ρ 1 and ρ (1) = ρ 2 , with ρ 2 > ρ 1 > 0 denoting the constan t densities of
the t w o in v olv ed fluids. The (nonlinear) viscosity function is η ≡ η ( ϕ ) > 0 and
satisfies η ( − 1) = η 1 and η (1) = η 2 , with η 1 , η 2 denoting the viscosities of the
in v olv ed fluids.
Remark 1 (Nonlinear densit y and viscosit y) . In gener al ther e is no quantitative
upp er b ound available for ϕ and thus in p articular | ϕ | > 1 is p ossible. Thus a
line ar r elation b etwe en ϕ and ρ c an le ad to ne gative densities in pr actic e. This
might app e ar for esp e cial ly lar ge density r atios, c omp ar e for example [11, R em.
4.1]. Note, that ϕ c an b e pr oven to b e b ounde d in L ∞ if W 00 is uniformly
b ounde d, se e e.g. [23].
It is a c ommon appr o ach to cut ϕ when inserting it into the line ar function
for ρ , se e e.g. [18]. This le ads to a nonsmo oth r elation b etwe en ϕ and ρ . How-
ever, as we r e quir e differ entiability of ρ to define R and J this is not admissible
her e. A se c ond appr o ach is to clip ρ of at some p ositive value, se e e.g. [11, 24].
This le ads to a uniform b ound on ρ b ase d on the A two o d numb er At = ρ 2 − ρ 1
ρ 2 + ρ 1 .
Her e we use the latter appr o ach and define ρ as the fol lowing smo oth, monotone
4

and strictly p ositive function
ρ ( ϕ ) =















1
4 ρ 1 if ϕ ≤ − A t − 1 ,
1
ρ 1  ρ 2 − ρ 1
2 ϕ + ρ 2 + ρ 1
2  2 + 1
4 ρ 1 if − A t − 1 < ϕ < − 1 − ρ 1
ρ 2 − ρ 1 ,
ρ 2 − ρ 1
2 ϕ + ρ 2 + ρ 1
2 if − 1 − ρ 1
ρ 2 − ρ 1 ≤ ϕ ≤ 1 + ρ 1
ρ 2 − ρ 1 ,
− 1
ρ 1  ρ 2 − ρ 1
2 ϕ − ρ 2 + ρ 1
2  2 + ρ 2 + 3
4 ρ 1 if 1 + ρ 1
ρ 2 − ρ 1 < ϕ < A t − 1 ,
ρ 2 + 3
4 ρ 1 if A t − 1 ≤ ϕ.
(9)
F or a discussion we r efer to [24, R em. 2.1]. The nonline ar visc osity η ( ϕ ) c an
b e define d analo gously.
W e note, that the total mass R Ω ρ ( ϕ ) dx is only c onserve d if ρ ( ϕ ) is a line ar
function on the (a-priori unknown) image of ϕ , se e e.g., [9, R em. 1], while
R Ω ϕ dx is a c onserve d quantity,
As b oundary data w e use generalized Na vier b oundary conditions for the
v elo cit y field and dynamically adv ected con tact angle b oundary conditions for
the t w o-phase equation, see [7, Eq. 4.4, Eq. 4.5]. Here γ denotes the fluid-solid
in terfacial free energy , see [7, Sec. 4], l ( ϕ ) is a slip co efficien t for the generalized
Na vier b oundary condition applied to the tangen tial part of the v elo cit y v tan :=
v − ( v · ν Ω ) ν Ω , while L ( ϕ ) ∇ ϕ × ν Ω is the uncomp ensated Y oung stress and L is
the c hemical p oten tial at the solid surface. The static con tact angle is denoted
b y θ s and r ≥ 0 is a phenological parameter allo wing for nonequilibrium at the
con tact line. F or r ≡ 0 (7) reduces to σ  ∇ ϕ · ν Ω = − γ 0 ( ϕ ) , which means, that
a static con tact angle at the in terface is assumed. F urthermore, for γ 0 ( ϕ ) ≡ 0
(or rather θ s ≡ 90 ◦ , see Remark 3), (7) further simplifies to ∇ ϕ · ν Ω = 0 , whic h
is a no-flux condition for ϕ at the solid surface. The no-slip condition for v is
obtained from (6) b y L ≡ 0 and l → ∞ (or rather the slip length l s ≡ 0 , se e
Remark 12).
Concerning the existence of solutions to (1) to (5) and (8) together with
no-slip for v and a homogeneous Neumann (or no-flux) b oundary condition for
ϕ as w ell as with different assumptions on b and W , w e refer to [22, 24–26].
F or the b oundary conditions considered here we are not a ware of suc h results,
but refer to [27] for the Cahn–Hilliard Na vier–Stok es system with equal den-
sities, to [28] for analytical results for the Cahn–Hilliard system with dynamic
b oundary conditions, and to [11] for a Cahn–Hilliard Na vier–Stok es mo del with
dynamical con tact angle condition, but no-slip condition for the Navier–Stok es
equation. Concerning sh arp in terface limits, we refer to [8] for the bulk model
with homogeneous b oundary conditions. Sharp interface analysis for the model
with equal densities including con tact line dynamics is av ailable in [29].
F or (1) to (5) and (8) together with no-slip for v and no-flux for ϕ , sev eral
thermo dynamically consisten t discretization sc hemes were proposed in the last
y ears. Here, w e refer to [9, 10, 13]. Especially in [9] the influence of spatial
adaptivit y on the fully discrete energy law is discussed. W e further refer to [12],
where the b enefit of using fully coupled schemes is sho wn numerically , and to
5

[30] for an extensiv e discussion of sev eral discretization sc hemes for the bulk en-
ergy p oten tial W poly . F or the full mo del (1)–(8) thermo dynamically consistent
sc hemes are for example prop osed in [17] for the case of constan t density , and
in [18] for the general case. The case with no-slip b oundary condition for v and
dynamically adv ected b oundary condition for ϕ is n umerically and analytically
considered in [11].
Remark 2 (Bulk energy p oten tials) . Thr oughout this work we c onsider p oly-
nomial ly b ounde d p otentials for W . T o state the pr e cise assumptions we split
W = W + + W − with W + denoting the c onvex p art of W and W − denoting the
c onc ave p art. W e assume that W : R → R is c ontinuously differ entiable and
that W and its derivatives W 0
+ and W 0
− ar e p olynomial ly b ounde d, i.e., ther e
exists C > 0 such that
| W ( ϕ ) | ≤ C (1 + | ϕ | 4 ) , | W 0
+ ( ϕ ) | ≤ C (1 + | ϕ | 3 ) , | W 0
− ( ϕ ) | ≤ C (1 + | ϕ | 3 ) .
Note, that these b ounds on the p olynomial de gr e e might b e r elaxe d, se e [9, (A3)],
and that these assumptions ar e use d to show the existenc e of discr ete solutions.
These assumptions ar e for example fulfil le d by the c ommonly use d p olynomial
p otential
W poly ( ϕ ) := 1
4 (1 − ϕ 2 ) 2 , W poly 2 ( ϕ ) := ( 1
4 (1 − ϕ 2 ) 2 if | ϕ | ≤ 1 ,
( | ϕ | − 1) 2 else,
wher e W poly 2 is a mo dific ation of W poly that guar ante es an L ∞ b ound on ϕ , se e
[23].
A nother p otential that fulfil ls the assumptions is
W s ( ϕ ) := 1
2  1 − ( ξ ϕ ) 2 + sλ ( ξ ϕ ) 2  + θ ,
wher e λ ( x ) := max(0 , x − 1) + min(0 , x + 1) , θ := 1
2( s − 1) and ξ := s
s − 1 ar e
chosen such that W ( ± 1) ≡ 0 ar e the two minima of W s . Her e s  1 is a
p enalization p ar ameter. It app e ars as Mor e au–Y osida r elaxation of the double
obstacle p otential W ∞ , se e [20, 31]. In a synthetic rising bubble b enchmark,
[32], our r esults with this p otential ar e typic al ly closer to the r esults fr om sharp
interfac e metho ds than with the p otential W pol y , se e [9, T ab. 1].
In the fol lowing, whenever we use the letter W , we me an any of the thr e e
mentione d bulk ener gy p otentials.
In pr ep ar ation of later r esults, we state the splittings of the p otentials W into
W ( ϕ ) = W + ( ϕ ) + W − ( ϕ ) . These ar e
W poly
+ ( ϕ ) = 1
4 ϕ 4 − 1
4 , W poly
− ( ϕ ) = 1
2 (1 − ϕ 2 ) ,
W poly 2
+ ( ϕ ) = ( 1
4 ϕ 4 − 1
4 if | ϕ | ≤ 1 ,
( | ϕ | − 1) 2 − 1
2 (1 − ϕ 2 ) if | ϕ | > 1 , W poly 2
− ( ϕ ) = 1
2 (1 − ϕ 2 ) ,
W s
+ ( ϕ ) = s
2 λ ( ξ ϕ ) 2 + θ , W s
− ( ϕ ) = 1
2 (1 − ( ξ ϕ ) 2 ) .
6

These splittings ar e not unique, and we r efer for example to [33] for an al-
ternative splitting of W poly 2 . W e further r efer to [30] for a discussion on the
dissip ation that is intr o duc e d by the c onvex-c onc ave splitting and also for an
elab or ate d discussion on the dissip ation that in gener al is intr o duc e d by splitting
W . In our numeric al tests, splittings that have a quadr atic c onvex p art and thus
give line ar systems, typic al ly le ad to br o ader interfac es during the simulation
and r e quir e smal ler time steps to pr event this effe ct. Thus it is favor able to use
non-line ar systems as obtaine d by the pr op ose d splittings ab ove.
T o define the sc ale d surfac e tension σ we intr o duc e the c onstant c W as
c − 1
W = R ∞
−∞ 2 W (Φ 0 ( z )) dz = R ∞
−∞ ( ∂ z Φ 0 ( z )) 2 dz , wher e Φ 0 denotes the first or der
appr oximation of ϕ dep ending on W . It satisfies Φ 0 ( z ) z z = W 0 (Φ 0 ( z )) , se e [8,
Se c. 4.3.3]. Then σ = c W σ 12 , wher e σ 12 denotes the physic al value of the sur-
fac e tension b etwe en phase 1 and phase 2 . As the dynamics of the diffuse mo del
dep end on the p articular form of W , this sc aling is ne c essary to guar ante e that
the same sharp interfac e dynamic is appr oximate d indep endently of W . Using
W poly and W poly 2 it holds Φ 0 ( z ) = tanh( z / √ 2) and c W = 3
2 √ 2 . F or W s one
obtains by elementary c alculation
Φ 0 ( z ) = 




− Φ 0 ( − z ) if z < 0 ,
√ ξ − 1 sin( ξ z ) if 0 ≤ z ≤ z 0 := ξ − 1 arctan( √ s − 1) ,
1 − s − 1 exp( − ξ √ s − 1( z − z 0 )) if z > z 0 ,
and
c − 1
W = (1 − s − 2 ) arctan( √ s − 1) + s − 2 ( s + 2) √ s − 1 .
F or s → ∞ we r e c over the wel l-known sc aling c W = 2
π for the double-obstacle
p otential.
Remark 3 (Con tact line energy) . The b asic formula to derive the c ontact line
ener gy is given by Y oung’s law, namely
σ s 1 − σ s 2 = σ 12 cos θ s .
Her e σ s 1 and σ s 2 denote the physic al surfac e tensions b etwe en phase 1 ( ϕ = − 1 )
and the solid ( σ s 1 ) and phase 2 ( ϕ = 1 ) and the solid ( σ s 2 ). F urther σ 12
denotes the surfac e tension b etwe en phase 1 and phase 2 and θ s denotes the static
e quilibrium c ontact angle b etwe en the solid and the interfac e and is me asur e d in
phase 2.
W e use the ansatz
γ ( ϕ ) := σ s 1 + σ s 2
2 − σ 12 cos θ s ϑ ( ϕ )
and cho ose ϑ ( ϕ ) to fulfil l
γ ( − 1) = σ s 1 , γ (0) = σ s 1 + σ s 2
2 , γ (1) = σ s 2 , γ 0 ( ± 1) = 0 .
7

In p articular it holds, that ϑ ( − 1) = − 1
2 and ϑ (1) = 1
2 . Her e, the unsc ale d
value of the surfac e tension app e ars as c an b e shown by matche d asymptotic
exp ansions, se e [8, Se c. 4.3.4].
Common choic es for ϑ c ontain the sine function ϑ sin ( ϕ ) := 1
2 sin( π
2 ϕ ) , for
example pr op ose d in [7, Se c. 4], or a cubic p olynomial ϑ poly ( ϕ ) = 1
4 (3 ϕ − ϕ 3 ) , for
example pr op ose d in [29]. A n alternative is given in [34]. Her e the assumption
of e quip artition of ener gy, i.e., 
2 |∇ ϕ | 2 ≈ 1
 W ( ϕ ) , is use d to derive ( ϑ W ) 0 ( ϕ ) =
c W p 2 W ( ϕ ) . Final ly, we state a c ontact line ener gy, that is the sum of a c onvex
and a c onc ave function namely ϑ cc ( ϕ ) = ϑ cc
+ ( ϕ ) + ϑ cc
− ( ϕ ) with
ϑ cc
+ ( ϕ ) = 




− 1
2 if ϕ ≤ − 1 ,
1
2 ( ϕ + 1) 2 − 1
2 if ϕ ∈ ( − 1 , 0) ,
ϕ if ϕ ≥ 0 ,
ϑ cc
− ( ϕ ) = 




0 if ϕ ≤ 0 ,
− 1
2 ϕ 2 if ϕ ∈ (0 , 1) ,
1
2 − ϕ if ϕ ≥ 1 .
Her e ϑ cc
+ is c onvex and ϑ cc
− is c onc ave and ϑ cc ∈ C 1 , 1 ( R ) with ϑ 00 ∈ L ∞ ( R ) .
Note, that for any ϑ that has a b ounde d se c ond derivative, we c an define a
c onvex-c onc ave splitting via
ϑ + ( ϕ ) = ϑ ( ϕ ) + 1
2 max
φ ∈ R ( ϑ 00 ( φ )) ϕ 2 , ϑ − ( ϕ ) = − 1
2 max
φ ∈ R ( ϑ 00 ( φ )) ϕ 2 ,
c omp ar e [35]. This is very similar to the stabilization appr o ach, pr op ose d for
example in [16], that essential ly r esembles one of Eyr e’s line ar schemes [30]. In
the fol lowing we always assume a c onvex-c onc ave splitting of γ . This appr o ach
c an also b e use d for the p otential W .
Remark 4. T o the b est of our know le dge, ther e is no c onsent yet which c om-
binations of bulk ener gy p otential and c ontact line ener gy ar e most appr opriate
fr om b oth a physic al and numeric al p oint of view. F r om an analytic al p oint of
view, al l c ombinations ar e r e asonable that le ad to the c orr e ct sharp interfac e
limit, se e [29] for r esults on formal sharp interfac e asymptotics. Her e, the au-
thors use the c ombination of W poly and ϑ poly . However, this topic is subje ct
to futur e work. F urther note, that using the notation fr om [29] we ar e in the
setting L d = O (  ) , and V s = O (1) .
2. The w eak form ulation
W e next deriv e the weak form ulation that is the basis for our numerical
sc heme prop osed in Section 3. W e assume sufficien t regularit y of all app earing
functions. Multiplying (3) with ∂ ρ
∂ ϕ w e observe
∂ t ρ + div ( ρv + J ) = R. (10)
Note that if ρ is a nonlinear function R 6 = 0 holds and th us mass conserv ation
can b e violated as so on as a nonlinear function for ρ is used to guarantee ρ > 0 .
8

Note that the conserv ation of ϕ is not affected. Using (10) the momen tum
equation (1) can equiv alen tly b e written as
∂ t ( ρv ) + div ( v ⊗ ( ρv + J )) − R v
2 − div (2 η D v ) + ∇ p = − ϕ ∇ µ + ρg , (11)
see [22, Eq. (1.12)]. W e stress that this reform ulation is indep enden t of the
actual b oundary condition.
T o define the w eak formulation w e multiply both (1) and (11) by a solenoidal
test function 1
2 w that satisfies w | ∂ Ω · ν Ω = 0 and sum up the equations to achiev e
1
2 Z Ω
( ρ∂ t v + ∂ t ( ρv )) · w dx − Z Ω
div (2 η D v ) · w dx + Z Ω
( ϕ ∇ µ − ρg ) · w dx
+ 1
2 Z Ω
(( ρv + J ) · ∇ ) v · w dx + 1
2 Z Ω
div ( v ⊗ ( ρv + J )) · w dx = 0 .
Using in tegration by parts together with the boundary conditions v · ν Ω = 0 and
∇ µ · ν Ω = 0 w e observ e
1
2 Z Ω
(( ρv + J ) · ∇ ) v · w dx + 1
2 Z Ω
div ( v ⊗ ( ρv + J )) · w dx
= 1
2 Z Ω
(( ρv + J ) · ∇ ) v · w − (( ρv + J ) ∇ ) w · v dx
=: a ( ρv + J, v , w ) .
Note that a ( · , v , v )=0 holds. Using in tegration b y parts for the viscous stress
w e observ e
− Z Ω
div (2 η D v ) · w dx = Z Ω
2 η D v : D w dx − Z ∂ Ω
2 η D v ν Ω · w ds,
= Z Ω
2 η D v : D w dx + Z ∂ Ω
( l ( ϕ ) v tan + r B ∇ ϕ ) · w ds
where D v : D w := P n
ij =1 ( D v ) ij ( D w ) ij and w e use the b oundary conditions (6)
and (7).
The w eak form of (3)–(4) is deriv ed b y the standard pro cedure. Summarizing
the equations, w e obtain the following w eak form of (1)–(8):
Definition 5 (The w eak form ulation) . Find sufficiently smo oth v , µ, ϕ , with v
solenoidal, v · ν Ω = 0 , such that for al l w , ψ , φ , with w solenoidal, the fol lowing
9

e quations ar e satisfie d:
1
2 Z Ω
( ρ∂ t v + ∂ t ( ρv )) · w dx + a ( ρv + J, v , w ) + Z Ω
2 η D v : D w dx
+ Z ∂ Ω
( l ( ϕ ) v tan + r B ( ϕ t , ϕ, v ) ∇ ϕ ) · w ds − Z Ω
( − ϕ ∇ µ + ρg ) · w dx = 0 , (12)
Z Ω
ϕ t ψ dx − Z Ω
ϕv · ∇ ψ dx + Z Ω
b ∇ µ · ∇ ψ dx = 0 , (13)
Z Ω
σ  ∇ ϕ · ∇ φ + σ
 W 0 ( ϕ ) φ dx − Z Ω
µφ dx
+ Z ∂ Ω
( r B ( ϕ t , ϕ, v ) + γ 0 ( ϕ )) φ ds = 0 . (14)
The w eak form (12)–(14) allo ws us to deriv e the following energy iden tity .
Theorem 6 (The formal energy iden tit y) . Assume ther e exists a sufficiently
smo oth solution to (12) – (14) . Then the fol lowing ener gy identity holds
d
dt  Z Ω
1
2 ρ | v | 2 dx + σ Z Ω

2 |∇ ϕ | 2 + 1
 W ( ϕ ) dx + Z ∂ Ω
γ ds 
+ Z Ω
2 η | D v | 2 dx + Z Ω
b |∇ µ | 2 dx
+ Z ∂ Ω
l ( ϕ ) | v tan | 2 ds + r Z ∂ Ω | B ( ϕ t , ϕ, v ) | 2 ds = Z Ω
ρg · v dx.
(15)
Note that the ener gy in the system c an only incr e ase by the gr avitational ac c el-
er ation.
Pr o of. Use w ≡ v , Ψ ≡ µ , and Φ ≡ ∂ t ϕ as test functions in (12)–(14) and sum
up the resulting equations.
3. The n umerical sc hemes
F or a practical implemen tation in a finite elemen t scheme w e introduce a
time grid 0 = t 0 < t 1 < . . . < t m − 1 < t m < . . . < t M = T on I = [0 , T ] . F or
the sak e of notational simplicity let the time grid be equidistant with step size
τ > 0 . W e further in tro duce a triangulation T h of Ω in to cells T i , suc h that
T h = S N
i =1 T i co v ers Ω exactly .
On T h w e in tro duce the finite elemen t spaces
V 1 := { v ∈ C (Ω) | v | T i ∈ P 1 } ,
V 2 := { v ∈ C (Ω) d | v | T i ∈ ( P 2 ) 2 , v · ν Ω = 0 } ,
where P k denotes the space of p olynomials of order up to k . W e use V 1 to
define discrete appro ximations ϕ h , µ h , and p h of the corresp onding con tinuous
v ariables, and V 2 to define the discrete appro ximation v h of v . This means that
10

w e use standard T a ylor–Ho o d elemen ts for the Na vier–Stokes part and explicitly
denote the pressure v ariable in the follo wing.
The sc heme reads as follows:
Giv en ϕ m − 1 ∈ V 1 , µ m − 1 ∈ V 1 , and v m − 1 ∈ V 2 , find ϕ m
h ∈ V 1 , µ m
h ∈ V 1 , p m
h ∈ V 1
and v m
h ∈ V 2 , suc h that for all w ∈ V 2 , q ∈ V 1 , Φ ∈ V 1 , and Ψ ∈ V 1 the follo wing
equations hold
1
τ  ρ m + ρ m − 1
2 v m
h − ρ m − 1 v m − 1 , w 
+ a ( ρ m − 1 v m − 1 + J m − 1 , v m
h , w ) + (2 η m − 1 D v m
h , D w ) − ( div w , p m
h )
+( l ( ϕ m − 1 ) v m
h,tan + r B m
h ∇ ϕ m − 1 , w ) ∂ Ω
+( ϕ m − 1 ∇ µ m
h , w ) − ( g ρ m − 1 , w )=0 , (16)
− ( div v m
h , q )=0 , (17)
1
τ ( ϕ m
h − ϕ m − 1 , Ψ) − ( ϕ m − 1 v m
h , ∇ Ψ) + ( b ∇ µ m
h , ∇ Ψ) = 0 , (18)
σ  ( ∇ ϕ m
h , ∇ Φ) + σ
 ( W 0
+ ( ϕ m
h ) + W 0
− ( ϕ m − 1 ) , Φ) − ( µ m
h , Φ)
+ ( r B m
h , Φ) ∂ Ω +  γ 0
+ ( ϕ m
h ) + γ 0
− ( ϕ m − 1 ) , Φ  ∂ Ω = 0 , (19)
with J m − 1 := − b ∂ ρ
∂ ϕ ( ϕ m − 1 ) ∇ µ m − 1 , B m
h :=  ϕ m
h − ϕ m − 1
τ + v m
h · ∇ ϕ m − 1  , ρ m − 1 :=
ρ ( ϕ m − 1 ) , and η m − 1 := η ( ϕ m − 1 ) .
Using Brou w er’s fixed-p oin t theorem one can sho w the existence of at least
one solution follo wing [9, Thm. 2]. The uniqueness sta ys unclear due to the
nonlinearit y ρ m v m
h in (16). The sc heme fulfills a fully discrete v ariant of the
formal energy iden tit y (15).
Theorem 7 (The fully discrete energy inequalit y) . L et ϕ m
h ∈ V m
1 , µ m
h ∈ V m
1 ,
and v m
h ∈ V m
2 denote a solution to (16) – (19) . Then the fol lowing ener gy in-
e quality holds
1
τ  1
2 Z Ω
ρ m | v m
h | 2 + σ Z Ω

2 |∇ ϕ m
h | 2 + 1
 W ( ϕ m
h ) dx + Z ∂ Ω
γ ( ϕ m
h ) ds 
+ Z Ω
2 η m − 1 | D v m
h | 2 dx + b Z Ω |∇ µ m
h | 2 dx + Z ∂ Ω
l ( ϕ m − 1 ) | v m
h,tan | 2 ds + r Z ∂ Ω | B m
h | 2 ds
+ 1
τ  1
2 Z Ω
ρ m − 1 | v m
h − v m − 1 | 2 dx + σ 
2 Z Ω |∇ ϕ m
h − ∇ ϕ m − 1 | 2 dx 
≤ 1
τ  1
2 Z Ω
ρ m − 1 | v m − 1 | 2 + σ Z Ω

2 |∇ ϕ m − 1 | 2 + 1
 W ( ϕ m − 1 ) dx + Z ∂ Ω
γ ( ϕ m − 1 ) ds 
+ Z Ω
ρ m − 1 g · v m
h dx.
Pr o of. W e use w ≡ v m
h , q = p m
h , Ψ ≡ µ m
h and Φ ≡ ϕ m
h − ϕ m − 1
τ as test functions
11

in (16)–(19) and sum up to obtain
1
τ  1
2 Z Ω
ρ m | v m
h | 2 − 1
2 Z Ω
ρ m − 1 | v m − 1 | 2 + 1
2 Z Ω
ρ m − 1 | v m
h − v m − 1 | 2 dx 
+ Z Ω
2 η m − 1 | D v m
h | 2 dx − Z Ω
ρ m − 1 g · v m
h dx
+ Z ∂ Ω
l ( ϕ m − 1 ) v m
h,tan · v m
h ds + r Z ∂ Ω
B m
h ∇ ϕ m − 1 · v m
h ds
+ b Z Ω |∇ µ m
h | 2 dx
+ σ 
2 τ  Z Ω |∇ ϕ m
h | 2 − |∇ ϕ m − 1 | 2 + |∇ ϕ m
h − ∇ ϕ m − 1 | 2 dx 
+ σ
 Z Ω
( W 0
+ ( ϕ m
h ) + W 0
− ( ϕ m − 1 )) ϕ m
h − ϕ m − 1
τ dx
+ r Z ∂ Ω
B m
h
ϕ m
h − ϕ m − 1
τ ds + Z ∂ Ω
( γ 0
+ ( ϕ m
h ) + γ 0
− ( ϕ m − 1 )) ϕ m
h − ϕ m − 1
τ ds = 0 .
Using con v exit y and concavit y of W + and W − , and γ + and γ − it holds
Z Ω
( W 0
+ ( ϕ m
h ) + W 0
− ( ϕ m − 1 )) ϕ m
h − ϕ m − 1
τ dx ≥ 1
τ Z Ω
W ( ϕ m
h ) − W ( ϕ m − 1 ) dx,
Z Ω
( γ 0
+ ( ϕ m
h ) + γ 0
− ( ϕ m − 1 )) ϕ m
h − ϕ m − 1
τ ds ≥ 1
τ Z Ω
γ ( ϕ m
h ) − γ ( ϕ m − 1 ) ds.
Summing up and using v · ν Ω = 0 , w e obtain the desired result.
Remark 8 (A daptiv e meshing) . In gener al, in diffuse interfac e simulations it
is advantage ous to use adaptive meshes to r esolve the interfacial r e gion. Then in
every time step additional pr olongation op er ators b etwe en subse quent meshes ar e
r e quir e d. As a c onse quenc e, in this c ase the ener gy ine quality fr om The or em 7
only holds with the pr olongate d data for the ener gy fr om the old time instanc e.
W e further note that sp e cial c ar e has to b e taken for pr olongating the velo city
field, as the pr olongate d velo city field typic al ly is not solenoidal with r esp e ct to
the new mesh. W e r efer to [9, 36] for further discussion of this topic.
3.1. V ariants
Let us state v arian ts of the ab ov e discretization sc heme (16)–(19) for n u-
merical comparison. W e note, that (16)–(19) is a fully coupled and non-linear
sc heme.
3.1.1. A stable de c ouple d scheme
If r ≡ 0 the sc heme is only coupled by the transport term ( ϕ m − 1 v m
h , ∇ Ψ) in
(18). The same holds for l → ∞ , whic h results in the commonly used no-slip
condition for the Na vier–Stok es equation. In the case of no-slip conditions B is
indep enden t of v and th us again the only coupling is the transp ort term in (18).
12

In b oth cases w e can decouple the Na vier–Stokes equation and the Cahn–
Hilliard equation b y using an augmented v elo cit y field in (18), see for example
[11, 13, 18, 37]. Here w e substitute − R Ω ϕ m − 1 v m
h · ∇ Ψ dx in (18) b y
− Z Ω
ϕ m − 1 v m − 1 · ∇ Ψ dx + τ Z Ω
( ρ m − 1 ) − 1 | ϕ m − 1 | 2 ∇ µ m
h · ∇ Ψ dx. (20)
The resulting sc heme is decoupled; w e can first solv e (18) and (19) and thereafter
(16) and (17). This sc heme is also energy stable, as the additional in tegral
comp ensates terms arising from Hölder’s and Y oung’s inequalit y to balance the
first in tegral with the numerical dissipation 1
2 R Ω ρ m − 1 | v m
h − v m − 1 | 2 dx . This
sc heme with no-slip conditions for Navier–Stok es and r ≡ 0 is analyzed in [11]
for differen t treatmen ts of W 0 . W e also refer to [38] for an alternativ e decoupling
in the case of constan t densit y . Here the systems are decoupled by using v m − 1
in (18), and the energy stability is obtained b y in tro ducing a step size restriction
for the temp oral discretization.
If r > 0 , w e use v m − 1 in the definition of B m
h in (19) and v m
h in the cor-
resp onding term in (16) and can still deriv e an energy inequalit y containing
an error of order r R ∂ Ω ( v m
h − v m − 1 ) · ∇ ϕ m − 1 ds . In [11] a no-slip condition is
assumed for v to decouple the b oundary conditions. Then the decoupling pro-
p osed in (20) is sufficien t to decouple the Na vier–Stokes and the Cahn–Hilliard
equation.
W e note that this sc heme can b e applied for an y bulk energy p oten tial that
admits a con v ex-conca v e splitting.
3.1.2. A stable de c ouple d and line ar scheme
Using the decoupling prop osed in Section 3.1.1, the only nonlinearit y in
the sc heme arises from W 0
+ . In [16–18], a stabilized linear scheme is used and
the term W 0
+ ( ϕ m
h ) + W 0
− ( ϕ m − 1 ) is substituted b y W 0 ( ϕ m − 1 ) + S W ( ϕ m
h − ϕ m − 1 ) ,
where S W is a suitable stabilization parameter. F or smo oth W it satisfies S W ≥
1
2 max t | W 00 ( t ) | . It can be derived b y T aylor expansion of W at ϕ m − 1 , see for
example [16]. As W s is of class C 1 , 1 only , ( W s ) 00 jumps at ξ − 1 from − ξ 2 to
( s − 1) ξ 2 . In this case w e use S W ≥ s/ 2 . F or large v alues of s w e exp ect
that this stabilization will prev en t c hanges in ϕ and th us migh t hav e a deep
impact on the allo v er dynamics. This is inv estigated in Section 4 and esp ecially
discussed in Remark 11. T o linearize γ w e substitute γ 0
+ ( ϕ m
h ) + γ 0
− ( ϕ m − 1 )
b y γ 0 ( ϕ m − 1 ) + S γ ( ϕ m
h − ϕ m − 1 ) with S γ ≥ 1
2 max t | γ 00 ( t ) | and esp ecially S γ ≥
1
2 σ 12 | cos( θ s ) | in the case of γ cc . Here, again S γ is obtained b y T aylor expansion
of γ at ϕ m − 1 .
Remark 9 (F urther sc hemes) . F or further discr etization schemes of the bulk
ener gy density W we r efer for example to [11, 30, 33]. Se c ond or der schemes
for the Cahn–Hil liar d e quation ar e for example pr op ose d and analyze d in [30,
33, 39 – 41]. R e c ently the Invariant Ener gy Quadr atization appr o ach for W ≡
W poly was pr op ose d in [42]. It is use d in [43] for the Cahn–Hil liar d moving
c ontact line mo del to gether with a Cr ank–Nic olson and a BDF2 scheme in time.
However, typic al ly for these schemes either higher r e gularity than W s pr ovides
13

is r e quir e d for W , or the p articular W poly is assume d and ne c essary. Mor e over,
unc onditional ener gy stability is typic al ly not pr oven yet.
Remark 10 (Energy Consistency) . Considering the ener gy ine quality fr om The-
or em 7, the terms in the first line c orr esp ond to the discr ete ener gy of the system,
while the se c ond line c orr esp onds to the ener gy dissip ation of the system, and
the thir d line c orr esp onds to numeric al dissip ation of the scheme. Base d on this
we c an define four differ ent values to define the ener gies in our system. These
ar e the ener gy E m at time instanc e m , the physic al dissip ation ∆ m
p at time in-
stanc e m , the ener gy E m
g intr o duc e d fr om gr avity at time instanc e m , and the
numeric al dissip ation ∆ m
n at time instanc e m . They ar e define d by
E m := 1
2 Z Ω
ρ m | v m
h | 2 dx + σ Z Ω

2 |∇ ϕ m
h | 2 + 1
 W ( ϕ m
h ) dx + Z ∂ Ω
γ ( ϕ m
h ) ds, (21)
∆ m
p := τ Z Ω
2 η m − 1 | D v m
h | 2 dx + τ Z Ω
b |∇ µ m
h | 2 dx
+ τ Z ∂ Ω
l ( ϕ m − 1 ) | v m
h,tan | 2 ds + τ Z ∂ Ω
r | B m
h | 2 ds, (22)
E m
g := τ Z Ω
ρ m − 1 g · v m
h dx, (23)
∆ m
n := E m − 1 + E m
g − E m − ∆ m
p . (24)
W e c al l a scheme thermo dynamic al ly c onsistent if The or em 7 is fulfil le d without
the explicit form of the numeric al dissip ation, thus if
E m + ∆ m
p ≤ E m − 1 + E m
g (25)
holds, i.e., ∆ m
n ≥ 0 . W e investigate this ener gy ine quality numeric al ly in Se c-
tion 4.
4. Numerics
In this section w e n umerically in v estigate the three sc hemes under consider-
ation. In Section 4.1 w e briefly giv e results from the well-kno wn second b enc h-
mark in [32], where no contact line motion is included, to estimate the difference
of the sc hemes in the bulk. In Section 4.2 w e thereafter inv estigate the b eha vior
of the con tact line for a gravit y-driven droplet sliding on an inclined surface in
a t w o-dimensional setting.
W e implemen t the sc hemes in Python3 using FEniCS 2018.1.0 [44, 45]. F or
the solution of the arising nonlinear and linear systems and subsystems the soft-
w are suite PETSc 3.8.4 [46–48] together with the direct linear solver MUMPS
5.1.1 [49, 50] are utilized. Note, that we do not apply an y preconditioning or
subiterations except for the Newton iterations.
14

4.1. R ising Bubble
A t first, w e discuss the accuracy of the prop osed sc hemes without moving
con tact lines. Later on, this allows for an ev aluation of the influence of the
sc hemes on the mo ving con tact line. W e employ the quan titative benchmark
case prop osed in [32]. In [51] it is found, that three differen t diffuse interface
appro ximations together with the p olynomial p oten tial W pol y agree w ell with
the sharp in terface results from [32]. In [9] the b enchmark is used to compare
to a phase field mo del with a relaxed double obstacle p oten tial.
4.1.1. Setup
T able 1 lists the prop erties of our simulations, whic h corresp ond to the sec-
ond b enc hmark case in [32]. F or details on the setup w e refer to the references
ab o v e. Note, that σ 12 denotes the ph ysical surface tension, yielding σ ≈ 1 . 24
for W s =100 , σ ≈ 1 . 22 for W s =10 and σ ≈ 2 . 07 for W poly 2 . F ollo wing [32], w e
in tro duce a c haracteristic length scale L = 2 r 0 , where r 0 equals the initial radius
of the bubble, and a c haracteristic velocity scale U = √ Lg . T o classify our sim-
ulations w e indicate in T able 1 the dimensionless n um b ers Reynolds R e = ρ l U L
η l ,
Eötv ös (or Bond) Eo = ρ l g L 2
σ , Capillary Ca = η l U
σ , A tw o o d At = ρ l − ρ g
ρ l + ρ g , Cahn
Cn = 
L and P éclet Pe = LU 
bσ , see [52].
W e apply no-slip b oundary conditions for the v elo cit y on the top and b ottom
w alls, free-slip on the left and s ymmetry at the cen terline through the bubble at
x = 0 . 5 . Similar to [9], w e set b = 10 − 3  and  = 0 . 02 . The time discretization
step is set to differen t v alues and the final time is t = 3 . W e initialize the
sim ulations b y solving the Cahn–Hilliard equations without con vection un til a
steady state is reac hed. In total, w e p erform 7 distinct sim ulations using the
three sc hemes from Section 3 with W poly 2 and W s with s = 100 , and one
additional sim ulation with s = 10 for the fully linear and stabilized sc heme
with W s , see the first three columns in T able 2. T o get an impression of the
influence of the discretization parameters, w e use differen t v alues for τ and h min ,
see columns four and fiv e in T able 2.
In [32] a set of b enc hmark parameters is used, that we define in the phase
field setting as follo ws.
The cen ter of mass is calculated using
( x c , y c ) = R Ω ( x, y ) 1+ ϕ
2 dx
R Ω
1+ ϕ
2 dx , (26)
where 1+ ϕ
2 = 1 indicates the droplet.
W e define the mean v elo cit y in unit direction a ∈ R 2 as
v a = R Ω v · a 1+ ϕ
2 dx
R Ω
1+ ϕ
2 dx . (27)
If a denotes the unit v ector in rising direction, this is called rising v elo cit y v r ,
while if a p oin ts in sliding direction, w e call this v alue sliding velocity v s .
15

σ 12 ρ l ρ g η l η g g y R e Eo Ca A t
1.96 1000 1 10 0.1 -0.98 35 125 3.5 0.99
 b Cn Pe
2 × 10 − 2 2 × 10 − 5 0.04 178
T able 1: P arameters used in the rising bubble sim ulations.
Finally w e define the stretching of the in terface as
c = c W R Ω ( 
2 |∇ ϕ 0 | 2 + 1
 W ( ϕ 0 )) dx
c W R Ω ( 
2 |∇ ϕ | 2 + 1
 W ( ϕ )) dx . (28)
Here the denominator denotes an appro ximation to the length of the in terface
represen ted b y ϕ , and the numerator denotes the same for the initial phase field
ϕ 0 . If ϕ 0 denotes a sphere, this is equiv alen t to the circularit y as defined in [32]
as the v olume of the bubble is constant o ver time.
Remark 11 (Choice of s in W s ) . F or the choic e of the r elaxation p ar ameter s
in W s , se e R emark 2, sever al p oints must b e c onsider e d. T o r e duc e the inter-
mixing b etwe en the phases and incr e ase the r ate at which the e quilibrium pr ofile
of ϕ is r e establishe d after a deformation, it is desir able to exhibit a lar ge spino dal
r e gion and subse quently a smal l metastable r e gion [53]. The metastable r e gion
of the bulk ener gy p otential W s is lo c ate d b etwe en 1 > | ϕ | > ξ − 1 = 1 − 1
s , while
the metastable r e gion for W poly is lo c ate d b etwe en 1 > | ϕ | > √ 3 − 1 ≈ 0 . 577 .
Thus alr e ady for smal l values of s , say s = 10 , the metastable r e gion of W s is
signific antly smal ler than the metastable r e gion of W pol y . F urthermor e, r eferring
to [9, 54], the value of s c ontr ols the deviation of the L ∞ norm of ϕ fr om 1. Sinc e
ρ and η dir e ctly dep end on ϕ a smal l deviation is desir able, which is achieve d
by a lar ge value of s .
On the other hand, the stable de c ouple d and line ar scheme, Se ction 3.1.2,
includes a stabilization p ar ameter S W which has to b e chosen like S W > s/ 2
for W s . In this c ase a lar ge value of s has a sever e imp act on the over al l
dynamics as the stabilization c an b e interpr ete d as adding the quadr atic p otential
S W
2 k ϕ − ϕ m − 1 k 2 to W for given ϕ m − 1 . F or lar ge values of S W thus ϕ ≡ ϕ m − 1
is pr eferr e d. T o show the influenc e of S W in the c ase W ≡ W s we test the line ar
and de c ouple d scheme with two values of s .
4.1.2. R esults
The resulting b enc hmark v alues are listed in T able 2. As it is not ev en clear
in the sharp in terface simulations whether or not topological changes dev elop,
e.g. the separation of trailing gas filamen ts, w e compare our results only up to
time instance t = 2 , see [51] . Our results show that all the sc hemes giv e v ery
similar results compared to the sharp in terface solution even for the significan tly
larger time step τ = 0 . 001 and on a coarse mesh with h min = 0 . 0125 . In
16

general, decoupling the t wo systems has a v ery small impact on the b enc hmark
v alues. F or ev en larger τ = 0 . 008 the coupled scheme is adv an tageous against
the decoupled sc hemes. The latter migh t b e explained b y the fact, that the
decoupling adds artificial diffusion of order τ to the Cahn–Hilliard system, see
(20). Thus w e exp ect a stronger influence of this decoupling for larger v alues
of τ . As exp ected, the stabilized linear sc heme together with W s =100 hinders
the dynamics of the rising bubble. Ho w ev er, the results improv e significantly
with smaller s . All sc hemes together with W s =100 give sligh tly b etter results
compared to W poly 2 except the decoupled/linear sc heme. Ho wev er, for v ery
small τ and h min the results con v erge to wards similar v alues.
Concerning the computational effort the difference in using W s =100 or W poly 2
is insignifican t. The decoupled/nonlinear and decoupled/linear sc hemes are
around 1.4 resp ectiv ely 2.0 times faster than the coupled sc heme. In the non-
linear sc hemes 2-3 Newton iterations are needed p er time step. Note that the
p erformance results strongly dep enden t on the solv er and whether sophisti-
cated preconditioning is applied. F or an efficien t preconditioner for the cou-
pled/nonlinear system w e refer to [55].
4.2. Sliding Dr oplet
T o compare the influence of the n umerical sc hemes from Section 3 on the
mo ving con tact line, w e p erform sim ulations of single droplets sliding down an
inclined surface. Besides the effect of gra vity on the droplet mo vemen t, this test
case allo ws to observ e b oth an adv ancing and receding contact line.
4.2.1. Setup
In Figure 1 the initial configuration is sho wn and T able 3 lists the prop erties
of our sim ulations. T he fluid prop erties are c hosen to b e similar to the first
rising bubble test case in [32]. Note, that σ 12 denotes the ph ysical surface
tension, yielding σ ≈ 15 . 58 for W s =100 , σ ≈ 15 . 34 for W s =10 and σ ≈ 25 . 98
for W poly 2 . A liquid droplet with radius r 0 = 0 . 25 is placed in a 0 . 5 × 2 . 0
rectangular domain at ( 0 , 1 . 5 ) on a smo oth, solid surface with an initial con tact
angle of 90 ◦ . The inclination angle of the plate is 45 ◦ . The density of the
droplet is greater than that of the surrounding fluid. W e hav e no-slip b oundary
conditions for the v elo cit y on the left and righ t side and free-slip on the top
side. The conditions (6) and (7) are applied on the b ottom solid surface, see
Figure 1. The influence of the b oundary conditions (6) and (7) on the sliding
droplets are examined b y v arying the static con tact angle θ , the relaxation factor
r and the slip co efficien t l , see the fifth to sev enth column in T able 4. W e v ary
the con tact angle from sup er-h ydrophilic ( 5 ◦ ) to sup er-h ydrophobic ( 150 ◦ ) [56].
W e initialize the sim ulations by solving the Cahn–Hilliard equations without
con v ection and a con tact angle of 90 ◦ until a steady state is reac hed.
In a first step, w e compare 21 distinct sim ulations obtained with the three
sc hemes from Section 3 with W poly 2 and W s with s = 100 , and one additional
sim ulation with s = 10 for the fully linear and stabilized sc heme with W s ,
see the first t w o columns in T able 4. These sim ulation are p erformed with
17

Bulk p ot. Deco./Lin.? τ h min y c v max t v max c min t c min
W s =100 no/no 8 × 10 − 3 0 . 012 50 0 . 9024 0 . 2433 0 . 7520 0 . 6962 1 . 99
W s =100 no/no 1 × 10 − 3 0 . 012 50 0 . 9071 0 . 2479 0 . 7410 0 . 6636 2 . 00
W s =100 no/no 5 × 10 − 5 0 . 006 25 0 . 9059 0 . 2475 0 . 7349 0 . 6611 2 . 00
W s =100 y es/no 8 × 10 − 3 0 . 012 50 0 . 8816 0 . 2486 0 . 7600 0 . 7124 1 . 99
W s =100 y es/no 1 × 10 − 3 0 . 012 50 0 . 8994 0 . 2503 0 . 7460 0 . 6710 2 . 00
W s =100 y es/no 5 × 10 − 5 0 . 006 25 0 . 9060 0 . 2479 0 . 7349 0 . 6614 2 . 00
W s =100 y es/y es 8 × 10 − 3 0 . 012 50 0 . 7128 0 . 2066 1 . 9920 0 . 7355 1 . 99
W s =100 y es/y es 1 × 10 − 3 0 . 012 50 0 . 8642 0 . 2374 0 . 8710 0 . 8290 2 . 00
W s =100 y es/y es 5 × 10 − 5 0 . 006 25 0 . 8996 0 . 2457 0 . 7449 0 . 7455 2 . 00
W s =10 y es/y es 8 × 10 − 3 0 . 012 50 0 . 8475 0 . 2370 0 . 8800 0 . 8154 1 . 99
W s =10 y es/y es 1 × 10 − 3 0 . 012 50 0 . 8951 0 . 2402 0 . 7600 0 . 6965 2 . 00
W s =10 y es/y es 5 × 10 − 5 0 . 006 25 0 . 9030 0 . 2472 0 . 7349 0 . 6725 2 . 00
W poly 2 no/no 8 × 10 − 3 0 . 012 50 0 . 8826 0 . 2206 0 . 7680 0 . 6857 1 . 99
W poly 2 no/no 1 × 10 − 3 0 . 012 50 0 . 8865 0 . 2249 0 . 7610 0 . 6618 2 . 00
W poly 2 no/no 5 × 10 − 5 0 . 006 25 0 . 8941 0 . 2494 0 . 7517 0 . 6536 2 . 00
W poly 2 y es/no 8 × 10 − 3 0 . 012 50 0 . 8668 0 . 2278 0 . 7760 0 . 7092 1 . 99
W poly 2 y es/no 1 × 10 − 3 0 . 012 50 0 . 8815 0 . 2279 0 . 7680 0 . 6699 2 . 00
W poly 2 y es/no 5 × 10 − 5 0 . 006 25 0 . 8944 0 . 2429 0 . 7649 0 . 6556 2 . 00
W poly 2 y es/y es 8 × 10 − 3 0 . 012 50 0 . 8646 0 . 2272 0 . 7840 0 . 7193 1 . 99
W poly 2 y es/yes 1 × 10 − 3 0 . 012 50 0 . 8813 0 . 2278 0 . 7690 0 . 6713 2 . 00
W poly 2 y es/yes 5 × 10 − 5 0 . 006 25 0 . 8944 0 . 2429 0 . 7649 0 . 6559 2 . 00
ref. diffuse 4 × 10 − 3 0 . 8994 0 . 2503 0 . 7960 0 . 6684 1 . 98
ref. sharp 1 . 95 × 10 − 4 0 . 9154 0 . 2502 0 . 7313 0 . 6901 2 . 00
T able 2: Benchmark v alues for the second b enc hmark prop osed in [32]. Here y c denotes the
cen ter of mass at time t = 2 , v max denotes the maximum rising v elo city that appears at time
t v max , and c min denotes the minimal circularit y that app ears at time t c min . See (26)–(28) or
[32] for the definition of these v alues. As reference w e choose the results from the 3rd group
participating in [32] (ref. sharp) and for model 3 in [51] (ref. diffuse,  = 0 . 02 ). W e note that
in the latter piece wise quadratic finite elemen ts are used for ϕ and µ , whic h is the reason,
wh y we do not pro vide a v alue for h min . F urther, w e do not provide a v alue for h min for the
reference solution in the sharp setting b ecause here a differen t numerical approac h is used,
that can not directly b e compared with the presen t situation.
18

0 . 5
2 . 0
1 . 5
0 . 5
0 . 25
45 ◦
x
y
Droplet
W all with
eqns. (6),(7)
and s , r , l .
g
Figure 1: Initial configuration for the sliding droplet sim ulations.
σ 12 ρ l ρ g η l η g g R e Eo Ca A t
24.5 1000 100 10 1 -0.98 35 10 0.28 0.81
 b Cn Pe
2 × 10 − 2 2 × 10 − 5 0.04 14
T able 3: P arameters used in the sliding droplet sim ulations.
a relativ ely coarse mesh ( h min = 0 . 0125) and large time step ( τ = 0 . 001 ) to
discuss the practical applicabilit y of the solution sc hemes. Afterwards, w e sho w
the thermo dynamic consistency of the sc hemes and compare the ph ysical and
n umerical dissipation rates. T o discuss the influence of the time step size on
the results, w e p erform 14 additional simulations with τ b et ween 0.008 and
0.00025. Finally , we perform 8 simulations with v arying in terfacial thic knesses
 on a v ery fine mesh ( h min = 0 . 0002 ) to briefly discuss the con v ergence to the
sharp-in terface limit.
Remark 12 (Choice of r and l ) . F or me aningful values of the r elaxation p a-
r ameter r and the slip c o efficient l , we write (6) and (7) in non-dimensionalize d
form,
 Ca
Cn 2 ˆ η D ˆ v ν Ω + Ca
Cn
L
l s − ˆ
L ˆ
∇ ϕ  × ν Ω = 0 , (29)
Ca
Cn
r s
L ( ∂ ˆ
t ˆ ϕ + ˆ v ˆ
∇ ϕ ) + ˆ
L = 0 , (30)
in which Ca = η l U /σ and Cn = /L ar e the Capil lary numb er r esp e ctively
the Cahn numb er, and L and U ar e some char acteristic macr osc opic length
sc ale r esp e ctively velo city. W e cho ose r = r s η l and l = η l /l s such that the
dimensionless gr oups Ca
Cn L
l s and Ca
Cn
r s
L ar e of O (1) , se e [57].
19

ϕ = 0
h min
θ d

Figure 2: Measuremen t of the dynamic (or apparent) con tact angle.
As b enc hmark v alues w e again use the three v alues defined in Section 4.1
with minor mo difications. F or the center of mass, w e use a co ordinate system
that is aligned with the inclined plate, see Figure 1, and for the no w called sliding
v elo cit y , we use for a the unit v ector in tangen tial direction to the inclined plate.
The stretc hing of the interface is defined as before.
A dditionally , w e ev aluate tw o v alues that are sp ecific for the mo ving con tact
line setup. F or b oth the receding and adv ancing con tact line the p osition of the
con tact p oin ts and a dynamic (or apparen t) contact angle measured at some
heigh t ab o v e the con tact p oin ts are ev aluated. The p osition of a con tact p oin t
is defined b y
y p = y on ∂ Ω where ϕ = 0 , (31)
and the dynamic con tact angle θ d is calculated by linear in terp olation b etw een
y p and the in tersection y p +∆ where ϕ = 0 and ∆ = h min , see Figure 2 and [58].
4.2.2. R esults
Comp arison of dr oplet shap es and char acteristic values obtaine d on a c o arse
mesh and with a lar ge time step. In dep endence on θ s , r and l the droplets
sho w c haracteristic dev elopmen ts. The calculated shap es for different com bina-
tions of θ s , r and l at t = 0 . 0; 0 . 5; 1 . 0; 1 . 5; 2 . 0 are presen ted in Figure 3. All
the sim ulated droplets sho w the exp ected ph ysical b eha vior: on the hydropho-
bic surface (third ro w) the droplet con tracts, whereas the droplets spread on
the h ydrophilic surface (second row). In addition, the droplets slide down the
surface due to the densit y difference and gra vit y . The differen t b eha vior at the
adv ancing and receding con tact p oin ts is visible and one can observ e nonequi-
librium con tact angles in the second and third row. It is eviden t that there
are virtually no differences b et w een the coupled (solid blac k) and decoupled
sc hemes (crosses) for all con tact angles. In con trast, in the linearized sc heme
with s = 100 (dashed blac k) the dynamics are greatly reduced. Similar as in
the rising bubble case, a smaller s ( s = 10 , dashed gra y) leads to improv ements.
20

F or comparison, w e show the behavior of the droplet with the coupled sc heme
and W poly 2 (dotted line). Here, a slightly differen t droplet shap e is observ ed
esp ecially for later times and large con tact angles.
The ev olution of the slide v elo cit y v s , the p osition of the con tact p oin ts
y p along the surface and the dynamic con tact angle θ d are display ed in Fig-
ure 4. Again, w e sho w all the sc hemes together with W s =100 and in addition
the stabilized sc heme together with W s =10 and the coupled, nonlinear sc heme
with W poly 2 . T o allo w for a more quan titativ e comparison b etw een the solution
sc hemes, w e list the c haracteristic v alues at t = 2 in T able 4. As exp ected,
in sim ulations without equilibrium con tact angle relaxation ( r = 0 ) and slip
( l = 1 e 6 ) (first ro w) the apparen t con tact angles on b oth sides of the droplets
sta y near the equilibrium v alue θ s = 90 ◦ the whole time. In con trast, app lying
the full b oundary conditions (6) and (7) with r = 0 . 35 and l = 140 leads to
clearly visible adv ancing and receding con tact angles (third column). As b efore,
no difference is visible b et w een the coupled (solid blac k) and decoupled (blac k
crosses) nonlinear sc hemes for all the characteristic quan tities. The c haracter-
istic v alues at t = 2 differ only v ery sligh tly . The results with the decoupled,
stabilized sc heme with s = 100 (dashed blac k) are very far off and sho w very lo w
sliding v elo cities (left column) and a differen t con tact p oin t b eha vior (middle
column), esp ecially for θ s = 150 ◦ (last ro w). The sliding v elo cities at t = 2
differ greatly . In con trast to the comparison in the bulk only , see Section 4.1,
the usage of the coupled sc heme with W poly 2 (dotted black) giv es results which
are noticeable differen t from the results with W s =100 . This is most ob vious in
the sim ulations with θ s = 5 ◦ (middle ro w): the sliding v elo city (left column) is
slo w er and the terminal v elo cit y is reac hed later. In addition, the receding and
adv ancing con tact angles are b oth lo wer than in the sim ulations with W s =100 .
F or example, at t = 2 , the adv ancing con tact angle for W poly 2 is around 12 ◦
smaller than in the nonlinear sim ulations with W s =100 .
Thermo dynamic c onsistency and c omp arison of dissip ation r ates. W e reveal the
thermo dynamic consistency of the sc hemes b y calculating the evolution of the
energy inequalit y using (25). W e use γ cc and set θ s = 150 ◦ , r = 0 . 35 and
l = 140 . W e observ e, that ∆ m
n is p ositiv e for all times, whic h justifies that the
sc hemes are thermo dynamically consisten t, see (25). Note that for r > 0 w e
in tro duce an additional error as so on as w e use the decoupling strategy . W e
observ e, that the physical dissipation for the three nonlinear sc hemes are close
together, while the ph ysical dissipation for the linear mo del is strongly reduced.
This corresp onds to the reduced dynamics that are observ ed in Figure 3 for the
linear sc hemes, esp ecially for s = 100 . This influence can b e reduced by using
v ery small time steps and finer meshes, see the results for the rising bubble case
in T able 2. Comparing the n umerical and physical dissipation of the nonlinear
sc hemes, ∆ m
n only accoun ts for around 25% of the total dissipation ev en for large
time steps. F urthermore, b y halving the time step τ , the n umerical dissipation
∆ m
n relativ e to the total dissipation ∆ m
n + ∆ m
p is greatly reduced to around 12%,
see the grey plots in the b ottom figure of Figure 5.
21

0 . 5
1
1 . 5
θ s = 90 ◦ , r = 0 , l = 1 e 6
t = 0 . 5 t = 1 . 0 t = 1 . 5 t = 2 . 0
0 . 5
1
1 . 5
θ s = 5 ◦ , r = 0 . 35 , l = 140
0 . 1 0 . 3
0 . 5
1
1 . 5
θ s = 150 ◦ , r = 0 . 35 , l = 140
0 . 1 0 . 3 0 . 1 0 . 3 0 . 1 0 . 3
coupl., W s =100 decoupl., W s =100 decoupl./lin, W s =100 decoupl./lin, W s =10 coupl., W poly 2

Figure 3: Shap es of the sliding droplets calculated with the sc hemes from Section 3. Three
differen t surfaces ranging from sup erh ydrophilic ( 5 ◦ , middle) to superhydrophobic ( 150 ◦ , bot-
tom) are compared. The corresponding parameters can b e found in T ables 3 and 4.
22

0
0 . 1
0 . 2
0 . 3
0 . 4
θ s = 90 ◦ , r = 0 , l = 1 e 6
v s
0 . 5
1
1 . 5
2
y p
60
80
100
120
θ d
0
0 . 1
0 . 2
0 . 3
0 . 4
θ s = 5 ◦ , r = 0 . 35 , l = 140
0 . 5
1
1 . 5
2
0
20
40
60
80
0 0 . 5 1 1 . 5 2
0
0 . 1
0 . 2
0 . 3
0 . 4
time
θ s = 150 ◦ , r = 0 . 35 , l = 140
0 0 . 5 1 1 . 5 2
0 . 5
1
1 . 5
2
time
0 0 . 5 1 1 . 5 2
100
120
140
160
180
time
coupl., W s =100 decoupl., W s =100 decoupl./lin, W s =100 decoupl./lin, W s =10 coupl., W poly 2

Figure 4: Characteristic quan tities calculated with the sc hemes from Section 3. Three differen t
surfaces ranging from sup er-h ydrophilic ( 5 ◦ , middle) to sup er-hydrophobic ( 150 ◦ , b ottom) are
compared. The corresp onding parameters can b e found in T ables 3 and 4
23

Bulk p ot. Deco./Lin.? τ h θ r l y c y p,a y p,r v s c θ d,a θ d,r
W s =100 no/no
0.001 0.0125 90 0 1000000
1 . 1067 0 . 8298 1 . 4442 0 . 2593 0 . 9550 96 96
W s =100 y es/no 1 . 1048 0 . 8279 1 . 4398 0 . 2619 0 . 9570 90 96
W s =100 y es/y es 1 . 4087 1 . 1426 1 . 6926 0 . 0549 0 . 9921 90 96
W s =10 y es/y es 1 . 1904 0 . 9302 1 . 5115 0 . 1883 0 . 9723 90 96
W poly 2 no/no 1 . 1390 0 . 8765 1 . 4500 0 . 2231 0 . 9667 90 96
W poly 2 y es/no 1 . 1369 0 . 8757 1 . 4469 0 . 2239 0 . 9685 90 96
W poly 2 y es/y es 1 . 1437 0 . 8842 1 . 4548 0 . 2184 0 . 9696 90 96
W s =100 no/no
0.001 0.0125 150 0.35 140
0 . 9745 0 . 8818 1 . 1438 0 . 3829 0 . 8905 160 135
W s =100 y es/no 0 . 9749 0 . 8818 1 . 1438 0 . 3833 0 . 8906 160 135
W s =100 y es/y es 1 . 4053 1 . 2375 1 . 5876 0 . 0535 0 . 9368 148 146
W s =10 y es/y es 1 . 1340 1 . 0154 1 . 3091 0 . 2259 0 . 9149 154 140
W poly 2 no/no 1 . 0263 0 . 9030 1 . 1651 0 . 3257 0 . 9066 148 135
W poly 2 y es/no 1 . 0266 0 . 9046 1 . 1660 0 . 3236 0 . 9069 148 132
W poly 2 y es/yes 1 . 0410 0 . 9185 1 . 1823 0 . 3092 0 . 9093 148 135
W s =100 no/no
0.001 0.0125 5 0.35 140
0 . 9330 0 . 3551 1 . 8732 0 . 3872 0 . 5286 24 3
W s =100 y es/no 0 . 9331 0 . 3534 1 . 8736 0 . 3874 0 . 5282 24 3
W s =100 y es/y es 1 . 3550 0 . 8641 1 . 8510 0 . 1137 0 . 7592 17 10
W s =10 y es/y es 1 . 0188 0 . 4872 1 . 8750 0 . 3389 0 . 5726 24 3
W poly 2 no/no 0 . 9763 0 . 3253 1 . 8850 0 . 3494 0 . 5264 6 3
W poly 2 y es/no 0 . 9736 0 . 3252 1 . 8865 0 . 3527 0 . 5244 6 3
W poly 2 y es/y es 0 . 9808 0 . 3251 1 . 8849 0 . 3492 0 . 5318 6 3
T able 4: Parameters and c haracteristic v alues for the sliding droplets simulations. F or y c and
c see caption of T able 2. In addition, y p and θ d denote the p osition of the contact points and
the dynamic con tact angles. The first and second v alues corresp ond to the adv ancing and
receding con tact p oin t resp ectiv ely angle. The slide velocity is v s and all v alues are rep orted
at t = 2 .
24

0 . 5
1 . 5
· 10 − 2
Ph ysical diss. ∆ m
p
coupl., W s =100
deco., W s =100
deco./lin, W s =100
deco./lin, W s =10
coupl., W poly 2
0 . 5 1 . 5
0 . 5
1 . 5
· 10 − 2
time
Numerical diss. ∆ m
n
0
0 . 1
0 . 2

τ = 0 . 001
τ = 0 . 0005

∆ m
n / (∆ m
n + ∆ m
p )

Figure 5: V alidity of the energy inequalit y (25) (b ottom) and the physical dissipation (22)
(top) for the differen t schemes. All sim ulations are p erformed with τ = 0 . 001 , h min = 0 . 0125 ,
θ s = 150 ◦ , r = 0 . 35 and l = 140 . The coupled/nonlinear and decoupled/nonlinear scheme
matc h almost p erfectly and app ear as a single graph. The numerical dissipation relativ e to
the total dissipation is sho wn for tw o differen t time steps sizes in the b ottom figure.
Comp arison of char acteristic values obtaine d with smal ler time steps τ . W e
sho w the b eha vior of the sc hemes for differen t time step sizes in T able 5. F or
small time steps, b oth the nonlinear schemes (coupled and decoupled) con v erge
to the same c haracteristic v alues for the particular bulk energy p oten tials. Ho w-
ev er, by comparing the v alues b et ween the differen t bulk energy p oten tials, we
note, that the differences are still relativ ely large ev en for small time steps.
Again, the linear sc heme together with W s giv es results far a w a y from the so-
lution obtained with the coupled sc hemes.
Conver genc e to sharp-interfac e limit. In T able 6 we sho w s olutions obtained
with b oth bulk energy p oten tials and smaller  on a v ery fine mesh ( h min =
0 . 0002 ). T o reduce the computational effort, the inclination angle of the plate,
see Figure 1, is se t to zero and the sim ulation is already stopp ed at t = 0 . 2 .
As exp ected for b = O (  ) and r = O (1) , see [29], the rate of con v ergence for
25

Bulk p ot. Deco./Lin.? τ y c y p,a y p,r v s c θ d,a θ d,r
W s =100 no/no 8 × 10 − 3 1 . 1872 1 . 0606 1 . 3681 0 . 1926 0 . 9172 158 135
W s =100 no/no 1 × 10 − 3 0 . 9745 0 . 8818 1 . 1438 0 . 3829 0 . 8905 160 135
W s =100 no/no 2 . 5 × 10 − 4 0 . 9382 0 . 8539 1 . 1025 0 . 4186 0 . 8818 160 138
W s =100 y es/no 8 × 10 − 3 1 . 1913 1 . 0637 1 . 3687 0 . 1965 0 . 9169 157 138
W s =100 y es/no 1 × 10 − 3 0 . 9749 0 . 8818 1 . 1438 0 . 3833 0 . 8906 160 135
W s =100 y es/no 2 . 5 × 10 − 4 0 . 9383 0 . 8539 1 . 1025 0 . 4187 0 . 8818 160 138
W s =100 y es/y es 8 × 10 − 3 1 . 4833 1 . 2785 1 . 6958 0 . 0198 0 . 9657 140 140
W s =100 y es/y es 1 × 10 − 3 1 . 4053 1 . 2375 1 . 5876 0 . 0535 0 . 9368 148 146
W s =100 y es/y es 2 . 5 × 10 − 4 1 . 2442 1 . 1002 1 . 4177 0 . 1498 0 . 9251 152 140
W s =10 y es/y es 8 × 10 − 3 1 . 3802 1 . 2484 1 . 5842 0 . 0574 0 . 9297 148 142
W s =10 y es/y es 1 × 10 − 3 1 . 1340 1 . 0154 1 . 3091 0 . 2259 0 . 9149 154 140
W s =10 y es/y es 2 . 5 × 10 − 4 1 . 0011 0 . 9009 1 . 1648 0 . 3499 0 . 8966 157 138
W poly 2 no/no 8 × 10 − 3 1 . 2590 1 . 0789 1 . 3748 0 . 1663 0 . 9266 146 135
W poly 2 no/no 1 × 10 − 3 1 . 0263 0 . 9030 1 . 1651 0 . 3257 0 . 9066 148 135
W poly 2 no/no 2 . 5 × 10 − 4 0 . 9956 0 . 8748 1 . 1268 0 . 3564 0 . 8995 150 135
W poly 2 y es/no 8 × 10 − 3 1 . 1909 1 . 0895 1 . 3766 0 . 1630 0 . 9259 146 138
W poly 2 y es/no 1 × 10 − 3 1 . 0266 0 . 9046 1 . 1660 0 . 3236 0 . 9069 148 132
W poly 2 y es/no 2 . 5 × 10 − 4 0 . 9958 0 . 8748 1 . 1255 0 . 3558 0 . 8995 150 138
W poly 2 y es/y es 8 × 10 − 3 1 . 2285 1 . 1302 1 . 4226 0 . 1339 0 . 9284 146 135
W poly 2 y es/y es 1 × 10 − 3 1 . 0410 0 . 9185 1 . 1823 0 . 3092 0 . 9093 148 135
W poly 2 y es/y es 2 . 5 × 10 − 4 1 . 0000 0 . 8786 1 . 1318 0 . 3517 0 . 9007 150 135
T able 5: Characteristic v alues for the sliding droplet simulations obtained with differen t v alues
of τ ( h min = 0 . 0125 , θ = 150 ◦ , r = 0 . 35 , l = 140 ). F or details ab out the setup and the
c haracteristic v alues see the caption of T able 4.
26

Bulk p ot.  b y p
W s =100 0 . 040 4 × 10 − 5 0 . 1887
W s =100 0 . 020 2 × 10 − 5 0 . 1987
W s =100 0 . 010 1 × 10 − 5 0 . 2032
W s =100 0 . 005 5 × 10 − 6 0 . 2112
W poly 2 0 . 040 4 × 10 − 5 0 . 1902
W poly 2 0 . 020 2 × 10 − 5 0 . 1977
W poly 2 0 . 010 1 × 10 − 5 0 . 2027
W poly 2 0 . 005 5 × 10 − 6 0 . 2097
T able 6: Position of the con tact line for a receding droplet (similar to the sliding droplet case
with inclinication angle set to zero) obtained with differen t v alues of  on a v ery fine mesh
h min = 0 . 0002 ( τ = 0 . 001 , θ = 150 ◦ , r = 0 . 35 , l = 140 ). The decoupled/nonlinear solution
sc heme is used. F or details ab out the setup and the characteristic v alues see the caption of
T able 4.
b oth p oten tials is v ery slo w and the sharp-in terface limit is not reac hed y et.
Ho w ev er, from our sim ulations w e can conclude, that on the fine mesh b oth
p oten tials giv e v ery similar results and exbibit the same b ehavior for smaller
 . F or larger  , solutions obtained with W s =100 seem to div erge slightly faster
from the sharp-in terface solution than solutions obtained with W poly 2 .
5. Conclusion
W e compare the qualit y of the n umerical results with three differen t schemes
and t w o differen t bulk energy p oten tials. F or simulations without a mo ving con-
tact line (rising bubble case), w e find very similar results in the bulk independent
of the coupling and linearization for b oth p oten tials. Ho wev er, the linearization
of W s for large s hinders the dynamics to a great extend but gets b etter for
smaller s . F or the sim ulations including mo ving con tact lines (sliding droplet
case), the differences b etw een the p olynomial p otential W poly 2 and the relaxed
double-obstacle p oten tial W s are more pronounced. Again, w e observe a strong
truncation of the allo v er dynamics using W s together with the linear sc heme.
In b oth cases, the influence of the decoupling of the Navier–Stok es and Cahn–
Hilliard system sligh tly dep ends on the time step size. How ever, the decoupling
has a negligible influence on the all-o v er dynamics ev en for larger time steps.
Concerning the t wo tested bulk energy potentials, w e observe, that b oth giv e
in general ph ysically sound results, but differences are still exists even for small
time steps. The results and the b eha vior for smaller  on a fine mesh are almost
the same for b oth p oten tials. F or larger  , solutions obtained with W s =100 seem
to div erge sligh tly faster from a sharp-in terface solution than with W pol y 2 .
Summarizing our results, w e find that
• the decoupling strategy giv es excellen t results while the computational
effort is significan tly reduced compared to the fully coupled scheme,
27

• a further linearization of the Cahn–Hilliard system applying the stabiliza-
tion is not recommended together with W s for large v alues of s ,
• b oth bulk energy p oten tials pro duce sound and similar results in particular
for a smaller in terfacial thickness  .
T o further judge whether one of the p oten tials lead to more accurate results,
high fidelit y sharp in terface results on flo ws with mo ving con tact lines (similar to
the b enc hmark p erformed in [32]) are critical. It is our hop e, that the presented
w ork sparks further comparisons of diffuse and sharp interface models esp ecially
for the frequen tly observ ed and relev an t case of sliding droplets.
A c kno wledgment
The authors thank Marion Dziwnik for helpful discussions on the scaling of
the con tact line surface tensions.
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