The Rosensweig instability in isotropic magnetic gels

The Rosensweig instability in isotropic magnetic gels Von der Universit¨ at Bayreuth zur Erlangung des Grades eines ,,Doktors der Naturwissenschaften“ (Dr. rer. nat.) genehmigte Abhandlung vorgelegt von Stefan Bohlius geboren in Lichtenfels/Bayern 1. Gutachter: Prof. Dr. Helmut R. Brand 2. Gutachter: Prof. Dr. Harald Pleiner Tag der Einreichung: 28. M¨ arz 2008 Tag des Kolloquiums: 14. Juli 2008 Contents List of Figures v Zusammenfassung vii 1 Introduction 1 1.1 Ferrofluids.................................... 1 1.2 Ferrogels..................................... 2 1.3 The Rosensweig instability . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Nonlinear theoretical descriptions for the Rosensweig instability . . . . . . 5 1.4.1 The energy method . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4.2 Functional analysis approaches . . . . . . . . . . . . . . . . . . . . 6 1.4.3 The Swift-Hohenberg approach . . . . . . . . . . . . . . . . . . . . 6 1.4.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 The adjoint system and deformable surfaces . . . . . . . . . . . . . . . . . 7 1.6 The scope of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2 Macroscopic mathematical framework 9 2.1 The basic hydrodynamic equations . . . . . . . . . . . . . . . . . . . . . . 9 2.1.1 Different classes of macroscopic variables . . . . . . . . . . . . . . . 10 2.1.2 Continuity and balance equations . . . . . . . . . . . . . . . . . . . 10 2.1.3 The case of isotropic ferrogels . . . . . . . . . . . . . . . . . . . . . 12 2.2 Assumptions for the Rosensweig instability . . . . . . . . . . . . . . . . . . 16 2.2.1 The simplified bulk equations . . . . . . . . . . . . . . . . . . . . . 16 2.2.2 The boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . 17 3 Recalling the linear problem 21 3.1 Thegroundstate ................................ 21 3.2 Linear deviations from the ground state . . . . . . . . . . . . . . . . . . . . 22 3.3 Surface wave dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . 23 3.4 Rosensweig instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.5 The linear eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.6 On the normal stress boundary condition . . . . . . . . . . . . . . . . . . . 30 4 Nonlinear discussion using the energy method 31 4.1 Surface energy density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2 Linearstability ................................. 34 4.3 Stability of different Geometries . . . . . . . . . . . . . . . . . . . . . . . . 34 4.3.1 Stripe solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 i ii CONTENTS 4.3.2 Squares ................................. 35 4.3.3 Hexagons ................................ 37 4.4 Some drawbacks of the energy method . . . . . . . . . . . . . . . . . . . . 39 5 The amplitude equation 41 5.1 Introduction................................... 41 5.1.1 Nonlinear expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.1.2 The solvability condition for higher orders . . . . . . . . . . . . . . 43 5.2 The adjoint system for the Rosensweig instability . . . . . . . . . . . . . . 46 5.2.1 Dynamic surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.2.2 Basic equations and ground state . . . . . . . . . . . . . . . . . . . 46 5.2.3 The linear equations and the adjoint system . . . . . . . . . . . . . 47 5.2.4 Adjoint eigenvectors for the Rosensweig instability . . . . . . . . . . 51 5.3 The second perturbative order . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.3.1 The solvability condition in second order . . . . . . . . . . . . . . . 54 5.3.2 Solutions proportional to the main characteristic modes . . . . . . . 56 5.3.3 Solutions proportional to the higher harmonics . . . . . . . . . . . . 58 5.3.4 The normal stress boundary condition . . . . . . . . . . . . . . . . 61 5.4 The third perturbative order . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.5 Amplitudeequation............................... 66 5.6 On the Newell-Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.7 Discussion and comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6 Rosensweig instability in films and membranes 77 6.1 Motivation.................................... 77 6.2 Film properties in viscoelastic media . . . . . . . . . . . . . . . . . . . . . 77 6.3 Magnetic surface properties . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.4 Non-magnetic film modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.5 Ferrogel film surface modes . . . . . . . . . . . . . . . . . . . . . . . . . . 82 6.6 Rosensweig instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6.6.1 Stationary, asymmetric case without surface magnetism . . . . . . 84 6.6.2 Permanent-magnetic, symmetric case . . . . . . . . . . . . . . . . . 86 6.6.3 Thegeneralcase ............................ 86 6.6.4 Additional remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.7 Discussion.................................... 87 7 The adjoint system for the Marangoni convection 89 7.1 Introduction to Marangoni convection . . . . . . . . . . . . . . . . . . . . . 89 7.2 Basic equations and the adjoint system . . . . . . . . . . . . . . . . . . . . 90 7.3 The dimensionless representation . . . . . . . . . . . . . . . . . . . . . . . 92 7.4 The dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 7.5 The adjoint dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . . 95 7.6 Discussion of the dispersion relation . . . . . . . . . . . . . . . . . . . . . . 96 8 Conclusions 99 A Decoupling of the dynamic system 103 CONTENTS iii B Magnetic fields 105 B.1 The Heaviside-Lorentz system of electromagnetic units . . . . . . . . . . . 105 B.2 Expansion to higher perturbative orders . . . . . . . . . . . . . . . . . . . 107 B.2.1 The Maxwell equations . . . . . . . . . . . . . . . . . . . . . . . . . 107 B.2.2 The boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . 108 B.3 Solutions in linear order . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 B.4 Solutions in higher orders . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 B.5 Magnetic fields in the case of membranes . . . . . . . . . . . . . . . . . . . 112 B.5.1 The superparamagnetic case . . . . . . . . . . . . . . . . . . . . . . 112 B.5.2 The permanent-magnetic case . . . . . . . . . . . . . . . . . . . . . 114 C The hydrodynamic boundary conditions 117 C.1 Expansion of the boundary conditions . . . . . . . . . . . . . . . . . . . . . 117 C.2 The linear perturbative order . . . . . . . . . . . . . . . . . . . . . . . . . 117 C.3 The second perturbative order . . . . . . . . . . . . . . . . . . . . . . . . . 119 C.4 The third perturbative order . . . . . . . . . . . . . . . . . . . . . . . . . . 120 D Eigenvectors in the second order 123 E Usual ferrofluids 131 Literature 133 Acknowledgments 143 iv CONTENTS List of Figures 1.1 Sketches of a ferrofluid with and without an external magnetic field . . . . 2 1.2 Sketches of an isotropic and an anisotropic magnetic gel . . . . . . . . . . . 3 1.3 The Rosensweig instability in ferrofluids . . . . . . . . . . . . . . . . . . . 4 1.4 Qualitative geometry for the Kelvin-Helmholtz instability . . . . . . . . . . 8 2.1 Qualitative geometry for the Rosensweig instability . . . . . . . . . . . . . 18 3.1 Schematic sketch of the different surface wave regimes . . . . . . . . . . . . 24 3.2 Dispersion relation in ferrofluids for different external magnetic fields . . . 26 3.3 Dispersion relation in ferrogels for different elastic shear moduli . . . . . . 27 3.4 Stress distribution within the ferrogel . . . . . . . . . . . . . . . . . . . . . 29 4.1 Considered planforms for the energy method . . . . . . . . . . . . . . . . . 35 4.2 Graphs to estimate the validity regime for the energy method (squares) . . 36 4.3 Graphs to estimate the validity regime for the energy method (hexagons) . 37 4.4 Bifurcation scenario for the Rosensweig instability . . . . . . . . . . . . . . 38 5.1 The general bifurcation scenario for amplitude equations . . . . . . . . . . 44 5.2 The relative orientation of the different wave vectors under consideration . 59 5.3 Qualitative dynamical growth of the surface spikes in ferrogels . . . . . . . 70 5.4 Sketch of a physical system described by the sine-Gordon equation . . . . . 71 6.1 Qualitative geometry in the case of membranes and thin films . . . . . . . 78 6.2 Derivation of the surface stress boundary condition . . . . . . . . . . . . . 80 7.1 Qualitative geometry for the Marangoni instability . . . . . . . . . . . . . 90 v vi LIST OF FIGURES Zusammenfassung Die vorliegende Arbeit befasst sich mit der nichtlinearen theoretischen Analyse der Rosensweig Instabilit¨ at in isotropen magnetischen Gelen. Die Rosensweig Instabilit¨ at wurde erstmals im Jahr 1967 entdeckt und bezeichnet den ¨ Ubergang einer zun¨ achst flachen Grenzfl¨ ache zwischen einer magnetischen Fl¨ ussigkeit und einem nicht-magnetischen Medium zu einer hexagonal geordneten Stacheloberfl¨ ache, sobald ein senkrecht zur flachen Oberfl¨ ache angelegtes homogenes Magnetfeld einen bestimmten kritischen Wert ¨ uberschreitet. Magnetische Fl¨ ussigkeiten, auch Ferrofluide genannt, sind kolloidale Suspensionen ferromagnetischer Nanoteilchen in einer gew¨ ohnlichen, dem Anwendungszweck entsprechenden Tr¨ agerfl¨ ussigkeit, wie Wasser oder Benzol. Einem angelegten Magnetfeld ausgesetzt, verhalten sich Ferrofluide wie gew¨ ohnliche paramagnetische Stoffe, jedoch ist ihre Permeabilit¨ at bis zu einer Gr¨ oßenordnung h¨ oher als in ¨ ublichen paramagnetischen Stoffen, weshalb man sie auch als superparamagnetisch bezeichnet. Mit der Entdeckung der Rosensweig Instabilit¨ at wurde auch eine erste theoretische Beschreibung des Ph¨ anomens vorgestellt. An der freien Grenzfl¨ ache zwischen der Ferrofl¨ ussigkeit und dem dar¨ uber liegenden Vakuum ¨ uberwiegen f¨ ur niedrige Magnetfelder die stabilisierenden Kr¨ afte der Gravitation und der Oberfl¨ achenspannung die destabilisierende Kraft des Magnetfeldes. Zwar besitzt ein homogenes Magnetfeld keine Kraftwirkung auf die Oberfl¨ ache, jedoch unterliegt die Grenzfl¨ ache den immer vorhandenen thermischen Fluktuationen, die das Magnetfeld lokal st¨ oren und so eine resultierende Kraft erzeugen. Bei gen¨ ugend hohen Magnetfeldst¨ arken ¨ ubertrifft diese Kraft die Gravitation und die Oberfl¨ achenspannung und das Rosensweigmuster bildet sich aus. Startet man den Vernetzungsprozess in einer Mischung aus Polymeren, Vernetzungsreagenzien und einem Ferrofluid, so erh¨ alt man ein isotropes Ferrogel, ein elastisches Medium, welches zus¨ atzlich superparamagnetisches Verhalten aufweist. Ferrogele bilden eine neue Materialklasse, von der man sich Anwendungen in vielen technischen und medizinischen Bereichen erhofft. So gelten sie zum Beispiel als vielversprechende Kandidaten zur Herstellung k¨ unstlicher Muskeln oder von außen regelbarer Medien zur gezielten Wirkstofffreisetzung im K¨ orper. Theoretisch l¨ asst sich zeigen, dass auch die Oberfl¨ ache dieser Medien in einem angelegten Magnetfeld instabil wird, wobei die typische Wellenl¨ ange im Vergleich zu gew¨ ohnlichen Ferrofluiden unver¨ andert bleibt, w¨ ahrend die kritische Magnetfeldst¨ arke mit wachsendem elastischen Schermodul steigt. Experimentell konnte dies bereits qualitativ best¨ atigt werden. Allerdings ist man in Experimenten auf sehr schwach vernetzte Gele angewiesen, da die kritische Magnetisierung anderenfalls gr¨ oßer als die S¨ atigungsmagnetisierung des elastischen Mediums ist. Nach dem einf¨ uhrenden Kapitel und der Diskussion der grundlegenden hydrodynamischen Gleichungen zur Beschreibung isotroper magnetischer Gele in Kapitel 2, werden im dritten Kapitel die linearen Eigenschaften der Rosensweig Instabilit¨ at in isotropen Fervii 4Introduction Figure 1.3: The experimental realization of the Rosensweig instability. On the left hand side the applied magnetic field (applied normal to the surface) is below the critical value whereas the picture on the right hand side is taken beyond the critical value. The containers diameter is of the order of 20 cm and the height as well as the diameter of the spikes is in the order of 1 cm. 1.3 The Rosensweig instability The Rosensweig or normal field instability describes the phenomenon of a flat ferrofluid surface becoming unstable in an external magnetic field. The experimental setup consists of a Petri-dish filled with a ferrofluid which develops a flat surface in earth’s gravitational field (as given in the photograph on the left of fig. 1.3). If one applies a homogeneous magnetic field oriented parallel to the surface normal, this flat surface becomes unstable beyond a certain critical magnetic field strength and a regular pattern of surface spikes arises (as seen on the right of fig. 1.3). Experimentally one observes a hexagonal arrangement of these surface spikes at the linear threshold [19]. With its discovery in 1967 [19] a first theoretical description for the normal field instability was given which allowed the determination of the critical magnetic field and the most unstable mode at onset in terms of the fluid properties. The model considered the force balance at the free surface between the stabilizing forces of surface tension and gravitation and the destabilizing magnetic force. Although the homogeneous magnetic field does not generate a force in the first place, the occurrence of surface perturbations render the local magnetic field inhomogeneous resulting in a local Kelvin force that drives the instability. For a magnetic gel rather than a ferrofluid, the surface was also predicted to become unstable [20]. Additionally, however, the stabilizing elastic force has to be overcome by the magnetic field, which is why its critical value is shifted towards higher field strengths. The characteristic mode at onset, however, remains the same compared to usual ferrofluids and is given by the capillary mode. Recently the threshold shift has qualitatively been shown experimentally for a thermoreversible magnetic gel at the University of Bayreuth [21]. Since, however, the threshold magnetic field increases with increasing shear modulus, one is restricted to very weak gels, otherwise the threshold magnetization is higher than the saturation magnetization of the medium. To maintain a finite and constant shear modulus of the thermoreversible gel and to avoid creep flow, a time dependent magnetic field was applied which complicates the comparison between the experiments and theory. At the moment more accurate experimental results can be obtained using inverse ferrofluids [22, 23, 24]. One calls a ferrofluid inverse, if non-magnetic particles are also dispersed in the ferrofluid. Usually the particles’ diameter is of the order of micrometers and they are typically made of polystyrene. The onset of the Rosensweig instability in these fluids is shifted to higher magnetical field strengths, which cannot be solely explained 1.4 Nonlinear theoretical descriptions for the Rosensweig instability 5 by dilution effects [25], but probably requires the assumption of a finite effective shear modulus, which may be responsible for the threshold shift. Fig. 1.3 shows that a ferrofluid is a darkish brown, non-transparent medium and for this reason quantitative experimental results of nonlinear patterns are difficult to obtain optically. For supercritical magnetic field strengths the properties of the most unstable mode and its growth rate have been discussed theoretically as well as experimentally in [26, 27]. In 1984 Bacri and Salin [28] discovered the hysteretic nature of the transition between the flat surface and surface spikes. They used a very thin container (≈200 µm) where the arising pattern was quasi one dimensional. This allowed the observation of the instability from the side exploiting the optical contrast between the magnetic fluid and the medium above. Using radioscopic methods [29], the hysteretic region between the flat surface and hexagons was accurately measured [30]. If the magnetic field is increased further, the hexagonal pattern is found to be unstable with respect to regular squares. Also this transition is accompanied by a hysteretic region and is experimentally discussed in [31, 32]. Experiments on the nonlinear regime for the Rosensweig instability in ferrogels have recently been started, but up to now no publications are available. 1.4 Nonlinear theoretical descriptions for the Rosensweig instability Since its discovery, the Rosensweig instability has attracted the attention of experimentalists and theoreticians, alike. The work of the experimental scientists together with an intuitive linear description has been introduced in the previous section. A linear analysis, however, gives us no information about the amplitude and the spatial structure of the arising pattern nor on the nonlinear dynamic behavior. This is why nonlinear discussions of the governing basic equations are needed which will be the aim of this thesis. In the following the reader will find an introduction to previous nonlinear discussion of the normal field instability in ferrofluids. 1.4.1 The energy method The energy method was the first attempt to theoretically access the nonlinear regime of the Rosensweig instability and was published in 1977 by Gailitis [33]. Gailitis discussed laterally unbounded ferrofluid layers of infinite depth. The application of this method to systems with a finite depth was later done by Friedrichs and Engel in [34]. The general idea of this energy-based approach is to find the dependence of the surface energy density of the fluid as a function of the surface deflection, the applied magnetic field, and the material parameters of the ferrofluid except for the viscosity, which has to be neglected completely in this approach. This energy density functional can then be minimized with respect to prescribed surface patterns. The patterns considered by Gailitis were regular stripes, squares and hexagons. The relative stability of which can then be given as a function of the applied magnetic field. Gailitis found, that stripe patterns are always unstable with respect to one of the other two patterns. Below the linear threshold the flat surface is always a stable configuration whereas at the linear onset hexagons turn out to be energetically favored. Upon further 6Introduction increase of the magnetic field, however, the hexagons become unstable with respect to squares and the previous hexagonal pattern transforms into a square pattern. Both transitions are found to be accompanied by hysteretic regions. In that respect the theoretical results qualitatively match the experimental findings. For finite fluid layer depths, the critical values for the magnetic field are shifted to higher strengths and the hysteretic behavior of both transitions becomes more pronounced [34]. In chapter 4 we will apply the energy method to the Rosensweig instability in isotropic magnetic gels to obtain a first estimate of the nonlinear patterns arising in ferrogels. However, there are severe drawbacks to this method. Therefore one has to use more fundamental methods to discuss the nonlinear regime. 1.4.2 Functional analysis approaches The first approaches considering the basic hydrodynamic equations have been discussed by Twombly and Thomas [35, 36] and later on by Silber and Knobloch [37]. Both groups considered only static hydrodynamic equations under the condition that the velocity vanishes, reducing the Navier-Stokes equation to the hydrostatic pressure contribution. In this approximation the stress free surface is governed by the normal stress boundary condition whereas the tangential boundary conditions are trivially satisfied. Additionally, the static Maxwell equations were considered with the corresponding boundary conditions. This set of fundamental equations and boundary conditions is then expanded in terms of ǫ(the normalized difference between the applied magnetic field and the critical one) following the ideas of [38]. Since no time derivative is involved in the set of equations, it turns out to be self-adjoint and the solvability conditions in the higher orders of the expansion can be fulfilled. As stated by Silber and Knobloch [37], no stable pattern was found at the linear onset for realistic magnetic permeabilities. Another drawback of this method is, since it rests on the static assumption, that it cannot give predictions for the nonlinear dynamics in terms of an amplitude equation and that it neglects the possibility of oscillatory instabilities. A different approach, using again the static approximation for the macroscopic set of equations, was presented by Friedrichs and Engel in 2003 [39]. In their discussion they focused on the normal stress boundary condition only. Inspired by [40], where it is shown that for a strong enough tangential component of the magnetic field two dimensional patterns can be suppressed and only stripes are the stable solution, they focused on the nonlinear discussion of stripes only. Malik and Singh [41, 42, 43] were the first to discuss an ǫ−expansion of the fundamental hydrodynamic equations allowing for dynamic processes. To circumvent the general solvability condition for the higher orders of the expansion, where one needs to know the adjoint system of equations, they restricted their discussion to potential flow only. This is an unphysical approximation, as we will see later on in our discussion, since only the vorticity contributions to the flow can guarantee that the tangential boundary conditions at the free surface are satisfied. 1.4.3 The Swift-Hohenberg approach To compensate for the lack of dynamic descriptions, Kubstrup, Herrero and P´erez-Garc´ıa discussed the normal field instability in terms of a phenomenological Swift-Hohenberg 1.5 The adjoint system and deformable surfaces 7 equation [44, 45]. In particular, they discussed the dynamics of stable fronts between hexagons and squares. A general ansatz for the surface deflection, which is not so different from the one used in the energy method approach, is substituted into a generalized SwiftHohenberg equation and a set of amplitude equations is obtained. These amplitude equations are then solved numerically. This approach nicely reveals the stability dynamics between hexagons and squares in the nonlinear regime, but only on a phenomenological basis. The main disadvantage of this study is, that the coefficients in the amplitude equation have no relation whatsoever to real material properties. Nevertheless, a good qualitative agreement can be obtained fitting the phenomenological coefficients to the experimental results [31]. 1.4.4 Numerical results The methods discussed so far describe the dynamics of the Rosensweig instability in the weakly nonlinear regime. These methods allow one to discuss the dynamics and the arising pattern close to the threshold as long as the amplitudes stay small. A prediction of the final shape of a single surface spike cannot be obtained by these methods. Using a finite element method, Lavrova et al. determined the shape of one of the surface spikes [46] by integration of the basic hydrodynamic equations. In [30] the experimental and numerical results were compared and a very good agreement was observed. 1.5 The adjoint system and deformable surfaces At a closer inspection of section 1.4.2 and the nonlinear approaches discussed therein, it is clear that there is still a crucial piece missing in the description of the nonlinear regime of the Rosensweig instability in the spirit of [38, 47, 48], namely the knowledge of the adjoint system of equations together with its boundary conditions. A first attempt to derive this was made by Lange in [49] who used a particular form of a scalar product known from discussions of the Marangoni instability. The attempt failed since this scalar product rests on the assumption of an undeformable surface; an assumption not appropriate for the case of the Rosensweig instability. In the presence of a deformable surface and for a dynamic system, the adjoint system with its boundary conditions was unknown. A detailed discussion and the derivation is given in this thesis in section 5.2. As indicated already in the previous paragraph, the Rosensweig problem is not the only instability that involves a deformable surface. Another very prominent phenomenon, the Marangoni convection, sensitively depends on the deformability of the surface [50, 51]. In the Marangoni convection, small temperature fluctuations at the interface between the underlying fluid and the medium above induce fluctuations of the local surface tension that in turn cause the surface to deform. An analytical weakly nonlinear description for the Marangoni convection accounting for the deformability of the surface is still missing. Many authors [52, 53, 54, 55, 56] treated the nonlinear system assuming a flat undeformed boundary between two fluids. The reason is mainly due to the missing solution of the adjoint system in the presence of a deformable surface. The case of the Marangoni instability is of particular interest to us, since in contrast to the Rosensweig instability, the driving force of the instability acts purely tangentially to the surface. The method we used to derive the adjoint system for the Rosensweig instability can therefore be used for 8Introduction fluid 1 fluid 2 ~v1 ~v2 Figure 1.4: A qualitative sketch of the geometry appropriate for the KelvinHelmholtz instability. Two fluids move at different velocities with respect to each other. The initially flat interface becomes unstable against deformation beyond a critical velocity difference. any arbitrary direction of the driving force. A further situation where a deformed boundary becomes important for the nonlinear regime and more precisely, where the actual position of the boundary depends on the dynamics of the system as a whole, is the Faraday instability. Certain modes become unstable upon periodic normal vibrations of the medium. The weakly nonlinear analysis of this problem also crucially depends on the knowledge of the adjoint system. We will not deal with this phenomenon since a comprehensive nonlinear study of this problem has been given recently by Skeldon and Guidoboni in [57]. Two further examples where the deformability of the boundary is essential in the nonlinear regime are given by the Rayleigh-Taylor instability and the Kelvin-Helmholtz instability. In the Rayleigh-Taylor problem the stability of a denser liquid on top of a lighter liquid, both subject to a gravitational field, is analyzed. In the Kelvin-Helmholtz problem the stability of the interface between two fluids that move with different velocities with respect to each other is discussed (cf. fig. 1.4). The latter problem is of particular interest for the creation of low pressure systems in the global weather system as it occurs in the atmosphere at the boundary between the air at the cold polar caps and the west wind zone. The method we applied in finding the adjoint system for the Rosensweig and the Marangoni case can also be applied to these systems. 1.6 The scope of this thesis The nonlinear behavior of the Rosensweig instability either in ferrofluids or in magnetic gels still contains many unsolved questions. In this thesis I will focus on the nonlinear analytic description of these phenomena in ferrogels putting particular attention on the derivation of the amplitude equation. A crucial step for this derivation will be the determination of the adjoint system with its boundary conditions. To set the stage for the special properties of the Rosensweig instability, we will extensively discuss the linear behavior and the possible patterns arising in the nonlinear regime using the energy method. In addition we will discuss the obtained amplitude equation for the special case of ferrofluids, because also in this case the amplitude equation derived from the basic hydrodynamic equations is still unknown. Since the boundary conditions play an important role in the discussion, we will also deal with thin films and membranes and discuss their linear stability properties in external magnetic fields. To conclude, our considerations are applied to derive the adjoint system of equations for the Marangoni instability. Chapter 2 Macroscopic mathematical framework In order to give a comprehensive theoretical description of the Rosensweig instability in isotropic ferrogels, we first have to define the physical and mathematical framework. We know, from experimental results, that the typical length scale of the instability is of the order of centimeters and the typical growth of the surface spikes takes place on a rather long time scale, say seconds (the growth can be followed by the naked eye). This suggests a viewpoint where we consider the medium continuous and macroscopic. The most suitable theory we can use is thus the generalized hydrodynamic theory [58, 59]. In this chapter the hydrodynamic approach will be introduced and we discuss the basic set of hydrodynamic equations one obtains for isotropic magnetic gels. This part is mainly based on, and summarizes, the results of the work of Jarkova et al. [60] and will be the basis for the nonlinear discussion. Additionally we specify the simplifications and extensions of this general set of equations that are appropriate for the Rosensweig instability. 2.1 The basic hydrodynamic equations The generalized hydrodynamic approach utilizes simple symmetry and thermodynamic arguments to derive a general set of dynamic equations for certain macroscopic variables. These variables have to be identified for the particular system under consideration and this particular choice dramatically reduces the number of degrees of freedom one takes into account. In a microscopic description, for instance, one might model atoms of a certain species as point masses interacting with each other via specified potentials. To consider systems of macroscopic size like a glass of water, however, the number of point masses needed to model this system is of the order of Avogadro’s constant. In fact, too many degrees of freedom to be handled. In a macroscopic theory one considers the mass density field instead, which reduces the number of free variables drastically. Calculating macroscopic properties of the system becomes feasible. One big advantage of the hydrodynamic method is given by its generality, which allows its application to a vast number of systems as long as we are able to treat these systems macroscopically. However, since we average over very many degrees of freedom, phenomenological coefficients have to be introduced that are specific for the particular system. One of these phenomenological coefficients is, for example, the well known heat 9 10 Macroscopic mathematical framework conductivity. These coefficients contain the information of all the microscopic processes taking place in the medium, and they could theoretically be determined via Green-Kubo relations if all microscopic processes were known, but practically this is only possible in some limiting cases, as, for example, small densities. Therefore these coefficients are usually taken from experimental measurements. 2.1.1 Different classes of macroscopic variables One can distinguish three different classes of macroscopic variables. The first class includes variables associated with global conservation laws. Assume a system with a conserved quantity, which by definition cannot decay or grow locally but which is allowed to be transported. The equilibration of this conserved quantity at different positions in space takes the longer, the farther these two positions are separated from one another. In reciprocal space this statement can be expressed as: the frequency of a process tends to vanish (ω→0) if its wavenumber vanishes (k→0). This is exactly the mathematical formulation of the hydrodynamic limit which rests on the fact, that historically hydrodynamics dealt with conserved quantities. The second class of variables, as a first generalization of hydrodynamics, includes variables that are connected to spontaneously broken continuous symmetries. What we refer to as a broken symmetry is the possibility that the Hamiltonian of a system is of higher symmetry than its eigenstates. One of the best known examples for this phenomenon is ferromagnetism, where the Hamiltonian is indeed invariant under rotation although there is an easy axis assigned to the system which is reflected in the eigenstates. In general there is no conserved quantity connected to this kind of variables (ferromagnetism is an exceptional case where the magnetization is a conserved quantity) [59], but they allow for excitations with infinite lifetime in the long wavelength limit, the so called Goldstone modes [59]. Therefore it seems reasonable to additionally include these variables into a macroscopic description. The third and final class accounts for variables connected to microscopic degrees of freedom whose dynamics takes place on a timescale large enough that it enters the macroscopic regime. Consequently one can include these specific microscopic degrees of freedom into the macroscopic description. Otherwise the validity of the description would be restricted to time scales even larger to guarantee that this specific variable has relaxed to its equilibrium value. But it is worth mentioning, that these variables do not show the hydrodynamic limit, ω6→ 0 if k→0, and long wavelength excitations with a finite frequency may be retained. While the identification of the variables of the first two classes is completely systematic, the identification of variables of this last class is not straightforward and involves a deeper knowledge of the system. 2.1.2 Continuity and balance equations If the macroscopic variables of a particular system are specified, one can turn to the derivation of equations that describe the dynamics of these variables. Assume first a macroscopic system in global thermodynamic equilibrium. The state of the system is then completely given by the numerical values of the macroscopic variables and one can define a thermodynamic potential that is a function of these macroscopic variables. Changes of the potential as a function of the macroscopic variables are related by the first law of 2.1 The basic hydrodynamic equations 11 thermodynamics. In the following formulation Erepresents the internal energy, Tthe temperature, Sthe entropy, pthe pressure, Vthe volume, µthe chemical potential and Nthe number of particles dE =TdS −pdV +µdN (2.1) The conjugated fields T,pand µcan be obtained by partial differentiation of the energy density with respect to the associated variable while the other variables are kept constant. With the help of Euler’s relation we can take the thermodynamic limit V→ ∞ and eliminate the volume from (2.1) which gives the local manifestation of the first law of thermodynamics [58] dε =Tdσ +µdρ (2.2) where εand σdenote the energy density and the entropy density, respectively. In the scope of a generalized hydrodynamic approach where we additionally account for variables associated with broken continuous symmetries and slowly relaxing variables, the Gibbs relation (2.2) also has to be generalized. This can be done by exploiting the fact that (2.2) is a total differential which allows one to introduce conjugated fields associated to the additional macroscopic variables. However, we then need the functional dependence of the energy density on the additional macroscopic variables. In our discussion we will always assume, that the system is close to thermodynamic equilibrium. This allows us to expand the energy density in terms of the macroscopic variables and their gradients. In order to do so, we have to consider the characteristics of the energy density. The equilibrium state is a stable state so that we have to provide a convex functional dependence. Furthermore the energy density should be invariant under inversion of space and time, under rigid translation and rigid rotation and it should be covariant upon Galilean transformation. If the system is close to thermal equilibrium, it will try to achieve the equilibrated state by dynamical processes. For the first class of variables the corresponding expressions can be derived easily exploiting the fact, that they represent conserved quantities in the system. Let us assume a scalar field αwhich is the volume density of a conserved quantity with its corresponding flux density jα. Since the amount of this particular quantity in an arbitrary volume Vis conserved, temporal changes of this amount have to be balanced by a flux through the closed bounding surface of that volume. Mathematically speaking one observes d dt Z V αdV =−I ∂V jα·df(2.3) where dfrepresents the surface area element of the closed surface1. Using Gauss’ theorem one can transform the last expression into its local form and obtain the continuity equation for the macroscopic variable α ∂ ∂tα+∇·jα= 0 (2.4) 1Throughout this thesis vectors are displayed in bold and their components using latin letters as indices, ∇denotes the vector ∇= (∂x, ∂y, ∂z) and we will imply summation over repeated indices except otherwise stated. In the latter context δij is the Kronecker symbol and ǫijk the Levi-Cevit`a tensor. 12 Macroscopic mathematical framework What we are left with is to find an explicit expression for the flux density jαwhich can be constructed as a power series in terms of the thermodynamic forces (usually the gradients of the conjugated fields), where the same symmetry arguments have to be applied as in the case of the energy density. Usually one can distinguish two different types of contributions to the currents: One contribution that accounts for reversible processes preserving the entropy density and one contribution due to the irreversible processes that lead to an increase of entropy. For a set of macroscopic variables usually cross-coupling contributions are allowed, for example in a binary fluid mixture an applied temperature gradient not only causes a heat flux but also a concentration flux. Onsager stated [61, 62, 63], that in these cases the corresponding symmetric contributions have to be present as well. Picking up the last example, this corresponds to a heat flux caused by an applied concentration gradient. For the other two kind of variables a straightforward derivation of the dynamic equations is not possible. However one can assume a similar dynamical behavior balancing the temporal change of the variable with a so called quasi current ∂tβ+Xβ= 0 (2.5) The quasi current itself can be constructed in the same way as the currents for the conserved macroscopic variables. 2.1.3 The case of isotropic ferrogels We can now apply the hydrodynamic method discussed in the previous section to the special case of isotropic magnetic gels. The first derivation of the generalized hydrodynamic equations was given by Jarkova et al. [60]. We will follow their work and give their results needed for the discussion of the Rosensweig instability. The energy functional and the Gibbs relation We start with the identification of the macroscopic variables. In the case of isotropic magnetic gels, the first class of variables consists of the mass density ρ, the momentum density g, the energy density εand the concentration of the magnetic particles c. To account for the elastic degrees of freedom we introduce the elastic strain field ǫij which belongs to the second class of variables. The strain field in amorphous solids is derived from crystals, where the long ranged positional order gives rise to the displacement vector field uas a hydrodynamic symmetry variable. In our description we will restrict ourselves to linear elasticity ǫij =1 2(∂iuj+∂jui). In usual ferrofluids the magnetization relaxes to the equilibrium value set by the external magnetic field. The appropriate relaxation time is much larger than all the other microscopic time scales. The same is true in magnetic gels. Therefore the magnetization Mis taken as an additional macroscopic variable belonging to the third class of macroscopic variables2. The transformation behavior under time ǫT and spatial ǫPinversion is summarized in table 2.1. In thermodynamic equilibrium all macroscopic variables are relaxed to their equilibrium values and one finds the Gibbs relation that relates infinitesimal changes of the 2The dynamics of the Rosensweig instability takes place on a time scale larger than the time scale of the dynamics of the magnetization. In that special case it is sufficient to exclude the magnetization again from the macroscopic dynamics as will be done in section 2.2.1. 2.1 The basic hydrodynamic equations 13 macroscopic variable time inversion ǫTspatial inversion ǫP ρ+1 +1 ε+1 +1 c+1 +1 gi−1−1 ǫij +1 +1 Mi−1 +1 Table 2.1: Table of the macroscopic variables important for the description of isotropic magnetic gels with their transformation behavior under time and spatial inversion macroscopic variables to infinitesimal changes of the entropy density σ dε =Tdσ +µdρ +µcdc +vidgi+HidBi+hM idMi+ Ψijdǫij (2.6) The corresponding thermodynamic conjugated fields are the temperature T, the chemical potential µ, the relative chemical potential µc, the velocity v, the magnetic molecular field hM iand the elastic stress Ψij and are defined as partial derivatives of the energy density with respect to the appropriate variable whilst the others are kept constant. The magnetic flux density Btogether with the magnetic field Hhave been introduced to account for the static Maxwell equations in our discussion. In order to give explicit expressions for the thermodynamic conjugated variables introduced above and to determine the thermodynamic forces we have to give an explicit expression for the energy density. Assuming an expansion around the equilibrium value one finds ε=ε0+1 2B2−B·M+1 2µijklǫijǫkl −1 2γijklMiMjǫkl +1 2αM2 i +ǫii(χρδρ +χσδσ +χcδc) (2.7) where ε0represents the energy density of a binary fluid mixture. The coefficient αaccounts for the dependence of the induced magnetization on the state of the medium, for example its temperature. Additionally αis a function of the applied magnetic field modeling the nonlinear magnetization behavior. In eq. (2.7) one can clearly distinguish the contributions due to the magnetic energy, the elastic energy, the cross coupling of the latter and the coupling between compression and the scalar field variables. Truncated at the quadratic order, this expansion is only valid for small elastic deformations of the medium. For large deformations one should extend the expansion to higher orders of ǫij accounting for nonlinear elastic deformations. The elasticity tensor µijkl and the magnetostrictive tensor γijkl take the isotropic form where we give, as an example, the elasticity tensor µijkl =µ1δijδkl +µ2δikδkl +δilδjk −2 3δijδkl(2.8) with its two invariants given by the elastic compressibility µ1and the elastic shear modulus µ2. 20 Macroscopic mathematical framework Chapter 3 Recalling the linear problem In this chapter we will focus on the linear aspects of the Rosensweig instability in isotropic magnetic gels. Parts of this chapter can be understood as a summary of previous works [70, 20], however it will also provide an easy introduction to the rather special nature of the kinematic boundary condition and the possible mathematical problems in the presence of dynamical deformable surfaces. This will help us in understanding the nonlinear regime and especially the way we treat it mathematically. 3.1 The ground state The system of equations and boundary conditions (2.32-2.34) and (2.41-2.42) always has the trivial ground state solution, where the surface is flat (ξ(x, y, t)≡0,n0=ez), flow and deformations are absent (v= 0, ǫij = 0), and the fields are constant (M0=M0ez with M0= (1 −1/µ)B0). The continuity equation (2.32) and the dynamic equation for the strain field (2.34) are then satisfied identically whereas the Navier-Stokes equation reads ∂jp0δij −B0iH0j+1 2B0kH0kδij=−ρGδiz (3.1) For the lateral dimensions in x−and y−direction these equations are easily satisfied by any pressure p0=p0(z), which is obtained by using i=zas p0(z) = −ρGz +p0(z= 0) (3.2) Furthermore we have to guarantee a stress free surface. The tangential stress boundary conditions are identically satisfied for this ground state solution, whereas the normal boundary condition reads p0(z= 0) = −1 21−1 µB2 0(3.3) which results upon substituting into (3.2) the final ground state pressure p0(z) = −ρGz −1 21−1 µB2 0(3.4) 21 22 Recalling the linear problem The system of equations therefore requires a non-zero, constant stress contribution due to the magnetic field, −(1/2)(1 −1/µ)B2 0, to the hydrostatic pressure, which is of minor relevance, since in an incompressible1system the pressure has no physical meaning anymore and merely serves as an auxiliary quantity that guarantees ∇·v= 0 for all times, if flow is present. It is worth mentioning here, that the ground state is the only state where the gravitational force contributes via the bulk equations. For the perturbed states, the gravitation only enters the analysis via the boundary conditions. 3.2 Linear deviations from the ground state For finite temperatures the system will be subject to thermal fluctuations which cause the surface to undulate randomly. These fluctuations can be viewed as a spectrum of propagating and damped surface waves (cf. fig. 2.1, p. 18) with a wave vector k= (kx, ky,0) and with the frequency consisting of a real ωand an imaginary part −σ ξ(x, y, t) = ˆ ξe−ikxx−ikyy+iωt+σt (3.5) and where ˆ ξdenotes the amplitude which is undetermined in the linear theory. In case of ω= 0, a stationary spatially periodic pattern is obtained. Generally ωis a complex function of k. Fourier modes of the type (3.5) can be superimposed as appropriate, and deviations from the ground state of all the other variables have to be proportional to ξ(x, y, t). Linear deviations of the surface normal from the ground state due to undulations are given by2n(1) ≡n−n0= (−∂xξ, −∂yξ, 0). The fact that the systems of hydrodynamic bulk equations decouples from the magnetic bulk equations enables us to solve the two bulk systems separately. A detailed derivation of the magnetic fields can be found in appendix B whereas here we just repeat the final results. The linear deviations of the magnetic field and induction from the ground state value, b(1) ≡B(1) −B0and h(1) =H(1) −H0, both for the ferrogel and the vacuum, still obey the linear electrostatic equations, b(1) =µh(1), divb(1) = 0 = curlh(1). This allows for the introduction of a magnetic scalar potential [69] h(1) =−∇Φ(1) that is determined by the Laplace equation with the appropriate solutions Φ(1) =−M0 1 + µξ(x, y, t)ekz (3.6) Φ(1)vac =µM0 1 + µξ(x, y, t)e−kz .(3.7) for the lower (ferrogel) and upper (vacuum) half plane, respectively and k2=k2 x+k2 y. 1In section 2.2.1 we still allowed for a compressible superparamagnetic medium. This assumption is necessary to derive the set of adjoint linear equations with its corresponding boundary conditions as we will see in section 5.2. For the discussion of the Rosensweig instability, however, we can safely assume an incompressible medium. 2The superscript (1) is just added for a consistent notation with chapter 5 and describes the deviations from the ground state in linear order. 3.3 Surface wave dispersion relation 23 The system of hydrodynamic bulk equations reads in linearized form ρ∂tv(1) i+∂ip(1) −ν2∂j(∂iv(1) j+∂jv(1) i)−2µ2∂jǫ(1) ij = 0 (3.8) ∂tǫ(1) ij −1 2(∂iv(1) j+∂jv(1) i) = 0 (3.9) ∂iv(1) i= 0 (3.10) The usual way to solve this system is to distinguish the irrotational flow contributions from the rotational ones, v(1) =v(1)pot +v(1)rot, where both parts can be deduced from a scalar potential ϕ(1) and a vector potential Ψ(1), respectively v(1)pot =∇ϕ(1) and v(1)rot =∇×Ψ(1).(3.11) The incompressibility of the medium requires ∆ϕ(1) = 0 and leads to the ansatz ϕ(1) = ˆϕ(1) ξ(x, y, t)ekz (3.12) for the scalar velocity potential. The vector velocity potential can be written as Ψ(1) =ˆ Ψ(1) ξ(x, y, t)eqz (3.13) where the amplitudes ˆϕ(1) and ˆ Ψ(1) and the decay length q−1are still undetermined. Since only two of the three amplitudes ˆ Ψ(1) can be independent, we set ˆ Ψ(1) z= 0 without loss of generality resulting in v(1)rot = (−qΨ(1) y, qΨ(1) x,−ikxΨ(1) y+ikyΨ(1) x). The strain ǫ(1) ij can be expressed by the velocity via eq. (3.9) and the linear pressure deviation, p(1) ≡p−p0is determined by eq. (3.8). With the help of eq. (3.9), iω∂jǫ(1) ij = (1/2)∆v(1) i, eq. (3.8) takes the linear form iωρv(1) i+∂ip(1) −ν2+µ2 iω∆v(1) i= 0 (3.14) Taking div and curl of eq. (3.14) we get [71] p(1) =−iωρϕ(1) + const.(3.15) and q2=k2−ρω2 µ2+iων2 (3.16) respectively, where the unimportant constant in the pressure can be ignored. 3.3 Surface wave dispersion relation We are left with three amplitudes, ˆϕ(1),ˆ Ψ(1) x,ˆ Ψ(1) y, that have to be related to the undulation amplitude, ˆ ξ, by the stress boundary conditions (2.41) and (2.42) and the kinematic boundary condition (2.44). For the linear analysis we could, without loss of generality, choose the in-plane wave vector kto be parallel to the x−axis, as done in [20]. With the nonlinear analysis in mind, it is worth considering an arbitrary direction of the wave vector. For linear deviations from the ground state and with the solutions obtained for 24 Recalling the linear problem ω∝ik2 ω∝k ω∝k3/2 ln k kc ω∝k1/2 ln µ2 ω∝k ln k ω∝k1/2 unstable ln M Figure 3.1: Schematic plots of the different surface wave regimes described by eq. (3.20) as in [71]. One encounters gravitational waves with ω∝k1/2, Rayleigh elastic waves with ω∝kand capillary waves with ω∝k3/2as depicted on the left. On the right an unstable region develops for strong magnetic fields. the magnetic fields, the stress boundary conditions can be written in terms of the flow potentials as (cf. appendix C) ˜µ2(∂2 z−∂2 y)Ψ(1) x+ ˜µ2(∂y∂x)Ψ(1) y+ 2˜µ2∂y∂zϕ(1) = 0 (3.17) ˜µ2(∂x∂y)Ψ(1) x+ ˜µ2(∂2 z−∂2 x)Ψ(1) y−2˜µ2∂x∂zϕ(1) = 0 (3.18) −(2˜µ2∂z∂y+Gρ∂y+σTk2∂y−µ 1 + µM2 0∂z∂y)Ψ(1) x +(2˜µ2∂z∂x+Gρ∂x+σTk2∂x−µ 1 + µM2 0∂z∂x)Ψ(1) y +(2˜µ2∂2 z+Gρ∂z+σTk2∂z−ρω2−µ 1 + µM2 0∂2 z)ϕ(1) = 0 (3.19) all taken at z= 0 and with the frequency dependent ˜µ2(ω)≡µ2+iων2describing (kinematic) elasticity and viscosity. To have a nontrivial solution for equations (3.17-3.19) the determinant of coefficients must vanish. This leads to the dispersion relation of surface waves for ferrogels ρω22˜µ2(ω)k2−ρω2+ρω2σTk3+ρGk + 2˜µ2(ω)k2−µ 1 + µM2 0k2 −4˜µ2 2(ω)k4"1−1−ρω2 ˜µ2(ω)k21/2#= 0 (3.20) In the absence of an external magnetic field (M0= 0) eq. (3.20) reduces to the dispersion relation for non-magnetic gels [71]. It also contains, as a special case, the surface wave dispersion relation for ferrofluids (in an external field) by choosing ˜µ2=iων2. It can be generalized to viscoelastic ferrofluids, whose elasticity relaxes on a time scale τ−1, by replacing µ2with iωτµ2/(1 + iωτ) [71]. The dispersion relation (3.20) is very complicated and it is impossible to solve it analytically for ω(k). For non-magnetic gels it is known that there are basically three wave regimes (neglecting dissipation or damping) (cf. fig. 3.1): ρω2=σTk3(capillary waves), ρω2= ˜αµ2k2(Rayleigh elastic waves), and ω2=Gk (gravity water waves) for 3.4 Rosensweig instability 25 small wavelengths (k≫µ2/σT,pρG/σT), intermediate ones (ρG/µ2≪k≪µ2/σT), and large ones (k≪ρG/µ2,pρG/σT), respectively, where ˜αis a number of order unity. For typical material values (µ2≈1 kPa, σT≈0.02 kg/sec2) waves at wavelengths of 10−4m and below (with frequencies of 50 kHz and above) are of purely capillary type, while for wavelengths above 1 m (and frequencies below 10 Hz) the gravity character dominates; this regime is, thus, irrelevant for usual ferrogel samples. In between, for typical wavelengths of 10−2m and frequencies of 100 - 1000 Hz the elastic nature of the wave is prevailing. This scenario also applies to isotropic ferrogels in the absence of a field. The effect of a normal external magnetic field on the surface is a destabilizing one [3]. From eq. (3.20) it is evident that an external field leads to an effective reduction of the surface stiffness (provided by surface tension, gravity or elasticity) and decreases the frequency (squared) of the propagating waves in all regimes by ∼M2 0k2. If the field is large enough, this reduction is the dominating effect and can lead to ω= 0 and thus, to the breakdown of propagating waves. In the next section it is shown that this is indeed related to the Rosensweig instability. 3.4 Rosensweig instability As mentioned already, eq. (3.20) is a complicated relation between the frequency of a surface wave implicitly given as a function of its wave vector. For a better understanding of the Rosensweig instability it is worth considering the simplification of eq. (3.20) to the case of an inviscid (ν2= 0) magnetic fluid ρω2=ρGk −µ 1 + µM2 0k2+σTk3(3.21) as has been done in [3, 19]. This is not a physical assumptions which we will have to correct later on, but it already reveals the static nature of the Rosensweig instability and the resulting dispersion relation (3.21) can be treated analytically. Upon minimizing (3.21) with respect to the frequency ωand the wave vector kone straightforwardly obtains the linear threshold of a static instability (ω≡0) M2 c= 21 + µ µpρGσT(3.22) beyond which the flat surface is unstable with respect to periodic patterns with a characteristic wave vector (cf. fig. 3.1) kc=rρG σT (3.23) The fact that the instability is static is, however, rather singular. This can be observed by plotting (3.21) as is shown in fig. 3.2 (with the material properties taken from the commercial ferrofluid EMG 901 as given for example in [72]). Without a magnetic field one obtains the behavior of an ideal fluid where the gravitational wave regime can be distinguished very clearly as well as the transition to the capillary regime. Upon increasing the magnetic field, the contribution due to the magnetization becomes more and more dominant and at a magnetization of about 6600A/m a branch of anomalous dispersion arises. More interesting for the Rosensweig instability is the necessary minimum that 26 Recalling the linear problem M= 0 A/m M= 6100 A/m M= 6600 A/m M= 6800 A/m M= 6931 A/m k[m−1] ω2[Hz2] 10008006004002000 8000 7000 6000 5000 4000 3000 2000 1000 0 Figure 3.2: The dispersion relation of surface waves in a usual magnetic fluid for different values of the externally applied magnetic field. The numerical parameter values are taken from the ferrofluid EMG 901 (ρ= 1.53·103kg m3,σT= 29.5·10−3N m and µ= 28.0·10−7T m). comes along with it. This relative minimum is shifted to lower ωvalues as the external magnetic field is increased and eventually it becomes the absolute one touching the abscissa at the characteristic wave vector for the critical magnetic field. In physical terms we can interpret this in a slightly different way. Below the critical magnetic field all modes show a finite oscillation in time. As soon as we approach the critical magnetic field the oscillation of the characteristic mode dies out resulting in the growth of a static pattern. The interpretation of the Rosensweig instability as the limiting case of surface waves with a vanishing frequency will be of importance for the nonlinear regime as we will see later. With the same procedure we can discuss the dispersion relation for magnetic gels (3.20). The characteristic mode turns out to be the same as for the case of ferrofluids, given by the capillary mode (3.23). This can be understood recalling the fact that the elastic contributions enter the dispersion relation with the same k−order as the magnetic contributions. The critical magnetic field is instead shifted towards higher magnetic fields [20] according to M2 c= 21 + µ µpρGσT+µ2(3.24) which is less surprising, since elasticity increases the surface stiffness. In the special cases of realistic ferrofluids with a finite viscosity (µ2= 0 and ν26= 0) and of ferrorubbers (ν2= 0 and µ26= 0) it can be shown analytically that only a static instability ω= 0 is possible at the linear onset [70, 20]. For the general case (3.20) of realistic magnetic gels with a finite viscosity and a finite shear modulus numerical calculations show the absence of an oscillatory instability at onset. Also in the general case of ferrogels, the static character of the instability is the limiting case of dynamical processes at the surface. To illustrate this, we can plot the dispersion 3.5 The linear eigenvectors 27 µ2= 2 Pa µ2= 1 Pa µ2= 0 Pa k[m−1] ω2[Hz2] 10008006004002000 8000 7000 6000 5000 4000 3000 2000 1000 0 Figure 3.3: The dispersion relation of surface waves on an isotropic magnetic gel at a fixed external magnetic field of 0.98Mcrit.and for different values of the elastic shear modulus. The numerical parameters for the other material properties are taken from the ferrofluid EMG 901 (ρ= 1.53 ·103kg m3,σT= 29.5·10−3N m, µ= 28.0·10−7T mand ν2= 6.5·10−6m2 s). relation in the general viscoelastic case (fig. 3.3). The different graphs have been obtained as numerical solutions of (3.20) where the material parameters are again taken from the ferrofluid EMG 901 with a varying value for the elastic shear modulus µ2and where the magnetic field is kept constant at about 98% of the critical magnetic field. Due to the numerical resolution we only had access to elastic moduli up to 2 Pa. However, it remains illustrative that the elastic contributions act opposite to the magnetic field and that the limiting case of a static instability is approached either for increasing magnetic fields or for decreasing shear moduli. An elastic shear modulus of 2 Pa is extremely weak, but fig. 3.3 additionally illustrates that already a low shear modulus influences the dynamic behavior of the surface drastically close to the linear threshold. 3.5 The linear eigenvectors The condition to find a solution of the linearized set of hydrodynamic equations has been discussed in the previous two sections. A solution can be found by first considering the two tangential stress boundary conditions (3.17,3.18) which allow one to express the amplitudes for the vector potential Ψ(1) in terms of the amplitude of the scalar potential ˆ Ψ(1) x=−2k(−iky) q2+k2ˆϕ(1) and ˆ Ψ(1) y= +2k(−ikx) q2+k2ˆϕ(1) (3.25) The kinematic boundary condition can then be used to find the explicit expression for the amplitude of the scalar potential ˆϕ(1) =iω q2+k2 k(q2−k2).(3.26) 28 Recalling the linear problem In the limiting case of ω→0, which corresponds to the approach of the onset of the Rosensweig instability if k=kc, the amplitudes of the potentials diverge with 1/ω. Since, however, the potentials are a mathematical tool to solve the system of equations, we need not worry about that, yet. Substituting the expression for the potentials into eq. (3.11), we eventually obtain the three components of the velocity field v(1) x=X i (−ikix)ekiz−2qiki q2 i+k2 i eqiziω(q2 i+k2 i) ki(q2 i−k2 i)ξi(3.27) v(1) y=X i (−ikiy)ekiz−2qiki q2 i+k2 i eqiziω(q2 i+k2 i) ki(q2 i−k2 i)ξi(3.28) v(1) z=X i kiekiz−2k2 i q2 i+k2 i eqiziω(q2 i+k2 i) ki(q2 i−k2 i)ξi,(3.29) where the index iaccounts for the fact that in a linear analysis of the instability the direction of the unstable mode remains degenerate and a whole set of characteristic modes with different directions may grow. We can exploit eq. (3.9) and eventually find the expressions for the components of the strain tensor, where again the index iaccounts for the different possible modes of the same modulus ǫ(1) zz =X i (q2 i+k2 i)ekiz−2qikieqiz q2 i−k2 i kiξi(3.30) ǫ(1) xz =X i (−ikix)ekiz−eqizq2 i+k2 i q2 i−k2 i ξi(3.31) ǫ(1) yz =X i (−ikiy)ekiz−eqizq2 i+k2 i q2 i−k2 i ξi(3.32) ǫ(1) xy =−X i kixkiy q2 i+k2 i ki(q2 i−k2 i)ekiz−2qiki q2 i+k2 i eqizξi(3.33) ǫ(1) xx =−X i k2 ix q2 i+k2 i ki(q2 i−k2 i)ekiz−2qiki q2 i+k2 i eqizξi(3.34) ǫ(1) yy =−X i k2 iy q2 i+k2 i ki(q2 i−k2 i)ekiz−2qiki q2 i+k2 i eqizξi.(3.35) We realized previously, that the introduced potentials diverge when approaching the linear onset. The velocity and strain field, as the observables of the system, should instead acquire a physical solution in the limit of a stationary instability. Taking the limit ω→0, 3.5 The linear eigenvectors 29 k= 2 k= 1 strain ǫzz depth z 0.80.70.60.50.40.30.20.10 0 -2 -4 -6 -8 -10 Figure 3.4: The absolute value of the zz−component of the strain field for two particular modes, k= 1 and k= 2 respectively, according to (3.37) as a function of depth and for an arbitrary infinitesimal deflection of the surface at a given point (x, y). The surface modes are measured in units of the characteristic wave vector kc, lengths in units of its inverse, k−1 c. we obtain the following eigenvectors at the linear onset v(1) i= 0 (3.36) ǫ(1) zz =−X i k2 izekizξi(ω= 0) (3.37) ǫ(1) xz =−X i (−ikix)kizekizξi(ω= 0) (3.38) ǫ(1) yz =−X i (−ikiy)kizekizξi(ω= 0) (3.39) ǫ(1) xy =X i kixkiyzekizξi(ω= 0) (3.40) ǫ(1) xx =X i k2 ixzekizξi(ω= 0) (3.41) ǫ(1) yy =X i k2 iyzekizξi(ω= 0) (3.42) The velocity field vanishes identically whereas the strain field acquires a finite stationary value. Fig. 3.4 gives the strain (and correspondingly the stress) distribution in the medium for two different surface modes as a function of the depth starting from the surface. The wave vectors are measured in units of the characteristic wave vector kc(3.23) and the depth is measured in units of the inverse characteristic wave vector k−1 c. At this point, the crucial difference between the Rosensweig instability and other commonly discussed instabilities becomes obvious. Mediated by the kinematic boundary condition, the velocity field vanishes. What one observes is not a stationary finite flow field 36 Nonlinear discussion using the energy method µ2= 0.100 µ2= 0.010 µ2= 0.001 µ2= 0.000 magnetic susceptibility χ control parameter ˜ǫ 32.521.510.50 1 0.1 0.01 0.001 Figure 4.2: Graphs separating regions in the χ−˜ǫ−plane in which the amplitudes of squares are smaller or higher than 0.25 for different values of the elastic shear modulus µ2. The critical magnetic susceptibility is enhanced with higher shear modulus. where we have again neglected ˜ǫin the denominator of the fourth order terms. Minimizing eq. (4.22) leads to the amplitude AS=AS(˜ǫ, µ2, η) AS=s4(˜ǫ−µ2)(1 + 4µ2)3−2√2(1 −µ2) NS (4.23) with the denominator NSgiven by NS= (1 + 4µ2)74√2−103 + (26√2−64)µ2 +16η212 −11√2 + (48 −46√2)µ2+ 16(2√2−3)µ2 2(4.24) Obviously ˜ǫ−µ2>0 is a necessary condition for the stability of square solutions. In fig. 4.2 the value of the control parameter ˜ǫis plotted as a function of the magnetic susceptibility χfor AS= 0.25 and different values of the shear modulus µ2. The graphs separate configurations in the parameter space with amplitudes smaller (below the curve) and higher (above the curve) than AS= 0.25 and the divergence for finite magnetic susceptibilities indicates that for a infinitesimal small control parameter ˜ǫthe amplitudes are already infinitely large. In [34], for µ2= 0, the plot has been used to estimate the maximum magnetic susceptibility at which the method diverges. As can be seen in fig. 4.2, for finite shear modulus the validity range of the method is increased to larger magnetic susceptibilities. For hexagons (cf. sec. 4.3.3) a similar result is obtained. The plot additionally illustrates the fact already stated by Gailitis himself that this energy method is rigorously valid only in the limit of a vanishing magnetic susceptibility. Following the method of Gailitis [33] i.e. starting with a more general ansatz for the wave vectors that include stripes as a special case, we find that even for µ2>0 stripes are always unstable with respect to squares. 4.3 Stability of different Geometries 37 µ2= 0.100 µ2= 0.010 µ2= 0.001 µ2= 0.000 magnetic susceptibility χ control parameter ˜ǫ 21.510.50 1 0.1 0.01 0.001 Figure 4.3: Graphs separating regions in the χ−˜ǫ−plane in which the amplitudes of hexagons are smaller or higher than 0.25 for different values of the elastic shear modulus µ2. The critical magnetic susceptibility is enhanced with higher shear modulus. 4.3.3 Hexagons We now discuss a regular hexagonal pattern generated by the three main wave vectors with angles of 2π/3 (cf. fig. 4.1). The difference in surface energy density with respect to the flat surface, eq. (4.14), becomes U=−1 2(˜ǫ−µ2)(A2 1+A2 2+A2 3)−3 4ηA1A2A3+1 45 16 −η2 1 + 4µ2(A4 1+A4 2+A4 3) +"−15 32 +3 8√3 + 11 8−7√3 8!η2−3 4−√32η2 1−√32+ 2√3µ2#(A2 1A2 2+A2 2A2 3+A2 1A2 3) (4.25) In the following we will refer to the expressions written in the first and second square bracket as β(0, µ2) and β(2π/3, µ2), respectively, since in the limit of vanishing elasticity they are identical to the ones given by Gailitis [33]. For the same reason we will also refer to 3/4ηas γ. For the regular hexagonal pattern we obtain from (4.25) by minimization AH=A1=A2=A3=γ±pγ2+ 4(˜ǫ−µ2)[β(0, µ2) + 4β(2π/3, µ2)] 2[β(0, µ2) + 4β(2π/3, µ2)] (4.26) The hexagonal solutions exist only if the square root in eq. (4.26) is real, and they are stable, if the second derivative of eq. (4.25) with respect to the amplitudes is negative, leading to the conditions −1 4[β(0, µ2) + 4β(2π/3, µ2)] <˜ǫ−µ2 γ2<2β(2π/3, µ2) + β(0, µ2) [2β(2π/3, µ2)−β(0, µ2)]2(4.27) 38 Nonlinear discussion using the energy method ˜ǫ−µ2 AH, AS AS AS AH AH Figure 4.4: Qualitative sketch (not to scale) of the evolution of the amplitudes for squares (AS) and hexagons (AH). The dashed lines correspond to the case of a ferrofluid while the solid lines qualitatively describe the behavior for ferrogels with finite shear modulus µ2. The dotted lines represent the energetically unstable branches. Since the β-values are positive (at least for χ≤1), hexagons can exist already below the linear threshold ˜ǫ=µ2. This existence range shrinks, however, for ferrogels compared to ferrofluids, since β(θij, µ2)> β(θij,0) for both θij = 2π/3 and 0. For the same reason the amplitude of the hexagonal pattern (4.26) decreases with increasing elastic modulus. Clearly, elasticity stabilizes a system against the Rosensweig instability, which is manifest not only in an increase of the (linear) threshold, but also in a decrease of the spike height. One can discuss the magnitude of the amplitudes as a function of the parameters ˜ǫand χfor the hexagonal solution in the same way as done for the square solution in section 4.3.2. Fig. 4.3 presents the corresponding plot for hexagons with the same qualitative result, that with increasing shear modulus the validity range of this method is extended towards higher magnetic susceptibilities. Following the method of Gailitis superposing hexagons and squares, we investigate the relative stability of hexagons and squares. We find that squares are unstable with respect to hexagons under the condition ˜ǫ−µ2 γ2<β(0, µ2) + 2β(π/2, µ2) [2β(2π/3, µ2) + 2β(π/6, µ2)−2β(π/2, µ2)−β(0, µ2)]2,(4.28) which is just Gailitis’ expression, but with µ2dependent β-abbreviations β(π/2, µ2) = −9 16 +1 √2+15 −13√2−4(3 −2√2)µ2 6−4√2 + 4√2µ2 η2(4.29) β(π/6, µ2) = 3 32(4√6−7) + 116 −41√6−(64 −28√6)µ2 16(2 −√6−2µ2)η2(4.30) Figure 4.4 shows the magnitude of the amplitudes as a function of the control parameter for a finite and a vanishing shear modulus, respectively. We note a decrease in size of the hysteretic region (for negative ˜ǫ−µ2) with increasing shear modulus. While in the case of no elasticity the lower boundary is at −0.25, it is shifted to −0.24 for a 4.4 Some drawbacks of the energy method 39 shear modulus of µ2= 0.1 (both values taken for a magnetic susceptibility of χ= 0.1)2. The second hysteretic region for the transition between squares and hexagons also shrinks with increasing µ2. For instance, the lower boundary of the hysteresis loop at 5.7 (for the right hand side of eq. (4.28)) for µ2= 0 increases to 6.9 for µ2= 0.1, while the upper boundary of the hysteresis loop at 540 for µ2= 0 is reduced to 480 for µ2= 0.1 (χ= 0.1). This result should be experimentally detectable, at least qualitatively. 4.4 Some drawbacks of the energy method One of the major drawbacks of this method is, that it is only valid in the asymptotic limit of a vanishing magnetic susceptibility χ. This can be easily realized by inspection of fig. 4.2. For high enough magnetic susceptibilities the amplitude diverges for arbitrarily small control parameters ˜ǫ. This is why we had to scale the stability boundaries given in eqs. (4.27,4.28) with γin order to compensate the divergence. For a nonlinear theory for magnetic fluids this validity limit is not satisfying. The method relies on the static energetic comparison of different surface deformations with respect to each other. It allows the determination of possible surface patterns but it does not tell us which pattern will be finally achieved. The selection process of a real pattern is additionally governed by dissipative processes. The Rosensweig instability, however, differs from other instabilities in that its final state is static and no dissipation occurs. But during the growth of the surface spikes energy is dissipated and this may play a role in the selection process. Additionally we realized in the discussions so far, that the occurrence of static surface spikes is the limit of a previously dynamic surface mode that freezes in its dynamics. Furthermore, as soon as the control parameter is beyond its critical value and as long as the final state is not achieved, the system is out of equilibrium and the definition of the potential (as is the energy) is not straightforward. As mentioned already, the energy method gives us first hints to become familiar with the Rosensweig phenomenon. What we would like to have instead is a full weakly nonlinear analysis of the basic hydrodynamic equations that also captures the dynamical processes during growth and that additionally considers dissipation. The derivation of a dynamic amplitude equation that may solve the addressed problems will be the subject of the following chapter 5. 2Recall that we introduced dimensionless units on page 32. 40 Nonlinear discussion using the energy method Chapter 5 The amplitude equation The physical, chemical, and biological systems [. . . ] are often quite complicated and the equations and boundary conditions describing them are not always known precisely. Even when they are known, as is the case for many hydrodynamic instabilities, a linear analysis already requires numerical evaluation and a direct analytical approach is impossible beyond threshold. The perturbation methods described below are a partial response to this situation, though calculation of the appropriate coefficients can be difficult even if the starting equations are known precisely. M. C. Cross and P. C. Hohenberg [48] The discussion from the previous chapters provides us with a first understanding of the Rosensweig instability. Chapter 3 taught us, that we should interpret the stationarity of the normal field instability rather as a limiting process where the frequency of the characteristic mode vanishes. In chapter 4 we discussed the energetically favored surface patterns, but we realized some problems with the energy method. In this chapter we will discuss the nonlinear regime starting from the fundamental hydrodynamic equations and use an ǫ−expansion to access the weakly nonlinear regime1. In this context ǫdenotes the normalized difference between the actually applied magnetic field and its critical value. The information we obtained so far will be of great importance on how we finally access this regime. 5.1 Introduction We will perform a weakly nonlinear analysis of the basic set of hydrodynamic equations as pioneered by Schl¨ uter, Lortz and Busse [38] and Newell and Whitehead [47] for the Rayleigh-B´enard system to obtain a set of amplitude equations in the case of the Rosensweig instability. The weakly nonlinear analysis is a perturbative approach and rests on the assumption that close to the linear threshold the nonlinear state can be expressed by a small perturbation from the ground state which is expanded in terms of the control parameter ǫ. The lowest order in ǫdetermines the linear stability as already discussed in chapter 3. The amplitudes of these disturbances, which are still undetermined in the 1Parts of this chapter have been published in [74], others are prepared for publication [75]. 41 42 The amplitude equation linear perturbative order (cf. chapter 3), have to fulfill certain equations in the higher perturbative orders to guarantee the solvability of the nonlinear hydrodynamic equations. These amplitude equations in turn are nonlinear dynamic differential equations describing the cooperative dynamics of a set of critical modes subject to nonlinear interactions. In this introductory section we will introduce this method and will focus on key problems one encounters when applying this method to the Rosensweig instability. For a comprehensive introduction to the weakly nonlinear analysis and on amplitude equations, the reader is referred to [48, 76, 77]. 5.1.1 Nonlinear expansion Performing a weakly nonlinear analysis of the stationary state evolving slightly beyond the linear threshold Mc, we have to expand the macroscopic variables in terms of ǫ, the normalized difference between the actual applied magnetic field and the critical one {p, B,H,M}={p0,Bc,Hc,Mc}+ǫ{p(1),B(1),H(1),M(1)}+... (5.1) {v, ǫij, ξ}= 0 + ǫ{v(1), ǫ(1) ij , ξ(1)}+... (5.2) The magnetic field, however, is an externally given parameter acting as the control parameter. The series expansion of H(5.1) can therefore be reinterpreted as the definition of ǫ. Note, that this definition of ǫis not the same definition as used in chapter 4, which is supposed to be the appropriate definition in the case of the Rosensweig instability. We will see later that the definition of ǫused here leads consistently to the control parameter as used in the previous sections. In our linear discussion (chapter 3), we modeled the surface deflection ξ(x, y, t) using plane waves ξ(x, y, t) = ˆ ξeiωt−ik·r, where the amplitude ˆ ξis a common factor in all contributions of the linearized hydrodynamic equations and therefore remains undetermined. In a nonlinear discussion, we have to extend this ansatz to address the possibility of nonlinear interactions between a set of critical modes. The most general ansatz as a starting point for a nonlinear discussion is to assume Nof these characteristic modes with different orientations. Each of these modes iconsists of a right and a left traveling contribution denoted by the subscripts Rand L, respectively. Since the surface deflection as an observable has to be real, we have to add the corresponding complex conjugate which is denoted by an asterisk ξ(1) = N X i ξiR +ξiL +ξ∗ iR +ξ∗ iL = N X i ˆ ξiReiωit−iki·r+ˆ ξiLe−iωit−iki·r+ˆ ξ∗ iRe−iωit+iki·r+ˆ ξ∗ iLeiωit+iki·r(5.3) Besides the expansion of the hydrodynamic variables, we can also consider to rescale time and space in order to further separate the dynamics to capture the long wavelength and the long time scale cooperative dynamics of the individual characteristic modes. One can interprete this rescaling in the sense that we permit the amplitude ˆ ξto be slowly dependent on time2and space, respectively. In the present discussion we will discard the 2The time tas used in eq. (5.3) is then considered the “fast” time scale of the surface waves (or of 5.1 Introduction 43 possible rescaling of space and focus on surface patterns that arise homogeneously and that do not show any long wavelength variation. As mentioned already, the rescaling of time can be interpreted in the sense that the dynamics of the amplitudes ˆ ξitself takes place on the slow timescales t(1) =ǫt and t(2) =ǫ2t(5.4) so that ˆ ξiR →ˆ ξiR(t(1), t(2),...) and correspondingly for the left traveling contributions and their corresponding complex conjugates. This rescaling of time will lead to the substitution for the time derivative ∂t−→ ∂(0) t+ǫ∂(1) t+ǫ2∂(2) t+... (5.5) These are, as we will see later, the time scales of the growth of the surface spikes. 5.1.2 The solvability condition for higher orders Fredholm’s theorem and the adjoint system With the rescaling of time and the expansion of the macroscopic variables in terms of ǫ, the whole system of differential equations can be expanded in terms of ǫ. Let L0be the linear differential operator and |φi=|φ(0)i+ǫ|φ(1)i+... the macroscopic state vector. The basic hydrodynamic equations as given by eqs. (2.32-2.34) then read in general form L0|φ(1)i= 0 (5.6) L0|φ(2)i=|N(φ(1), φ(1))i+|T(∂(1) tφ(1))i(5.7) . . . = . . . where every order in ǫneeds to be satisfied separately. The first equation (5.6) represents the linearized set of equations as used in the linear stability analysis of chapter 3, where the explicit translation of equation (5.6) is given by eqs. (3.8-3.10). Furthermore, eq. (5.6) defines the kernel of the linear operator L0, given by the linear eigenvectors |φ(1)i. In the second perturbative order the set of equations (5.7) becomes inhomogeneous due to the nonlinear nature of the basic set of equations (represented by N(·,·)) and due to the rescaling of time (represented by T(·)). In the case that these inhomogeneities reproduce elements of the kernel of the linear operator L0, equation (5.7) cannot be solved. The necessary condition that the inhomogeneities need to be orthogonal to the subspace spanned by the linear eigenvectors |φiprovides us with an additional solvability condition. This condition is named after Fredholm and reads for the second order hφ| N(φ(1), φ(1))i+hφ| T(∂(1) tφ(1))i= 0 (5.8) where h·|·i denotes a suitable scalar product about which we will talk shortly. general thermodynamic fluctuations in the case of other hydrodynamic instabilities) even though it is already a macroscopic time scale. The dynamics of the amplitudes ˆ ξis in turn assumed to take place on an even slower time scale. The same arguments hold if we rescale spatial coordinates. 44 The amplitude equation ǫ AH, AS AS AH ǫAǫSǫB Figure 5.1: The general bifurcation scheme for amplitude equations of the form (5.9) according to [78, 48]. The analytical expressions for the limits of stability ǫA,ǫBand ǫSare given in the main text. The solid lines represent the stable branches whereas the dotted lines correspond to the unstable ones. Amplitude equations in general The solvability condition (5.8) provides an additional equation and the still under-determined system of equations is closed to fix the amplitude ˆ ξ. If this condition, valid in the second order, is combined with the corresponding condition in the third order, one obtains the so called amplitude equations which define possible nonlinear solutions for the amplitudes of the critical modes that are necessary to satisfy the basic hydrodynamic equations. In the absence of the inversion symmetry ˆ ξi−→ −ˆ ξi(which is the case for the Rosensweig instability), the usual structure of this amplitude equation is, written in appropriate units, given by [48] ∂tˆ ξ1=ǫˆ ξ1−ˆγˆ ξ∗ 2ˆ ξ∗ 3−h|ˆ ξ1|2+g1(θij)|ˆ ξ2|2+|ˆ ξ3|2iˆ ξ1(5.9) together with its cyclic permutations 1 →2→3→1 and where θij denotes the angle between two different critical modes that become unstable at the linear threshold. The quadratic coefficient ˆγin eq. (5.9) is nonzero only for the hexagonal pattern (θij = 2π/3). In this case this contribution dominates close to the threshold and renders the bifurcation transcritical. In 1990 Ciliberto et al. [78] discussed the set of equations given by (5.9) using a linear stability analysis to determine the stability regimes of a hexagonal pattern with respect to a stripe pattern. Later, additionally the possibility of a square pattern was considered [79, 80]. For a nonzero quadratic coefficient ˆγ, the hexagon solution is the stable surface pattern at the linear threshold. The bifurcation from the flat surface state is transcritical and hexagons remain stable even below the linear threshold as long as ǫ > ǫA=−ˆγ2 4(1 + 2g1(θij = 2π/3)) (5.10) The hexagon solution is stable for all control parameters ǫ > ǫAif g1(θij = 2π/3) <1 and 1 + 2g1(θij = 2π/3) < g1(θij =π/2) + 2g1(θij =π/6). For g1(θij = 2π/3) >1 they loose stability with respect to either stripes or squares at ǫB=ˆγ2(2 + g1(θij = 2π/3) (1 −g1(θij =π/2))2(5.11) 5.1 Introduction 45 whereas if 1 + 2g1(θij = 2π/3) > g1(θij =π/2) + 2g1(θij =π/6) this happens at ǫB=ˆγ2(g1(θij =π/2) + 2g1(θij =π/6)) (1 + 2g1(θij = 2π/3) −g1(θij =π/2) −2g1(θij =π/6))2(5.12) Stripes and squares turn out to be mutually exclusive patterns. In the case that g1(θij =π/2) <1, g1(θij =π/6)+g1(θij = 2π/3) <1+g1(θij =π/2) and for large enough control parameters ǫ > ǫS=ˆγ2(1 + g1(θij =π/2)) (1 + g1(θij =π/2) −g1(θij = 2π/3) −g1(θij =π/6))2(5.13) squares are the stable surface pattern. Otherwise the stripe pattern turns out to be stable for control parameters larger than ǫS=ˆγ2 (1 −g1(θij = 2π/3)) (5.14) A schematic bifurcation diagram according to these considerations is drawn in fig. 5.1, where AHdenotes the amplitude of hexagons and ASthe amplitude of either squares or stripes, depending on which of these patterns is the preferred one. The scalar product We have some freedom to choose a scalar product that is suitable for our discussion of the nonlinear regime. In chapter 2 we found, that the bulk magnetic equations completely decouple from the bulk hydrodynamic equations, leaving us with no control parameter in the bulk for the nonlinear regime. At a first glance, this circumstance impedes the derivation of an amplitude equation as known for instance for the Rayleigh-B´enard convection, if we used the usual scalar product given by h·|·i = lim L→∞ 1 4L2 L Z −L dx L Z −L dy ξ Z −∞ dz τ Z0 dt ¯ ·· (5.15) To circumvent this problem, an extended scalar product can be introduced that, additionally to the bulk equations, is applied to certain boundary conditions, in particular to those containing a control parameter [54]. With a scalar product like that, Lange [49] tried to derive the adjoint system in the case of the Rosensweig instability. He failed in doing so, since he could not translate the surface contributions at the deformable surface into a set of adjoint boundary conditions. The kind of scalar product used by Lange was previously introduced by Dauby et al. [54] to describe the nonlinear regime of purely surface tension driven convection, the Marangoni instability. The approach was successful, since the authors assumed a flat, undeformable surface. In the case of the Marangoni instability this assumption is comprehensible, since what one is after is the flow field that develops in the bulk and not the deformation of the surface. An assumption instead, that is not appropriate for the Rosensweig instability since the flow field for the static nonlinear regime vanishes identically and the only observable is the deformed surface. 52 The amplitude equation case of a stationary instability ¯vx= ¯ωkx kekz −2¯qk ¯q2+k2e¯qz¯q2+k2 ¯q2−k2¯ ξ(5.48) ¯vy= ¯ωky kekz −2¯qk ¯q2+k2e¯qz¯q2+k2 ¯q2−k2¯ ξ(5.49) ¯vz=i¯ωekz −2k2 ¯q2+k2e¯qz¯q2+k2 ¯q2−k2¯ ξ(5.50) For the adjoint strain field we get ¯ǫzz = 2µ2kekz −2¯qk ¯q2+k2e¯qz¯q2+k2 ¯q2−k2¯ ξ(5.51) ¯ǫxx = 2µ2 k2 x kekz −2¯qk ¯q2+k2e¯qz¯q2+k2 ¯q2−k2¯ ξ(5.52) ¯ǫyy = 2µ2 k2 y kekz −2¯qk ¯q2+k2e¯qz¯q2+k2 ¯q2−k2¯ ξ(5.53) ¯ǫxy =−4µ2 kxky kekz −2¯qk ¯q2+k2e¯qz¯q2+k2 ¯q2−k2¯ ξ(5.54) ¯ǫxz =−4iµ2kxekz −e¯qz¯q2+k2 ¯q2−k2¯ ξ(5.55) ¯ǫyz =−4iµ2kyekz −e¯qz¯q2+k2 ¯q2−k2¯ ξ(5.56) Obviously the adjoint strain components have the same structure as the components in the original case and they also show a finite static limit. However, they do not have the same units. While the strain field in the original case is dimensionless, the adjoint strain field is proportional to the shear modulus µ2. This is consistent with the scalar product (5.23), where all contributions need to have the same dimension. One could avoid the dimension of the adjoint strain field by defining a scalar product with a metric containing units in the fifth to the tenth component. The reasons why previous attempts to solve the adjoint problem have failed are, at least, twofold. One crucial part in our discussion is to treat the medium as compressible. This ensures e.g. the presence of the contribution ∼∂j∂iv(1) jin the Navier-Stokes equation. During the process of adjoining, commutativity of gradients in this term requires that the surface terms ∼¯vi∂iv(1) jand ∼¯vj∂iv(1) iare equivalent, which would be violated if incompressibility is applied before. The assumption of an incompressible fluid is therefore too strong a restriction. An even more important point is to treat the system as a dynamic one. The subtle reason for that is manifest in the dynamic boundary condition of the surface deflection. Assuming stationarity from the beginning would imply an always undeformed surface because the vertical velocity at the surface would vanish in any case. However, this velocity component needs to be finite to allow the surface to deform. The marginal point where the spikes are about to develop (or the final point where the spikes have fully developed) are then obtained as the static limit ω→0 of the full dynamic behavior. 5.3 The second perturbative order 53 5.3 The second perturbative order The fact that within our assumptions the magnetic bulk equations completely decouple from the hydrodynamic bulk equations, has two important consequences. On the one hand this allows us to discuss and solve these two systems subsequently, i.e. we first solve the magnetic part in a given perturbative order for a given surface deflection ξ, and feed back this solution into the respective order of the hydrodynamic system. The detailed discussion of the magnetic field is given in appendix B. On the other hand, however, we have to face the problem that the control parameter (the magnetization or the magnetic field in our case) does not occur in the hydrodynamic bulk equations and that the bulk equations for the magnetic system are homogeneous in all perturbative orders, which makes it impossible to obtain the control parameter in the next order by Fredholm’s theorem, only. The coupling between these two systems is, however, mediated by the surface, and more precisely by the normal stress boundary condition. Satisfying the normal stress boundary condition provides us with an additional condition supplementing Fredholm’s theorem as we will see in section 5.3.4. According to the general expression (5.7) the set of hydrodynamic bulk equations is given in the second perturbative order by ρ∂(0) tv(2) i+∂ip(2) −2µ2∂jǫ(2) ij −ν2∂j∂iv(2) j+∂j∂jv(2) i =−ρ∂(1) tv(1) i−∂jρv(1) iv(1) j−2µ2ǫ(1) jk ǫ(1) ki (5.57) ∂(0) tǫ(2) ij −1 2∂iv(2) j+∂jv(2) i=−∂(1) tǫ(1) ij −v(1) k∂kǫ(1) ij (5.58) ∂iv(2) i= 0 (5.59) The structure of these equations suggests two kind of solutions. One contribution is proportional to the main characteristic modes ξ(1) and a second one proportional to the second harmonics ξ(2) given by ξ(2) =kcX i,j (ξiξj+ξiξ∗ j+c.c.) (5.60) The corresponding boundary conditions at the surface z=ξare expanded in the same manner as the bulk hydrodynamic equations (for a detailed discussion cf. app. C). Additionally, however, one has to consider that the linear eigenvectors are dependent on z with contributions either ∼ekczor ∼eqz (eqs. (3.27-3.35)). The boundary conditions have to be evaluated at z=ξand since ξitself is expanded in terms of ǫ, one has to expand the exponential functions ekczand eqz first with respect to zto substitute afterwards the series expansion of ξ(eq. (5.2)). As a result we obtain effective boundary conditions that have to be evaluated at z= 0. For the tangential contributions these effective boundary conditions read 2µ2ǫ(2) yz +ν2∂zv(2) y+∂yv(2) z= Ω(2) yz (5.61) 2µ2ǫ(2) xz +ν2∂zv(2) x+∂xv(2) z= Ω(2) xz (5.62) where the inhomogeneities are abbreviated by Ω(2) ij and are listed in app. C, eqs. (C.11) and (C.10). The inhomogeneities for the tangential stress boundary conditions are solely 54 The amplitude equation proportional to the second harmonics ξ(2) which is different for the normal stress boundary condition 2µ2ǫ(2) zz + 2ν2∂zv(2) z−p(2) +Gρξ(2) −µHc∂zΦ(2) +µ0Hvac c∂zΦ(2)vac = Ω(2) zz −σT∆ξ(2) +µ 1 + µM(1)Mckcξ(1) (5.63) with Ω(2) zz given in eq. (C.12). Finally, the kinematic boundary condition describing explicitly the deformable surface reads in second order ∂(0) tξ(2) +∂(1) tξ(1) + (v(1) i∂i)ξ(1) =v(2) z+ξ(1)∂zv(1) z(5.64) The last contribution in eq. (5.64) is due to the fact, that in second order the surface, at which the boundary conditions have to be evaluated, is already deflected. 5.3.1 The solvability condition in second order The general solvability condition discussed in section 5.1.2 is applied to the set of second order equations (5.57-5.59) and explicitly reads h¯vi|−∂(1) t(ρv(1) i)−∂j(ρv(1) iv(1) j−2µ2ǫ(1) jk ǫ(1) ki )i+h¯ǫij |−∂(1) tǫ(1) ij −v(1) k∂kǫ(1) ij i= 0 (5.65) At this point one might be tempted to use the fact that the Rosensweig instability is a static one (in linear approximation) and substitute ω(0) =σ(0) = 0 as well as the static limits of the adjoint and original eigenvectors into condition (5.65). The solvability condition would then reduce to h¯ǫij |−∂(1) tǫ(1) ij i= (±iω(1) +σ(1))h¯ǫij |ǫiji= 0 (5.66) corresponding to the solution ω(1) = 0 = σ(1). Here, we have replaced ∂(1) tby ±iω(1) + σ(1) (for rightand left-traveling waves, respectively) implying a normal mode ansatz for the time dependence of the amplitudes. Of course, ω(0) = 0 is the correct solution in the stationary limit. However, in that limit the connection between bulk equations and boundary conditions is lost (cf. eqs. (2.34) and (2.44)) and an amplitude equation cannot be derived. Therefore, one must still treat the system as fully dynamic at least at those places related to the kinematic boundary condition and to the velocity/strain relation, and satisfy Fredholm’s theorem with the time derivative ∂(0) tbeing finite. One can, however, at non-crucial instances simplify the calculations by the fact that ω(0) is small, but only at the very end one can take ω(0) ≡0. The solvability condition (5.65) consists of two different parts. One containing spatial derivatives and the other the (scaled) time derivative ∂(1) t. We first discuss the latter part. The integration upon xand yis straightforwardly done and only retains contributions that are proportional to δ(ki−kj). After integration with respect to zwe end up with 5.3 The second perturbative order 55 the following expression, h¯vi|∂(1) t(ρv(1) i)i+h¯ǫij |∂(1) tǫ(1) ij i =iω(1)n8µ2 kc(k2 c+q2)2 q(kc+q)3−ρ([ω(0)]2−[σ(0)]2)4k6 c+ 6k5 cq+ 6k4 cq2+ 6k3 cq3+ 2k2 cq4 qk3 c(kc+q)3o ×ˆ ξ∗ iL ˆ ξiRe2iωt −ˆ ξiL ˆ ξ∗ iRe−2iωte2σt +σ(1)n8µ2 kc(k2 c+q2)2 q(kc+q)3−ρ([ω(0)]2−[σ(0)]2)4k6 c+ 6k5 cq+ 6k4 cq2+ 6k3 cq3+ 2k2 cq4 qk3 c(kc+q)3o ×ˆ ξ∗ iL ˆ ξiRe2iωt +ˆ ξiL ˆ ξ∗ iRe−2iωt +ˆ ξiR ˆ ξ∗ iR +ˆ ξiL ˆ ξ∗ iLe2σt (5.67) For the second order contributions we finally get h¯vi|∂(1) t(ρv(1) i)i+h¯ǫij |∂(1) tǫ(1) ij i =iω(1)4µ2kc(ˆ ξ∗ iL ˆ ξiR −ˆ ξiL ˆ ξ∗ iR) + σ(1)4µ2kc(ˆ ξ∗ iL ˆ ξiR +ˆ ξiL ˆ ξ∗ iR +ˆ ξiR ˆ ξ∗ iR +ˆ ξiL ˆ ξ∗ iL) (5.68) where the static limit has safely been performed. Up to now it has been possible to do the calculations without specifying the actual number of modes contributing to the nonlinear pattern and the results are applicable for any value of Nand in particular for any angle between these modes. This is changed when the second part of eq. (5.65), containing the spatial derivatives, is considered. Two of these terms turn out to be irrelevant for the second order solvability condition since they are at least proportional to [∂(0) t]2and therefore vanish in the static limit. The only relevant term, 2µ2h¯vi|∂(0) j(ǫ(1) jk ǫ(1) ki )i, generally vanishes, except when three linear modes oriented at 2π/3 relative to each other are interacting. This hexagonal order is enforced by the integration upon xand y. Integrating with respect to zyields in lowest order of ω(0) and σ(0) 2µ2h¯vi∂(0) j(ǫ(1) jk ǫ(1) ki )i=−3iω(0)µ2k2 cˆ ξ1Rˆ ξ2Rˆ ξ3R−ˆ ξ1Lˆ ξ2Lˆ ξ3L+ˆ ξ1Rˆ ξ2Rˆ ξ3L+ˆ ξ1Rˆ ξ2Lˆ ξ3R −ˆ ξ1Lˆ ξ2Rˆ ξ3R−ˆ ξ1Lˆ ξ2Lˆ ξ3R−ˆ ξ1Lˆ ξ2Rˆ ξ3L+ˆ ξ1Rˆ ξ2Lˆ ξ3L−c.c. −3σ(0)µ2k2 cˆ ξ1Rˆ ξ2Rˆ ξ3R+ˆ ξ1Lˆ ξ2Lˆ ξ3L+ˆ ξ1Rˆ ξ2Rˆ ξ3L+ˆ ξ1Rˆ ξ2Lˆ ξ3R +ˆ ξ1Lˆ ξ2Rˆ ξ3R+ˆ ξ1Lˆ ξ2Lˆ ξ3R+ˆ ξ1Lˆ ξ2Rˆ ξ3L+ˆ ξ1Rˆ ξ2Lˆ ξ3L+c.c.(5.69) Eqs. (5.68) and (5.69) are the two parts that enter the solvability condition eq. (5.65), which we are now going to solve. The imaginary part yields the condition 4iω(1)(ˆ ξ∗ iL ˆ ξiR−ˆ ξiL ˆ ξ∗ iR) = −3iω(0)kc(ˆ ξ1Rˆ ξ2Rˆ ξ3R−ˆ ξ1Lˆ ξ2Lˆ ξ3L+ˆ ξ1Rˆ ξ2Rˆ ξ3L+ˆ ξ1Rˆ ξ2Lˆ ξ3R −ˆ ξ1Lˆ ξ2Rˆ ξ3R−ˆ ξ1Lˆ ξ2Lˆ ξ3R−ˆ ξ1Lˆ ξ2Rˆ ξ3L+ˆ ξ1Rˆ ξ2Lˆ ξ3L−c.c.) (5.70) This condition is identically fulfilled by the ansatz ˆ ξiL =ˆ ξiR =ˆ ξiand ˆ ξ∗ iL =ˆ ξ∗ iR =ˆ ξ∗ i(5.71) which is the solution one expects for the stationary case, since in that limit one cannot distinguish right from left traveling waves. Using this result for evaluating the real part, we obtain 2σ(1) X i ˆ ξiˆ ξ∗ i=−3σ(0)kc(ˆ ξ1ˆ ξ2ˆ ξ3+ˆ ξ∗ 1ˆ ξ∗ 2ˆ ξ∗ 3) (5.72) 56 The amplitude equation which obviously is solved by σ(1) ˆ ξ1=−σ(0)kcˆ ξ∗ 2ˆ ξ∗ 3and |ˆ ξ1|2=|ˆ ξ2|2=|ˆ ξ3|2(5.73) and all its cyclic permutations 1 →2→3→1 and their complex conjugates. Equation (5.73) tells us, that the slow variable σ(1) scales in the bulk with σ(0), indicating that σ(1)/σ(0) stays finite in the static limit. This behavior is mediated by the the kinematic boundary condition dtξ=vz(2.44). As a consequence, the velocity field as well as the adjoint velocity field are proportional to the time derivative as we realized in eqs. (3.273.29) and (5.48-5.50). This is physically reasonable, since in the case of the Rosensweig instability the velocity field vanishes if the surface pattern has fully developed and the hydrodynamic bulk equations are trivially fulfilled by v≡0, the same solution as for the initially undeformed ground state. This singular behavior, unique for the Rosensweig instability, is scaled out by the choice of a dimensionless time derivative ˜ ∂(1) T=σ(1)/σ(0) for the bulk hydrodynamic equations. Using this time derivative, eq. (5.73) can be rewritten as ˜ ∂(1) Tˆ ξ1=−kcˆ ξ∗ 2ˆ ξ∗ 3(5.74) Equation (5.74) gives the relation among the three amplitudes of the second order deflection, ξ(1), characteristic for hexagon patterns. For any other regular pattern the right hand side of eq. (5.69) is zero implying, that there is no nonlinear interaction between two different modes in the second order for those patterns. What is missing in eq. (5.74), which in a sense can be viewed as a primitive form of an amplitude equation, is a contribution proportional to the control parameter M(1). This is due to the fact, that the two bulk systems of magnetic and hydrodynamic equation decouple completely. The control parameter enters the amplitude equation via the normal stress boundary condition, the only way magnetic and hydrodynamic subsystems are interacting. 5.3.2 Solutions proportional to the characteristic modes ξ(1) Before we can exploit the normal stress boundary condition in section 5.3.4, we have to determine the solution of the hydrodynamic contributions, eqs. (5.57-5.62). From Fredholm’s theorem we learned, under what conditions we can find a solution to the system of equations in the second perturbative order. We distinguish solutions of the system of equations that are either proportional to ξ(1) or proportional to ξ(2). In this subsection we concentrate on the part proportional to ξ(1). Inspired by the linear discussion, we use a scalar ϕ(2,1) and a vector potential Ψ(2,1) for the potential and the vorticity flow, respectively. For the contributions proportional to the main characteristic modes ξ(1), the governing equations read ∆ϕ(2,1) = 0 (5.75) ρ∆∂(0) tϕ(2,1) + ∆p(2,1) =−ρ∆∂(1) tϕ(1) (5.76) ρ(∂(0) t)3Ψ(2,1) i−˜µ2∆∂(0) tΨ(2,1) i=−µ2∆∂(1) tΨ(1) i−ρ(∂(0) t)2∂(1) tΨ(1) i(5.77) with the abbreviation ˜µ2=µ2+ν2∂(0) t. On the right hand side of these equations the first order (linear) potentials act as inhomogeneities. 5.3 The second perturbative order 57 The appropriate boundary conditions for the flow potentials are derived in appendix C.3 and read for tangential stress ˜µ2∂2 z−∂2 yΨ(2,1) x+ ˜µ2∂y∂xΨ(2,1) y+ 2˜µ2∂z∂yϕ(2,1) = 0 (5.78) ˜µ2∂2 z−∂2 xΨ(2,1) y+ ˜µ2∂x∂yΨ(2,1) x−2˜µ2∂z∂xϕ(2,1) = 0 (5.79) The physical boundary conditions have to be taken at z=ξ(1) in the second order. This leads to additional contributions in ξ(1), which have already been taken into account in the effective boundary conditions eqs. (5.78) and (5.79). The latter therefore have to be taken at z= 0. The kinematic boundary condition now involves the slow timescale t(1) and reads v(2,1) z=∂(1) tξ(1) (5.80) We start with the particular inhomogeneous solutions of eqs. (5.76) and (5.77) for the vector potential Ψand the pressure p, respectively. It can be checked that the following fields satisfy the inhomogeneous bulk equations Ψ(2,1) i=ˆ Ψ(2,1)inhom iξ(1)zeqz and p(2,1)inhom =−ρ∂(1) tϕ(1) (5.81) with the amplitudes for the vector potential given by ˆ Ψ(2,1)inhom x=−µ2+ ˜µ2 ˜µ2q∂(1) t∂yand ˆ Ψ(2,1)inhom y=µ2+ ˜µ2 ˜µ2q∂(1) t∂x(5.82) The inhomogeneous solutions do not yet satisfy the boundary conditions (5.78) and (5.79). Substituting Ψinhom into eq. (5.78) results in an additional source of tangential stress at the boundary due to the inhomogeneous solutions, which can be balanced by the homogeneous ones ˜µ2∂2 z−∂2 yΨ(2,1)hom x+˜µ2∂y∂xΨ(2,1)hom y+2˜µ2∂z∂yϕ(2,1) =∂y˜µ2 µ2+˜µ2 ˜µ2 ∂(1) tξ(1)(5.83) If we use the following ansatz for the homogeneous solutions of the flow potentials Ψ(2,1)hom and ϕ(2,1) Ψ(2,1)hom x=−∂yˆ Ψ(2,1)eqzξ(1),Ψ(2,1)hom y=∂xˆ Ψ(2,1)eqzξ(1) and ϕ(2,1) = ˆϕ(2,1)ekczξ(1) (5.84) the amplitudes ˆ Ψ(2,1) are given by ˆ Ψ(2,1) =2kc q2+k2 c ˆϕ(2,1) −2µ2+ ˜µ2 ˜µ2(q2+k2 c)∂(1) t(5.85) Note that qis the inverse decay length of the linear transverse modes with q2=k2 c+ ρ[∂(0) t]2/(µ2+ν2∂(0) t) (chapter 3) and ∂(1) tis a short hand notation for ±iω(1) +σ(1), as before. The homogeneous solution of the pressure p(2,1)hom is straightforwardly given by eq. (5.76) p(2,1)hom =−ρ∂(0) tϕ(2,1) (5.86) 58 The amplitude equation and if we exploit the kinematic boundary condition (5.80), the solution of the scalar flow potential ϕ(2,1) can be determined as ˆϕ(2,1) =q2+k2 c kc(q2−k2 c)∂(1) t−2k2 c µ2+ ˜µ2 ˜µ2(q2+k2 c)∂(1) t(5.87) With the help of the flow potentials the velocity fields are determined v(2,1) z=1 q2−k2 cq2−2µ2+ ˜µ2 ˜µ2 k2 cekcz+ 2µ2 ˜µ2 k2 ceqz −µ2+˜µ2 ˜µ2 k2 c(q2−k2 c)zeqz q∂(1) tξ(1) i (5.88) v(2,1) x=iki,x ˜µ2(q2−k2 c)L(z)∂(1) tξ(1) iand v(2,1) y=iki,y ˜µ2(q2−k2 c)L(z)∂(1) tξ(1) i(5.89) with the abbreviation L(z) = ˜µ2(q2−k2 c)−2µ2k2 cekcz kc +2µ2q2−(µ2−˜µ2)(q2−k2 c)(1 + qz)eqz q(5.90) from which the strain fields follow ǫ(2,1) zz =−µ2+˜µ2 ˜µ2 k2 cL+(z)∂(1) t ∂(0) t ξ(1) i(5.91) ǫ(2,1) ab =µ2+˜µ2 ˜µ2 ki,aki,bL−(z)∂(1) t ∂(0) t ξ(1) i(5.92) ǫ(2,1) az =iki,a µ2+˜µ2 2˜µ22 q2−k2 c2k2 cekcz−(q2+k2 c)eqz+1+qz+k2 c qzeqz∂(1) t ∂(0) t ξ(1) i(5.93) for {a, b} ∈ {x, y}with L±=2 q2−k2 ckcekcz−qeqz±1 + qz qeqz (5.94) This concludes the derivation of the second order eigenfunctions that are proportional to ξ(1). These solutions satisfy every condition except the normal stress boundary condition. The latter will be used to determine the still unknown first order correction to the control parameter, M(1), which finally enters the amplitude equation as the linear contribution. We postpone the actual derivation of these contributions to section 5.3.4. 5.3.3 Solutions proportional to the higher harmonics ξ(2) We are left with solving the system of hydrodynamic equations in the second perturbative order, eqs. (5.57-5.59), for the higher harmonic contributions proportional to ξ(2). The appropriate set of bulk equations reads, if we use again the representation with a scalar potential and a vector potential, ∆ρ(∂(0) t)2ϕ(2,2) +∂(0) tp(2,2)=∂i−2µ2∂j(v(1) k∂kǫ(1) ij )−∂(0) t∂j(ρv(1) iv(1) j−2µ2ǫ(1) jk ǫ(1) ki )(5.95) ρ(∂(0) t)2−˜µ2∆∂i∂mΨ(2,2) m−∆Ψ(2,2) i(5.96) =ǫijk∂j−2µ2∂m(v(1) l∂lǫ(1) km)−∂(0) t∂l(ρv(1) kv(1) l−2µ2ǫ(1) lm ǫ(1) km) ∆ϕ(2,2) = 0 (5.97) 5.3 The second perturbative order 59 k1 k2 k3 k4 k5 k6 Figure 5.2: The sketch shows the relative orientation of the wave vectors under consideration in the amplitude equations (5.150,5.152). It allows to discuss the stability of hexagons and squares and their interaction. The first equation determines the pressure contribution p(2,2). Since the pressure appears only in the normal stress boundary condition, this is dealt with in section 5.3.4. Next we construct a particular inhomogeneous solution of eq. (5.97) for the vector potential Ψ. The most general ansatz necessary reads Ψ(2,2)inhom k=−ǫzkl X N,M X i,j ∂lΨinhom NMij(z)ξiN ξjM +˜ Ψinhom NMij(z)ξ∗ iN ξjM +c.c.(5.98) Here, summation over all relevant modes i, j is implied (e.g. {i, j} ∈ {1,2,3},{i, j} ∈ {1,5}, and i=j= 1 for hexagons, squares, and rolls, respectively, fig. 5.2) as well as over right and left traveling waves {N, M} ∈ {R, L}, cf. eq. (5.3). Substituting this ansatz into the dynamic equations and matching the coefficients with the inhomogeneous contributions of the vorticity equation (5.96) yields the functions Ψinhom NMij(z) and ˜ Ψinhom NMij(z). Since their general form is extremely bulky, in appendix D only the coefficients Ψinhom NMij(z) and ˜ Ψinhom NMij(z) for hexagonal (ij =ji = 13 = 23 = 31) and square patterns (ij =ji = 15) as well as for stripe solutions (ij = 11) are listed. The general solution is the sum of the particular inhomogeneous and a general homogeneous solution, Ψ(2,2) k= Ψ(2,2)inhom k+ Ψ(2,2)hom k. It has to satisfy the effective tangential boundary conditions (cf. appendix C.3) ˜µ2(∂2 z−∂2 y)Ψ(2,2) x+ ˜µ2∂y∂xΨ(2,2) y+ 2˜µ2∂z∂yϕ(2,2) = ∂yX N,M X i,j (ˆ F′ NMijξiN ξjM +˜ ˆ F′ NMijξiN ξ∗ jm +c.c.) (5.99) with suitably abbreviated amplitudes ˆ F′ NMij. The special form of the right hand side is obtained, if in eq. (C.14) the first order expressions for the variables are explicitly put in. Substituting the inhomogeneous solutions Ψ(2,2)inhom iinto eq. (5.99) a modified boundary 60 The amplitude equation condition for the homogeneous solution results ˜µ2(∂2 z−∂2 y)Ψ(2,2)hom x+ ˜µ2∂y∂xΨ(2,2)hom y+ 2˜µ2∂z∂yϕ(2,2) = ∂yX N,M X i,j (ˆ FNMijξiN ξjM +˜ ˆ FNMijξiN ξ∗ jm +c.c.) (5.100) since the inhomogeneous solution does not satisfy the boundary condition. In particular, on the right hand side the inhomogeneous part of the boundary conditions at z= 0 is modified ˆ FNMij =ˆ F′ NMij +ˆ Finhom NMij (5.101) with ˆ Finhom NMij ξiN ξjM =−˜µ2(2∂2 x−∂2 z)Ψinhom NMij(z)|z=0 ξiN ξjM (5.102) Similarly one obtains the ycomponent of the tangential boundary condition starting from eq. (C.15). Now the general homogeneous solutions of Ψ(2,2)hom kand ϕ(2,2) can be obtained by using the ansaetze ϕ(2,2) =X N,M X i,j ( ˆϕNMij ek1ij zkcξiN ξjM +˜ ˆϕNMij ek2ij zkcξ∗ iN ξjM +c.c.) (5.103) Ψ(2,2)hom k=−ǫzkl X N,M X i,j ∂l(ˆ Ψhom NMij eq1ij zkcξiN ξjM +˜ ˆ Ψhom NMij eq2ij zkcξ∗ iN ξjM +c.c.) (5.104) where the characteristic wave vector kcis just used to give the amplitudes ˆϕNMij,˜ ˆϕNMij, ˆ Ψhom NMij and ˜ ˆ Ψhom NMij the same physical units as the corresponding amplitudes in the linear discussion and where, again, the first summation is over right and left traveling waves and the second one over the fundamental modes involved. In order to satisfy the Laplace equation (5.97), the inverse decay lengths k1ij and k2ij of the second order scalar potential are given by k1ij =kcp2 + 2 cos θij (5.105) k2ij =kcp2−2 cos θij (5.106) and depend on the angle between the i-th and the j-th mode. The inverse decay length for the rotational flow contributions read correspondingly q2 1ij =k2 1ij +ρ[D(0) t]2 µ2+ν2D(0) t (5.107) and accordingly q2ij by substituting k2 2ij for k2 1ij in eq. (5.107). Here, D(0) tis an abbreviation for the Fourier transformed time derivative and takes the values iω(0) +σ(0),σ(0), and −iω(0) +σ(0) when applied to RR, RL or LR, and LL modes, respectively. The bulk equations and boundary conditions are fulfilled for the amplitudes ˆ Ψhom RRij =q2 1ij ˜µ2kc(q4 1ij +q2 1ijk2 1ij)ˆ FRRij −2˜µ2k1ijkcˆϕRRij(5.108) 5.3 The second perturbative order 61 and ˆϕRRijξiRξjR =q2 1ij +k2 1ij kck1ij(q2 1ij −k2 1ij)nkcD(0) tξiRξjR −k2 1ij ˆ FRRij ˜µ2(q2 1ij +k2 1ij)ξiRξjR −2ξ(1)∂zv(1) z +1 q2 1ij +k2 1ij h(k2 1ij∂2 z+q2 1ij[∂2 x+∂2 y])ˆ Ψinhom RRij ξiRξjRiz=0o(5.109) For the last expression we explicitly used the kinematic boundary condition for the second perturbative order, eq. (5.64). In appendix D these solutions for the flow potentials are specified for hexagons, eqs. (D.12) and (D.17), and squares, eqs. (D.13) and (D.18). The amplitudes with a tilde are obtained from those without one by replacing k1ij or q1ij by k2ij or q2ij, respectively. For ˜ ˆϕRRijξiRξjR this leads to a denominator ∼k2ij, which vanishes for i=jaccording to eq. (5.106). Nevertheless, all physical quantities derived from that potential, like velocities and strain components, stay finite. The amplitudes in eqs. (5.108) and (5.109) for the RL and LL (instead of RR) components are obtained by choosing the appropriate expressions for q1ij and D(0) t, according to the rules given above. The only remaining condition not yet satisfied is the normal stress boundary condition, which we will discuss in the next section. 5.3.4 The normal stress boundary condition To find the solutions to the hydrodynamic bulk equations (5.57-5.59), it was not necessary to use the normal stress boundary condition. The same situation appears in the derivation of the linear eigenvectors. There, substituting the eigenvectors into the normal stress boundary condition yields the dispersion relation restricting the linear solution to those with a specific ω(k) relation. The second order normal stress boundary condition, as will be shown below, leads to the determination of M(1), the first correction to the control parameter entering the final amplitude equation in linear order. The second order normal stress boundary condition has been derived in appendix C.3 and is given as eq. (C.12). It consists of two parts, one is proportional to ξ(1), eq. (5.110) and the other to ξ(2). The latter equation can easily be fulfilled by splitting the pressure p(2,2) =p(2,2)B+p(2,2)Sinto one part, p(2,2)B, that is determined by the bulk equation eq. (5.95) and the other, p(2,2)S, by the ξ(2)-boundary condition. This ansatz works, if ∆p(2,2)S= 0 in the bulk. Indeed, p(2,2)S∼ξiξjek1ij zor ∼ξiξ∗ jek2ij zleads to the required result. This additional pressure contribution is due to the inhomogeneities arising in the normal stress boundary condition, in particular the one due to surface tension. Since the surface tension always acts normal to the surface, this is the only point, where it can enter the nonlinear dynamics. It just contributes to the Laplace pressure, which is proportional to the curvature of the surface, a quite intuitive result. However, this additional pressure contribution is of no importance because of two reasons. First the pressure always enters linearly the hydrodynamic bulk equations and therefore it will never give rise to inhomogeneous contributions, which have to be accounted for by Fredholm’s theorem. Second the pressure enters only the normal stress boundary condition, which is actually the governing equation for the appropriate pressure contribution in the next order. In addition, also p(2,2)Bis not needed in the following and is therefore not shown here. 68 The amplitude equation and the second order equation (5.143) by ǫ2, we obtain (ǫ2˜ ∂(2) T+ǫ˜ ∂(1) T)ǫˆ ξ1+µ2kc 2(ρG+µ2kc)ǫ2[˜ ∂(1) T]2ǫˆ ξ1=kcµ(2ǫ2McM(2) +ǫ2[M(1)]2+2ǫMcM(1)) 4(1+µ)(ρG+µ2kc)ǫˆ ξ1 −kc 2ǫ2ˆ ξ∗ 2ˆ ξ∗ 3−A′ 32µ2kc ǫ3|ˆ ξ1|2ˆ ξ1 −B′(θij =2π/3) 64µ2kc ǫ3(|ˆ ξ2|2+|ˆ ξ3|2)ˆ ξ1(5.144) By the definition of ǫand the series expansion of the magnetization M2−M2 c=Mc+ǫM(1) +ǫ2M(2) +...2−M2 c = 2ǫMcM(1) + 2ǫ2McM(2) +ǫ2[M(1)]2+... (5.145) we define the control parameter ˜ǫin the usual way (M2−M2 c) = M2 c˜ǫ(5.146) Substituting the series expansion of the time derivative in terms of ǫ(cf. eq. 5.5) ǫ˜ ∂(1) T+ǫ2˜ ∂(2) T−→ ∂T(5.147) [ǫ1˜ ∂(1) T]2−→ ∂2 T(5.148) and using the standard scaling ǫkc√Aˆ ξi−→ ξi(5.149) the amplitude equation can be written as4 ∂Tξ1+δ 2∂2 Tξ1=1 2˜ǫξ1−1 2√Aξ∗ 2ξ∗ 3− |ξ1|2ξ1−B120 A(|ξ2|2+|ξ3|2)ξ1(5.150) where we introduce the dimensionless parameter δ=µ2kc(ρG +µ2kc)−1and where the abbreviations Aand B120 are given by A=A′ 32µ2k3 c≈5.750 and B120 =B′(θij =2π/3) 64µ2k3 c≈3.544 (5.151) Starting from eq. (5.141) instead of eq. (5.140) we obtain the corresponding amplitude equation for the square pattern ∂Tξ1+δ 2∂2 Tξ1=1 2˜ǫξ1− |ξ1|2ξ1−B90 A|ξ5|2ξ1(5.152) where the cubic coefficient B90 is analogously given as B90 =B′(θij =π/2) 64µ2k3 c≈4.021 (5.153) 4Recall that ρGk−1 c=√ρGσT 5.5 Amplitude equation 69 The fact that the linear contribution on the right hand side of eqs. (5.150) and (5.152) is only proportional to the control parameter ˜ǫjustifies a posteriori our choice of the typical time scale τ0. Let us first consider the static solutions of eq. (5.150). The quadratic contribution gives rise to a transcritical bifurcation from the flat surface to a hexagonal pattern at the linear threshold. As discussed for the phenomenological amplitude equation (5.9), a bistable regime exists for negative control parameter values ˜ǫwith its lower boundary given by ˜ǫA=−1 8(A+ 2B120)(5.154) The solution for the hexagonal pattern takes the form ξi=−|ξi|eiΦifor i∈ {1,2,3}, where the magnitude of the amplitudes reads |ξi|=1+p1+8(A+2B120)˜ǫ 4√A(1 + 2B120/A)(5.155) and where the phases have to fulfill the condition PiΦi= 0. Investigating the values of the cubic coefficients we realize, that B120/A < 1 indicating that the hexagon solution is always stable with respect to stripe solutions at the linear threshold. Stripes and squares are mutually exclusive pattern and since B90 + 2B30 < A+ 2B120 and B90/A < 1, the hexagons are loosing stability with respect to squares at the critical control parameter ˜ǫBgiven by (cf. section 5.1.2) ˜ǫB=B90 + 2B30 2(A+ 2B120 −B90 −2B30)2(5.156) where the cubic coefficient B30 ≈4.188 describes the nonlinear interaction between the hexagonal and the square pattern. The square pattern is stable for control parameters larger than ˜ǫS=A+B90 2(A+B90 −B120 −B30)2(5.157) Since ˜ǫS<˜ǫB, also a bistable regime between the hexagons and squares exists. Let us now focus on the dynamical behavior of the patterns beyond the linear threshold. We assume that the hexagonal pattern with the amplitude |ξi|, eq. (5.155), has developed and disturb it homogeneously in space by a small excess amplitude r,|ξi|→|ξi|+r. The linearized amplitude equation (5.150) for the disturbances rthen reads ∂Tr+δ 2∂2 Tr=1 2˜ǫ−1 √A|ξi| −31 + 2B120 A|ξi|2r(5.158) Substituting the solution (5.155) in the right hand side of eq. (5.158) it can be simplified to −(˜ǫ/2+ |ξi|/(2√A))r, which is always negative above the linear threshold. This reflects the fact that the exponential growth of the infinitesimal disturbances of the flat surface above the linear threshold gets nonlinearly saturated by the cubic coefficients and 70 The amplitude equation dimensionless time T Amplitude |ξi| 543210 1.4 1.2 1 0.8 0.6 0.4 0.2 0 Figure 5.3: Qualitative time dependent behavior (not to scale) of the surface spikes according to eq. (5.158). The time Tas well as the amplitudes |ξi|are dimensionless variables. If the control parameter ˜ǫis slightly beyond the critical threshold the plot may be considered as the qualitative dynamics from the flat surface |ξi|= 0 to the spike surface |ξi|= 1. a stable pattern develops. Eq. (5.158) therefore takes the form of a damped harmonic oscillator which can be solved by using the ansatz r=|r|eλT with the eigenvalues λ1/2=−1 δ±s1 δ2−˜ǫ√A+|ξi| √Aδ (5.159) where the eigenfrequency Ω of the oscillator is given by Ω2=˜ǫ√A+|ξi | √Aδ . These last results are still in dimensionless units. If we choose the time scale ν2/µ2to compare dissipative and oscillatory processes as suggested by eq. (5.132), the eigenvalues read λ1/2=−(√ρGσT+µ2) ν2±s(√ρGσT+µ2)2 ν2 2−µ2(˜ǫ√A+|ξi|)√ρGσT+µ2 √Aν2 2 (5.160) This result is intuitive, since the damping rate is inversely proportional to the dissipative processes, given by ν2, whereas the eigenfrequency increases with increasing shear modulus. We also realize that the relaxation towards the equilibrium pattern becomes faster in a stronger gravitational field as well as for larger surface tensions and elastic higher shear moduli of the medium. The bifurcation from the flat surface towards hexagons is transcritical and therefore involves a non continuous transition. If the control parameter is slightly above its critical value, the still flat surface (at T= 0) can be interpreted as a disturbance to the stable stationary solution (5.155). The dynamics towards hexagons from the flat surface is then described by equation (5.158) giving rise to an overshoot and a damped oscillation towards the equilibrium value (cf. fig. 5.3). 5.6 On the Newell-Operator 71 u(x, t) ~ G Figure 5.4: Sketch of a physical situation which is described by the sine-Gordon equation (5.161). The line of pendulums connected by a torsion wire is exposed to the gravitational field, which is directed downwards. The angle from the normal is denoted by u(x, t). 5.6 On the Newell-Operator In the previous discussion on the nonlinear properties of the Rosensweig instability we assumed spatially homogeneous patterns with no long wavelength variations. By additionally rescaling the spatial coordinates in the same way as the time coordinate (5.4), one could also implement these possible variations in space as we mentioned already in section 5.1.1. As a consequence, the amplitude equation additionally contains derivatives of the amplitudes with respect to the scaled spatial coordinates. For typical nonlinear differential equations these linear contributions to the amplitude equation can be obtained systematically by a standard method exploiting the linear properties of the system [81, 82]. This method is described in [83] and we summarize some of the ideas before we discuss a possible application to the Rosensweig instability. We illustrate this standard method by assuming the following nonlinear model equation [83], the so-called sine-Gordon equation ∂2 tu−c2∂2 xu+ω2 psin u= 0 (5.161) A physical system that is described by this equation is e.g. a line of pendulums that are connected by a horizontal torsion wire, where the twist angle is denoted by u(x, t) (cf. fig. 5.4). The second contribution in eq. (5.161) is then given by the twist of the wire while, the third contribution is due to gravity. Expanded in terms of u, the sine-Gordon equation reads ∂2 tu−c2∂2 xu+ω2 pu=1 6ω2 pu3+O(u5) (5.162) The linear parts can be solved by a sinusoidal ansatz u=ae−iωt+ikx +c.c., where the frequency ωand the wave vector kare not independent from each other, but related by the dispersion relation ω2=ω2 p+c2k2(5.163) 72 The amplitude equation Note, that the dispersion relation is the Fourier transform of the linearized equation (5.161) and therefore contains only the linear properties of the basic equation. To account for the nonlinear contributions of the model equation (5.161) we can apply the same ideas as in the case of the Rosensweig instability and expand the torsion angle uin terms of a small parameter ǫ u=ǫ(u0+ǫu1+ǫ2u2+...) (5.164) where, however, ǫhas a different meaning as in the previous sections. In particular it is not connected to a control parameter, but merely models an expansion for small amplitudes of the pendulums. In addition we can also rescale time according to T1=ǫt and T2=ǫ2tand obtain by solving the different orders in ǫsuccessively the following nonlinear solution5 u=ǫaeikx−it„ω−ω2 p 4ωǫ2aa∗«−ǫ2a3 48e3(kx−ωt)+c.c. (5.165) where the solvability conditions in the second and third order in ǫread, respectively ∂T1a= 0 (5.166) ∂T2a=iω2 p 4ω|a|2a(5.167) Until now we followed the same approach as previously in our discussion of the Rosensweig instability. Additionally we will now rescale space according to X1=ǫx. To avoid resonant growth in the second order in ǫ, the amplitudes have to fulfill ∂T1a+c2k ω∂Xa= 0 (5.168) which states that the amplitude ahas to travel with the group velocity ω′=∂kω(k) = c2k ω(5.169) where one uses the dispersion relation ω(k), eq. (5.163). In the third order the solvability condition reads ∂a ∂T2 =iω′′ 2 ∂2a ∂ζ2+iω2 p ω|a|2a(5.170) with ζ= (X−ω′T1), which is also known as the nonlinear Schr¨ odinger equation. The linear contributions arising in eq. (5.170) have been calculated by using the linear properties of the basic equation (5.161), in particular the dispersion relation (5.163). The structure of these linear contributions is universal and can be collected into an operator LN=∂ ∂T2−iω′′ 2 ∂2 ∂ζ2(5.171) The question that arises is, whether we can similarly calculate the corresponding coefficients for the scaled spatial derivatives in the case of the Rosensweig instability by 5Note, that eq. (5.165) is already the nonlinear solution where the second order correction to the frequency (given by eq. (5.167)) has already been substituted. 5.7 Discussion and comparison 73 simply taking the derivative of the known dispersion relation (3.20) with respect to the wave vector k. But there is a fundamental difference between the system of equations we used to describe the Rosensweig instability (cf. section 2.2.1) and the sine-Gorden equation (5.161). We realize, that for the latter case the dispersion relation is solely determined by the sine-Gordon equation itself, in particular it is nothing but the Fourier transform of the linearized sine-Gordon equation. In the case of the Rosensweig instability we additionally have to satisfy boundary conditions. If these were only determined by the stress balance at a fixed surface, the determination of corresponding contributions to the amplitude equation could be done by just using the operator LN. In the case of the Rosensweig instability, however, we additionally have to take into account the deformability of the surface and along with it the kinematic boundary condition. If the surface deforms, also its normal vector nchanges in the course of time, which we took into account in our previous discussions by explicitly expanding the latter in terms of the surface deflection ξ. All the different orders of ninvolve gradients of the surface deflection ξas can be seen in eqs. (B.17-B.19). Upon rescaling the spatial coordinates we also must expand the gradients appearing in nin terms of ǫ, which leads to additional contributions to the higher order boundary conditions solely due to the large scale variations of the normal vector. These contributions are not contained in the linear dispersion relation and, of course, cannot be implemented into it by any means, since the dispersion relation only considers the linear properties of the system of equations and therefore assumes a still flat surface. One rather has to expand the set of boundary conditions with the scaled spatial coordinates from the beginning. The contributions to the second spatial derivative in the amplitude equation may then be separated into those due to gradients in the stress tensor (for example ∂jvi), which are the ones that follow directly from the dispersion relation, and those solely due to the deformability of the surface. Furthermore we have to evaluate the boundary conditions at the physical boundary, z=ξ. In our calculations we accounted for this fact by expanding the eigenvectors in term of ξaround z= 0. This again involves gradients with respect to z, which have to be rescaled as well and which are not contained in the dispersion relation. Additionally, one has to expect contributions to the second order spatial derivatives in the amplitude equation that are due to the bulk equations. In the case of the scaled time derivative we showed that possible contributions due to the bulk scale out in the static limit, but it is far from obvious that this is also the case for the spatial derivatives. The previous discussion suggests that also in the case of a deformable surface the linear contributions to the amplitude equation are of a common structure and can be collected into an extended operator similar to LN. The determination of the latter is, however, beyond the scope of this thesis and will be left for future work. 5.7 Discussion and comparison In this chapter we succeeded in deriving an amplitude equation for the Rosensweig instability in isotropic magnetic gels based on the fundamental hydrodynamic equations. An important step was to find the adjoint linear system of equations together with its corresponding boundary conditions in the presence of a deformable surface. Two assumptions turned out to be very important in order to find the adjoint system. Besides the dynamic treatment of the Rosensweig instability, the medium has to be considered compressible for 74 The amplitude equation the adjoining process. The reason for the latter assumption is to maintain the symmetry of the stress tensor during the adjoining process. While we can assume an incompressible medium after the adjoining process, the dynamic treatment of the system of equations turns out to be also important in the discussion of the higher perturbative orders. With the help of the adjoint system we were able to satisfy Fredholm’s theorem and to perform a weakly nonlinear analysis. It turned out that due to the decoupling of the magnetic bulk equations from the hydrodynamic ones, Fredholm’s theorem does not contain a control parameter. We solved this problem by observing that the normal stress boundary condition consists of two parts. One is proportional to the higher harmonics of the characteristic wavelength and merely increases the hydrostatic pressure in the medium. The other one is proportional to the main characteristic wave vector and serves as an additional solvability condition providing the dependence between the scaled growth rate and the control parameter. Both solvability conditions show qualitative different behavior in the static limit. While the solvability condition obtained from the normal stress boundary remains finite, the bulk contributions scale with the linear growth rate. The latter behavior is mediated by the kinematic boundary condition and has been taken into account while combining both solvability conditions into one. Furthermore it reveals the fact that both states, the initial flat surface and the final spiked one, are motionless states where the velocity field vanishes identically. While combining the bulk solvability condition with the normal stress boundary one has some freedom to choose the relative weight of the boundary with respect to the bulk via the two differently scaled time derivatives. It seems reasonable to weigh these single contributions equally with respect to each other, which is implicitly also done, for example, in the nonlinear discussions using an extended scalar product [54, 56]. Upon combining the second and the third order solvability condition following the standard procedure, we obtained a set of amplitude equations for the special cases of stripes, squares and hexagons. The latter contains a quadratic coefficient that renders the bifurcation from the flat surface to the hexagonal pattern transcritical. The calculated cubic coefficients additionally reveal that at the linear onset hexagons are the stable surface pattern. For high magnetic field strengths instead, hexagons become unstable and a square pattern develops. Both transitions, from the flat surface to hexagons and from hexagons to squares, involve bistable regions. We obtained qualitatively the same results in the case of ferrofluids, where the derivation of the corresponding amplitude equation and the determination of the nonlinear coefficients has been discussed in appendix E. The results for the static patterns in this chapter are in qualitative accordance with the bifurcation scenario obtained with the energy method (chapter 4). The cubic coefficients in this chapter, however, are independent from the elastic shear modulus and the magnetic susceptibility. This is due to the assumptions of chapter 2, where we modeled the magnetic gel as a linear elastic and a linearly magnetizable medium and where we neglected magnetostrictive effects. The results in this chapter are therefore valid for a finite magnetic susceptibility and for finite shear moduli. As we realized in chapter 4, this was different for the energy method even though the same approximations were used. However, we minimized the energy density with respect to the higher harmonics and the main characteristic modes independently. As a consequence, the forth order coefficients of the energy method (these coefficients qualitatively correspond to the cubic coefficients in an ǫ−expansion) showed an inverse proportionality on the control parameter ˜ǫ. This dependence is omitted in the subsequent discussions of the energy method for simplicity, 5.7 Discussion and comparison 75 which renders this approach valid in the asymptotic limit of a vanishing magnetic susceptibility only. In retrospect this minimization procedure and the simplification afterwards seems to be unsystematic. In addition to the static properties of the surface patterns, the analysis in this chapter provides us with nonlinear dynamical processes. We obtain the typical first order time derivative that describes the growth of the surface spikes beyond the linear threshold but that also accounts for the dissipative processes in the medium. The typical time scale of the growth (or relaxation) processes increases for increasing viscosities and becomes smaller for increasing surface tension and shear moduli. Additionally, however, we find a second order time derivative in the case of magnetic gels. The analysis in this chapter elucidated the main aspects of the underlying mechanism that lead to the Rosensweig instability. But it also unraveled that for a better quantitative understanding additional phenomena have to be taken into account. Two nonlinear properties have been neglected. The nonlinear magnetization behavior, that already effects the linear threshold, and nonlinear elastic properties. Additionally, the magnetostrictive effect might influence the bifurcation behavior in magnetic gels. 76 The amplitude equation Chapter 6 Rosensweig instability in films and membranes 6.1 Motivation In the previous chapters we have emphasized, that in the scope of our assumptions the driving force of the Rosensweig instability is manifest in the boundary only, i.e. the bulk equations for the hydrodynamic variables and for the magnetic field are decoupled. On the one hand this enabled us to find the eigenvectors for the magnetic field and the hydrodynamic variables separately, but on the other hand we had to find a solution on how to implement the driving force into the amplitude equation. With the previous discussions on the Rosensweig instability in mind the question arises, how the characteristics of the Rosensweig instability will change if we reduce the elastic medium to be a boundary layer only, namely if we deal with thin films or membranes made of a magnetic gel. Rannacher and Engel focused in [84] on a thin but still finite film thickness allowing for peristaltic perturbations of the initial state where both surfaces were parallel. Here, however, we want to discuss the linear stability of the membrane in the limit of a macroscopically vanishing film thickness treating a quasi-two dimensional elastic magnetic medium. This restricts us to modes where both surfaces are distorted in phase keeping the film thickness constant1. Before we start with the Rosensweig instability in magnetic membranes, we elaborate on the thin film limit in order to obtain the viscoelastic properties of the membrane. This part briefly summarizes the work of Harden and Pleiner [86]. 6.2 Film properties in viscoelastic media If we discuss thin films or membranes sandwiched between two fluids, it is reasonable to start with three media, the membrane mof thickness d, whose mid-plane is placed at z= 0, the fluid aabove the membrane (z > d/2) and the fluid bbelow the membrane (z < −d/2) as depicted schematically in fig. 6.1. In our discussion of the Rosensweig instability, we will allow the fluids aand bto be ferrofluids with the magnetic permeabilities µaand µb, respectively. Besides their superparamagnetic property, we assume that they behave as usual Newtonian liquids. We will first concentrate on the hydrodynamic degrees of 1The discussions in this chapter have been published in [85]. 77 84 Rosensweig instability in films and membranes 6.6 Rosensweig instability As done in section 3.4, eqs. (6.19-6.20), and (6.27) can be slightly reinterpreted: These are conditions for an external field strength B, at which a surface perturbation ξwith wave vector kand (real) frequency ω0relaxes to zero or grows exponentially for σnegative or positive, respectively (ω=ω0−iσ). For σ= 0 such a surface perturbation is marginally stable (or unstable) against infinitesimal disturbances, since eq. (6.19) has been obtained by linearizing the dynamic equations and the boundary conditions about the ground state. The functions ω0and Bstill depend on kand the latter has to be minimized with respect to kin order to get the true linear instability threshold. There is no guarantee that a threshold exists for a finite frequency due to the additional requirement ω2 0>0. We therefore discuss first the stationary case. Assuming ω0= 0 the threshold condition σ= 0 leads to ˜ C(z)(k, ω=0) = 0. We will further analyze this condition for the special cases, where the surface magnetism can be either neglected or has only a small influence in section 6.6.1, a permanently magnetized film with no magnetic contrast of the surrounding fluids in section 6.6.2, while the general case, when both destabilizing magnetic field effects are present, is discussed in section 6.6.3. The possibility of an oscillatory instability and the case of hydrodynamically symmetric configurations is discussed in the final subsection 6.6.4. 6.6.1 Stationary, asymmetric case without surface magnetism Dealing with the case of a strong magnetic contrast between the upper and lower bulk fluid (e.g. vacuum and a ferrofluid, respectively), we neglect the surface magnetic effect. Experimentally, this case can be realized by a ferrogel (or a non-magnetic gel) on top of a ferrofluid and vapor or vacuum above the film. In that case the threshold magnetic field is κ1B2(k) = ˜γk +ρ(b)G k+cbk3(6.28) and is finite for a non-zero magnetic contrast, χa6=χb, of the bulk fluids, only. Minimizing with respect to kleads to the critical wave vector k2 c=1 6cbp˜γ2+ 12ρ(b)Gcb−˜γ(6.29) and the critical magnetic field Bc=B(k=kc). Slightly above the minimum, the curvature of the marginal stability curve is given by κ1(B(k)2−B2 c) = (1/kc)p˜γ2+ 12ρ(b)Gcb(k−kc)2(6.30) The linear threshold conditions for this stationary instability are completely independent of the viscosities of both, the underlying fluid as well as the film itself, resembling the case of bulk free surface Rosensweig instabilities in ferrofluids and ferrogels (cf. chapter 3). In contrast to the latter case, here the critical wave vector does depend on the transverse elastic properties (c⊥) of the ferrogel (through ˜γ) as well as on the bending elastic modulus cb. The reason is that both effects enter the normal stress boundary condition with a k-dependence different from that of the magnetic field (cf. eq. (6.27)), or to 6.6 Rosensweig instability 85 phrase it differently, the magnetic field deformations do not introduce a specific internal length scale compared to ordinary 3 D elasticity, but they do in relation with surface elasticity. On the other hand, the linear growth rate σof the most unstable mode is completely determined by the (transverse) viscous properties of the film and the bulk fluid σ=κ1(B2−B2 c) ν⊥kc+νbk3 c+ 2ν(b) 2 (6.31) where the wave vector of the most unstable mode ku=kc(1 −˜ δ) with ˜ δ=κ1(B2−B2 c) 2˜γ+ 12cbk2 c ν⊥+ 3νbk2 c ν⊥kc+νbk3 c+ 2ν(b) 2 (6.32) is slightly smaller than the critical one. If the dissipation in the film or membrane can be neglected, the growth rate, σ=κ1(B2−B2 c)/(2ν(b) 2), is given by the bulk fluid viscosity as in the case of a bulk ferrofluid or ferrogel (cf. section 5.3.4), and the most unstable mode is the critical one, ku=kcin linear order [27]. The linear threshold conditions for the stationary instability are also independent of the longitudinal material properties (ǫ,ck) of the film and therefore indistinguishable from those of an incompressible film. Since we are operating in the long wavelength limit, usually the bending elasticity is less important than ordinary elasticity, except for very thin films, where c⊥and γare zero or can be neglected. In the former case, in particular for ρ(b)Gcb≪˜γ2the critical quantities can be simplified to k2 c=ρ(b)G ˜γ1−3ρ(b)Gcb ˜γ2(6.33) κ1B2 c= 2pρ(b)G˜γ1 + 1 2 ρ(b)Gcb ˜γ2(6.34) Of course, the critical wavelength and field increase with increasing elasticity and scale at onset with the relevant elastic modulus of the ferrogel c⊥with exponents 1/2 and 1/4, respectively. In the pure ferrofluid case, c⊥= 0 = cb, the critical values are identical to those of the usual Rosensweig instability, i.e. there is no difference between a bulk free surface and a film, except for a possible difference in the surface tension σTin the two cases. In the opposite, bending dominated regime, ρ(b)Gcb≫˜γ2the critical values are k4 c=ρ(b)G 3cb 1−˜γ p3ρ(b)Gcb!(6.35) κ2 1B4 c=16 9ρ(b)Gp3ρ(b)Gcb 1 + 3 2 ˜γ p3ρ(b)Gcb!(6.36) Here, the critical wavelength and field scale at onset with the bending elastic modulus of the ferrogel film cbwith exponents 1/4 and 1/8, respectively. 86 Rosensweig instability in films and membranes 6.6.2 Permanent-magnetic, symmetric case We now consider a film consisting of a permanent-magnetic gel with the intrinsic (surface) magnetization M′ 0to be rigidly anchored to the elastic degrees of freedom. In particular we choose it to be always antiparallel to the external field B. In this section we just discuss the case of a magnetic symmetry between the bulk fluids aand b, being either both non-magnetic or having the same magnetic susceptibility. For this case the magnetic contribution stemming from the left hand side of eq. (6.22) cancels (κ1in eq. (6.25) is zero) and only the divergence of the magnetic membrane stress tensor gives a field dependent contribution to the threshold condition for a stationary instability ˜ C(z)(k) = ∆ρ Gk−2+ ˜γ−M′ 0B+cbk2= 0 (6.37) Here, ∆ρis the density difference between the medium above and below the film or membrane. Eq. (6.37) leads to an instability with a characteristic mode k4 c=∆ρ G cb (6.38) when the applied critical field reaches the threshold value Bc=1 M′ 0˜γ+ 2pcb∆ρ G.(6.39) Note that the critical wave vector is independent of M′ 0, dominated by the bending elastic coefficient, and rather similar to eq. (6.35). The threshold field is inversely proportional to the magnitude of the intrinsic permanent magnetization. 6.6.3 The general case We now discuss the general case, where both destabilizing magnetic field effects are present, i.e. a uniaxial film with the permanent magnetization opposite to the field and a magnetic contrast between the two surrounding fluids. The condition for marginal stability against stationary convection, eq. (6.27), ˜ C(z)(k) = ∆ρ Gk−2+ ˜γ+cbk2−M′ 0B−κ1B2k−1= 0 (6.40) leads to the neutral curve B=B(k). In principle, one could expect a competition between the two different instabilities described in the two preceding subchapters, i.e. a transition from a stationary instability with a wave vector like that of eq. (6.29) to one like that of eq. (6.38). The minimum threshold condition dB/dk = 0 allows us the calculate the critical wave vector kcas a real root of κ1(3cbk4 c+ ˜γk2 c−∆ρ G)2+ 2M′2 0(cbk4 c−∆ρ G)k3 c= 0 (6.41) In dimensionless form eq. (6.41) contains three relevant numbers RB=cb/(∆ρ Gd4), RE= ˜γ/(∆ρ Gd2), and RM=M′2 0/(∆ρ Gκ1d3), if the wave vector is scaled by the film thickness d. For RM> RB, REthere are two different minimum solutions, kc1and kc2, possible. However, the critical fields associated with these wave vectors, Bc1=B(kc1) and Bc2= 6.7 Discussion 87 B(kc2), are never equal, except in the limit RM→ ∞, where kc1=−kc2and the case of section 6.6.2 is reached. For RM.RB, REthere is only one minimum solution of eq. (6.41), which tends for smaller RMto the solution of section 6.6.1. Thus, for a given set of material parameters there is always one definite instability at a minimum Bc, and never a competition between instabilities of different kc. 6.6.4 Additional remarks Finally we will explore the possibility of an oscillatory instability. If we assume that the film compressional modulus, ˜ε, and the longitudinal elastic modulus ckand viscosity νk can be neglected (incompressible film), one can show that the curve of marginal stability, B=B(k, ω) has its minimum at ω0= 0, and thus any oscillatory state would have a higher threshold than the stationary one. In the general case, the proof of the nonexistence of an oscillatory instability is much more involved. One can show (under the proviso that ν⊥+νbk2and νkare of the same order of magnitude) that there is no finite frequency possible if ˜ǫk2≤√3(˜γk2+ ∆ρ G +cbk4). In the opposite case the threshold of an oscillatory instability (if it exists) is higher than that for the stationary one. If the densities of the two bulk fluids above and below the film or membrane are identical, their gravitational influence on the interface undulations cancels. The thin film itself is not sensitive to gravity, since its volume is going to zero in the two-dimensional limit. Therefore, the gravity term is absent in the normal stress boundary condition and the linear instability criterion in the stationary case is C(z)= 0 (instead of ˜ C(z)= 0). The general marginal stability curve B=B(k) then has a minimum for a vanishing k2 c∼∆ρ G →0 leading to a vanishing threshold B4 c∼∆ρ G →03. The lowest wave vector for a finite experimental set-up of horizontal dimension L,kc= 2π/L gives κ1B2 c≈2π˜γ/L, since effects of bending and surface magnetization are negligible for large L. This means there is only one surface excitation (spike) in the whole sample, governed by the (effective) surface tension. This is a very well known scenario, theoretically and experimentally [92], for ordinary ferrofluid free surfaces under strongly reduced gravity conditions. 6.7 Discussion The driving force of the Rosensweig instability manifests itself in the boundary conditions, only, for ferrofluids as well as ferrogels (if magnetostriction is neglected). The question arises, how will the characteristics of the onset of the instability change, if the elastic medium itself is very thin so that it can be considered as a film or a membrane. In the present chapter we have addressed this question by extending previously obtained dispersion relations of surface waves at a half-space ferrogel boundary to those of the membrane surfaces. The very thin membrane is surrounded by two Newtonian fluids that can be ferrofluids with different magnetic properties. Possible generalizations to viscoelastic surrounding fluids and to viscoelastic (rather than elastic) membranes have been sketched. The magnetic film itself can be either a superparamagnetic isotropic magnetic gel, or an anisotropic ferromagnetic one having a finite intrinsic magnetization. 3Since the limits k→0 and κ1→0 are not interchangeable, the formulas of section 6.6.2 are not applicable to the case of vanishing gravity; rather, one has to establish relations between the smallness of ∆ρ G, the smallness of kc, and the smallness of κ1, in order to get a definite result for Bcin that case. 88 Rosensweig instability in films and membranes Apart from the material properties of the surrounding fluids, the derivation of dispersion relations in thin films makes use of certain effective (frequency and wave vector dependent) surface material parameters that describe the internal film properties. For surface waves an effective elastic surface modulus is introduced that contains the intralayer elastic and viscous properties. In the same manner we introduce in our discussion an effective surface permeability for the magnetic film describing the induced or permanent magnetic film properties, which generally are different from the bulk quantities. In recent experiments [93] this kind of difference between bulk and surface behavior in the magnetic properties has been seen when spin coating a ferrofluid. In our discussion we have restricted ourselves to modes where the upper and the lower surface of the membrane move in phase, resulting in an undulated membrane of constant thickness (in linear approximation). This is complementary to a previous discussion of films of finite thickness, where just peristaltic motions where taken into account [84]. For superparamagnetic films we get two different additional contributions to the dispersion relation. One is due to the magnetic asymmetry between the surrounding liquids. This contribution is of the same character as the magnetic part of surface waves in the halfspace case and vanishes in the symmetric case (no magnetic contrast between the two surrounding fluids). The second contribution comes from the magnetizability of the thin film itself. This last contribution, however, acts always stabilizing and effectively stiffens the membrane. Thus, a (symmetric) superparamagnetic membrane in air, for instance, will never become unstable to undulations of the type described here. An intuitive reason for this is the fact that in the symmetric case the magnetic field is not distorted in the limit of an infinitely thin membrane even if the membrane itself is subject to small perturbations. As a result, no destabilizing force acts on the magnetic dipoles in the film. In the present discussion we therefore focus on the case of high magnetic contrast between the surrounding fluids discussing the influence of the surface elastic properties to the characteristics of the Rosensweig instability. Due to the elastic and bending elastic surface properties, the characteristic mode at onset is shifted to higher wavelengths and the critical magnetic field towards higher field strengths. We can distinguish the limiting cases of a bending dominated regime and the regime where surface elasticity plays the important role. For an anisotropic magnetic thin film or membrane, its permanent magnetization can lead to the Rosensweig instability, if the applied field is strong enough and oriented antiparallel to it. In this case the magnetic asymmetry between the surrounding liquids is not needed and such a magnetic film surrounded by air can become unstable. Finally, the general case of an anisotropic magnetic membrane separating two liquids of different magnetic properties has been discussed. In principle, there is a competition between the previously discussed instability mechanisms (either based on the magnetic contrast or on the permanent film magnetization), which generally occur at a different wavelength. However, it turns out that such a pattern competition does not occur in the system under consideration, because the critical magnetic field according to one of the mechanisms is always smaller than the other one. Only in the limiting case of infinitely high intrinsic magnetization (infinitely low magnetic contrast) both critical fields can be equal. In this case, however, the different characteristic modes at onset are of the same magnitude, but of opposite sign, and no competition of two different spatial modes arises. Chapter 7 The adjoint system for the Marangoni convection In this chapter we will apply the method that we introduced in chapter 5 to derive the adjoint system for the Rosensweig instability to the case of the Marangoni instability. Also in the case of the Marangoni instability the adjoint system taking into account the deformability of the surface was unknown and nonlinear discussions where therefore restricted to flat surfaces1. 7.1 Introduction to Marangoni convection The Marangoni instability is a prominent example of a surface tension driven instability. If a temperature gradient is applied to a layer of a fluid with a free surface, the conducting state becomes unstable beyond a certain critical temperature gradient when heating is done from below and convection starts. For thick layers the instability is driven by buoyancy (classical Rayleigh-B´enard convection), but if the layer is smaller than about 1 mm, Pearson [94] proposed fluctuations of the surface tension, that arise due to temperature fluctuations at the free surface, being the mechanism driving the convection. The Marangoni instability was investigated extensively theoretically. Nield [95] first compared linearly the competition between the buoyancy and the surface tension driven instability mechanism, but both, Pearson and Nield, still considered a flat, undeformable surface. Scriven and Sternling [96] and later on Smith [97] accounted for a free deformable surface. In Ref. [96] only capillary effects have been considered and an always unstable conducting regime was obtained due to missing stabilizing gravitational contributions for the long wavelength limit. Smith discussed a layer model, a light fluid above a heavier one. A comprehensive linear study was first given by Takashima [50, 51] in 1981, who also discussed the possibility of an oscillatory branch that could arise for negative Marangoni numbers. P´erez-Garc´ıa and Carneiro [98] generalized this approach to the combination of both, surface driven and buoyancy driven convection, which matches the results of Takashima in the limit of negligible buoyancy forces. All nonlinear theoretical discussion up to now assumed a flat, undeformable surface. Rosenblat et al., for instance, discussed the nonlinear regime in a cylindrical container [52] in terms of an extended Galerkin method. This discussion was later on extended to rectangular vessels [53, 54], but for this 1This chapter is based on [74]. 89 90 The adjoint system for the Marangoni convection z ~n x, y z= 0 z=dξ ∇T ~ G Figure 7.1: Qualitative sketch of the geometry under consideration in the case of pure Marangoni convection. The fluid is confined between the rigid surface at z= 0 and the deformable surface initially at z=d. The deflection of the deformable surface with respect to the flat surface is denoted by ξwith its unit normal vector npointing upwards. The applied temperature gradient is always parallel, the acceleration due to gravity Galways antiparallel to the z−axis. approach no adjoint system is needed. The case of a horizontally infinite layer of fluid was studied in Refs. [55] and [99]. In Ref. [56] a two layer model was considered, where the adjoint system was derived using the ansatz of [99] provided the surface is flat. Inspired by the result of the case of the Rosensweig instability, we apply the same formalism (section 5.2) to the case of stationary Marangoni convection to find the adjoint system of equations for this case as well. However, there exists a crucial difference between these two instabilities. While in the case of magnetic fluids the external force acts normal to the free surface, in the case of Marangoni convection the external force is acting tangentially to the surface. We can therefore verify our formalism for any arbitrary direction of the driving force. This external force for the Marangoni instability is, as mentioned already, mediated by temperature fluctuations. The surface tension σT is therefore assumed to be temperature dependent and reads in a series expansion up to linear order in T σT(T) = σT(TR)−γ(T−TR) (7.1) with the change in surface tension due to temperature fluctuations γ=−(∂σT(T)/∂T)T=TR and where TRrepresents an arbitrary reference temperature. For the following discussion we will refer to σT(TR) as σT. 7.2 Basic equations and the adjoint system To find the adjoint system for the purely surface driven convection we assume a viscous Newtonian fluid. As done in the case of the Rosensweig instability, we assume it to be compressible with a barotropic equation of state at the beginning, but in the end we will again use the limit of an incompressible fluid. Additionally we have to incorporate the equation of heat transport with the temperature Tand the thermal diffusivity ˜χ. All the other variables are denoted in the same way as in the previous discussion. As we want to discuss the purely surface driven contribution of convection, all contributions due to 7.2 Basic equations and the adjoint system 91 buoyancy are neglected. The system of equations thus reads ∂tρ+∂k(ρvk) = 0 (7.2) ∂tgi+∂jTij =ρGi(7.3) ∂tT+vj∂jT= ˜χ∂j∂jT(7.4) The stress tensor Tij of the fluid under consideration takes the form Tij =vjgi+pδij −ν2(∂jvi+∂ivj)−ˆν(∂kvk)δij (7.5) We require the normal as well as the tangential stress at the free surface between the Newtonian fluid and the vacuum to be balanced, leading to the normal and tangential boundary conditions, respectively p−ρ0Gξ −2ν2∂zvz−ˆν(∂kvk) = −σT(∂2 x+∂2 y)ξ(7.6) ν2(∂yvz+∂zvy) = −γ∂yT+γβ∂yξ(7.7) ν2(∂xvz+∂zvx) = −γ∂xT+γβ∂xξ(7.8) where βdenotes the applied temperature gradient across the fluid. Additionally we have to specify the phenomenological boundary conditions at the surface. Again the kinematic boundary condition (2.44) for a free deformable surface is assumed to hold. Second, we assume the heat flux Qthrough the surface to be proportional to the local temperature gradient, where κdenotes the coefficient of (surface) heat conduction. Q(T) = −κ∂zT(7.9) At the bottom (z= 0) of the container we assume the usual rigid boundary conditions vi=∂zvz=T= 0 (7.10) The state vector |φinow becomes six dimensional and is defined by |φi= (vx, vy, vz, p, T, ρ) (7.11) so that the system of equations reads again in the general form L0|φi= 0 (7.12) We use the usual scalar product, however, now the z−integration is bounded between the bottom plate (z= 0) and the free surface (z=ξ). h¯ φ|φi= lim L→∞ 1 4L L Z −L dx L Z −L dy ξ Z 0 dz t Z 0 dt ¯ φφ (7.13) The adjoint linear operator then turns out to be L† 0= A C B D ! 92 The adjoint system for the Marangoni convection A=   −ρ∂t−ν2∂2 i−(ˆν+ν2)∂2 x−ˆν∂x∂y−ν2∂y∂x−ˆν∂x∂z−ν2∂z∂x −ˆν∂y∂x−ν2∂x∂y−ρ∂t−ν2∂2 i−(ˆν+ν2)∂2 y−ˆν∂y∂z−ν2∂z∂y −ˆν∂z∂x−ν2∂x∂z−ˆν∂z∂y−ν2∂y∂z−ρ∂t−ν2∂2 i−(ˆν+ν2)∂2 z    (7.14) B=   −∂x−∂y−∂z 0 0 0 0 0 0   C=   −∂x0 0 −∂y0 0 −∂z−β0   (7.15) D=   0 0 −1 ρ0∂t 0−∂t−˜χ∂2 i0 −1 ρ0∂t0−c2 ρ0∂t   (7.16) The surface contributions of the integration by parts should vanish to fulfill eq. (5.22) leading to the corresponding boundary conditions in the adjoint case. iω2ν2∂z¯vz+iωˆν(∂k¯vk) + iω¯p+ρG¯vz+σTk2¯vz= 0 (7.17) ¯vx(−ikx)ˆ T(z)−¯vxγβ(−ikx) + ˆvx(z)ν2(∂z¯vx+∂x¯vz) = 0 (7.18) ¯vy(−iky)ˆ T(z)−¯vyγβ(−iky) + ˆvy(z)ν2(∂z¯vy+∂y¯vz) = 0 (7.19) −˜χ¯ T∂zT+ ˜χT∂z¯ T= 0 (7.20) In the last set of equations we have used the fact, that every variable of the original system is modulated by ξ, in particular we used T(z) = ˆ T(z)ξand vx,y(z) = ˆvx,y(z)ξ. Actually eq. (7.20) just states, that the adjoint temperature may differ from the original one by just a constant. For the phenomenological boundary conditions we take the same form as for the original case, namely ¯vz=i¯ω¯ ξ(7.21) ¯ Q(¯ T) = −κ ∂z¯ T(7.22) The boundary conditions at the rigid bottom turn out to be self-adjoint, but are repeated here ¯vi=∂z¯vz= 0 (7.23) ¯ T= 0 (7.24) 7.3 The dimensionless representation For the further discussion we give the dimensionless version of the problem discussed in the previous section, because it is common in all the other discussion regarding convection. Following the usual steps [100], the linearized dynamical equations for the deviations from the conducting state of the temperature Θ and the vertical component of the velocity vz read (D2−k2)(D2−k2−iω)vz(z) = 0 (7.25) (D2−k2−iωP)Θ(z) = −vz(z) (7.26) 7.3 The dimensionless representation 93 The boundary conditions at the free surface using the stress balance then read (D2+k2)vz(z) = −Mk2Θ(z)−1 Pξ(7.27) CP(iω −D2+ 3k2)Dvz(z) = −(B−k2)k2ξ(7.28) And for the phenomenological boundary conditions we have vz(z) = iωξ (7.29) P(D+F)Θ(z) = Fξ(7.30) At the bottom, the equations reduce to vz=Dvz= Θ = 0 (7.31) While rescaling the variables we have introduced dimensionless numbers such as the Prandtl number P=ν2/˜χ, the Marangoni number M=γβd2/(ρ˜χν2), the Crispation number C=ρν2˜χ/(σTd), the Bond number B=ρGd2/σTand the Biot number F= (∂Q/∂T )d/κ as well as the dimensionless derivative with respect to z,D=d/dz. Using the same arguments for the adjoint set of equations we find (D2−k2)(D2−k2+i¯ω)¯vz(z) = −A¯ Θ(z) (7.32) (D2−k2+i¯ωP)¯ Θ(z) = 0 (7.33) It is worth mentioning here that in eq. (7.32) an additional number, A=β2d4/(˜χν2), arises. This is, however, consistent with condition (7.20), which allows the temperature in the adjoint case to differ from the original temperature by a constant factor. One could rescale the dimensionless adjoint temperature by exactly this number A, resulting in a dimensionalized adjoint temperature. This, however, is not surprising since also in the discussion of the adjoint system of the Rosensweig problem, the adjoint strain field acquired a different physical unit due to the dynamic coupling between velocity field and the strain field. The adjoint boundary conditions stemming from the adjoining process turn out to be −M(D¯vz(z))k2ˆ Θ−1 P= (Dˆvz)(D2+k2)¯vz(z) (7.34) CP(ω¯ω−iωD2+3iωk2)D¯vz(z) = −(B−k2)k2¯vz(z) (7.35) While the ones describing the free surface are ¯vz(z) = i¯ω¯ ξ(7.36) P(D+F)¯ Θ(z) = F¯ ξ(7.37) The self-adjoint boundary conditions at the bottom are repeated here in dimensionless form ¯vz=D¯vz= 0 (7.38) ¯ Θ = 0 (7.39) 100 Conclusions we found that the stripe pattern is never stable with respect to either of the other two patterns. At the linear onset, the hexagonal configuration of surface spikes turns out to be the energetically favored pattern. Upon further increase of the control parameter the hexagonal pattern in turn becomes energetically unstable and a square pattern develops. Both transitions, from the flat surface to hexagons and from hexagons to squares, are accompanied by hysteretic regions that become smaller for increasing elastic shear moduli. The energy method compares the energy of the possible surface patterns but does neither predict, which of these patterns can be dynamically attained, nor takes into account the dissipative processes in the medium that become important during the growth of the surface spikes. Furthermore, it is strictly valid only in the unphysical limit of a vanishing magnetic susceptibility. These drawbacks motivated us to discuss in the fifth chapter the nonlinear regime of the Rosensweig instability using an expansion of the fundamental hydrodynamic equations in terms of the normalized difference ǫbetween the applied magnetic field and the critical one. When expanding the fundamental hydrodynamic equations in terms of ǫ, the nonlinearities give rise to inhomogeneities in the second and higher order equations. To systematically guarantee the solvability of these equations using Fredholm’s theorem, the adjoint linear eigenvectors are needed. For systems involving a deformable surface in general and for the Rosensweig instability in particular, the set of adjoint linear equations with their corresponding boundary conditions were not known. For the derivation of the latter two assumptions turned out to be crucial. First, one has to treat the system dynamically and the static limit should only be used at the very end, and second, one has to start with the hydrodynamic set of equations describing a compressible medium. The incompressibility assumption can then be used once the system of equations is adjoint. The first assumption rests on the findings of the linear discussion, namely that the static character of the Rosensweig instability should be interpreted as a limiting process rather than as a static process from the beginning. The second assumption guarantees the symmetry of the stress tensor that is needed to consistently define the adjoint tangential boundary conditions. With the set of adjoint equations and the corresponding boundary conditions, the adjoint linear eigenvectors were obtained. Thereby right traveling waves in the original system transform into left traveling waves in the adjoint system and vice versa. Furthermore they show the same static properties as the original eigenvectors, namely that the adjoint velocity field vanishes in the static limit whereas the strain field acquires a finite static value. With the adjoint linear system for the Rosensweig instability at hand, we fulfilled the solvability conditions in the second and in the third perturbative order in terms of ǫ and finally obtained the amplitude equation for the Rosensweig instability. Within the scope of our assumptions, in particular because we neglected magnetostrictive effects, the hydrodynamic and the magnetic bulk equations decouple. Since we assumed a fast relaxing magnetic field which is governed by linear static Maxwell equations, the solvability condition for the magnetic equations is fulfilled trivially. As a consequence, Fredholm’s theorem applied to the hydrodynamic bulk equations does not contain the magnetic field variables, which act as the control parameter. Besides the bulk equations the Rosensweig instability crucially depends on the boundary conditions and in particular on the normal stress boundary condition. We showed in our analysis that the normal stress boundary condition cannot be fulfilled trivially in the higher perturbative orders, but rather acts as a supplement to Fredholm’s theorem, in order to determine the higher order corrections 101 to the control parameter. In the derived amplitude equation two contributions are important. We succeeded for the first time in deriving the quadratic coefficient in the amplitude equation from the fundamental hydrodynamic equations. The quadratic coefficient implies that hexagons are the stable configuration of surface spikes at the linear threshold and in addition that the bifurcation from the flat surface to the surface spikes is transcritical involving a bistable region between hexagonally ordered spikes and the flat surface below the linear threshold. Both results are experimentally verified properties of the Rosensweig instability. Additionally we derived a second order time derivative in the case of magnetic gels. The linearized amplitude equation therefore acquires the form of a damped harmonic oscillator. If a finite amplitude surface pattern is disturbed, this disturbance will relax in the form of a damped oscillation. In the case of the Rosensweig instability in ferrofluids, whose amplitude equation has also been derived in this thesis, this second order time derivative is not present. The amplitudes of the stable static surface patterns are determined by the cubic coefficients in the amplitude equation. They show that for high magnetic field strengths the hexagons become unstable and transform into a square pattern. A result that is in accordance with the energy method. The cubic coefficients calculated in this thesis are independent of the elastic shear modulus and the magnetic susceptibility, due to the assumption of a linear elastic as well as a linear magnetic medium. Our discussions revealed that the Rosensweig instability is a purely surface driven instability within the scope of our assumptions. A natural question is then to ask: What happens, if the superparamagnetic medium is just a surface, namely a thin film or a membrane. In chapter 6 we discussed this question assuming a membrane, either made of an isotropic or an anisotropic magnetic gel, floating on a Newtonian liquid or on a ferrofluid. In the first case we realized, that the film does not become unstable, if we assume an isotropic magnetic gel. An intuitive reason for this is given by the character of the driving force itself. Small surface fluctuations render the applied homogeneous magnetic field locally inhomogeneous, which causes a Kelvin force. In the case of membranes we showed, that in the limit of vanishing film thickness the magnetic field remains undistorted causing no force acting to the magnetic film. This changes if we assume an anisotropic magnetic gel, where the frozen-in magnetization is rigidly anchored to the elastic medium. In this case the film becomes unstable with respect to periodical disturbances if the frozen-in magnetization is oriented opposite to the applied magnetic field. A typical property of the Rosensweig instability in isotropic magnetic gels is that its characteristic wavelength is the same as for usual ferrofluids. If we assume a non-magnetic film floating on top of a usual ferrofluid, this changes and the characteristic mode depends on the elastic properties of the membrane. We realized in this thesis that the character of the Rosensweig instability is owed to the deformability of the surface between the magnetic medium and the vacuum above. Another very prominent example of an instability which involves a deformable surface is given by the pure Marangoni instability. In this case, temperature fluctuations at the free surface of a fluid cause fluctuations of the surface tension that in turn deform the surface and drive convection. In the case of the Marangoni convection, the adjoint system of linear equations that takes into account a deformable surface also was unknown and nonlinear discussions therefore had to assume flat and undeformable surfaces. Using the same arguments as for the Rosensweig instability, we were able to derive the adjoint system and the corresponding boundary conditions for the Marangoni instability. As shown in 102 Conclusions chapter 7, the adjoint boundary conditions involve the linear eigenvectors of the original system. The latter property is due to a bulk coupling between the temperature field and the velocity field, although this coupling does not drive the instability. As a consequence, the solution of the adjoint system for the Marangoni convection is rather involved and an easy interpretation in terms of a translation between right and left traveling waves, as it was the case for the Rosensweig instability, is not possible. The driving force in the case of the Marangoni instability acts purely tangentially, while for the Rosensweig instability the force is orthogonal. Thus the method used to derive the adjoint systems for both instabilities should also work for any arbitrary orientation of the driving force at the surface. Appendix A Decoupling of the dynamic system In this appendix we show explicitly the decoupling of the Maxwell equations and the bulk equations for the magnetic medium under the assumptions of linear magnetostatics and negligence of magnetostrictive effects. Separating the actual magnetic field into the applied magnetic field H0(respectively B0for the flux density) and the perturbations due to the deformed surface denoted as hrespectively b H=H0+h(A.1) B=B0+b(A.2) The latter can be expressed as the gradient of a scalar potential Φ h=−∇Φ (A.3) b=−µ∇Φ (A.4) with µdenoting the magnetic permeability of the medium. The magnetic field enters the dynamic bulk equations of the medium only through the stress tensor Tij given by (2.35). Concentrating on the magnetic contributions of the momentum conservation equation (2.33), we obtain, since the applied magnetic field is assumed be constant, the contributions ∂jBihj+Hjbi+bihj−1 2(Bkhk+Hkbk+bkhk)δij(A.5) Substituting the perturbed magnetic fields in terms of the scalar potential one can simplify (A.5) as ∂j−Bi∂jΦ−Bj∂iΦ + µ(∂iΦ)(∂jΦ) + Bk(∂kΦ)δij −1 2µ(∂kΦ)(∂kΦ)δij(A.6) Evaluating the partial derivative ∂jleaves us with −Bi∂j∂jΦ−Bj∂j∂iΦ + µ(∂iΦ)(∂j∂jΦ) + µ(∂jΦ)∂j∂iΦ + Bj∂j∂iΦ−µ(∂jΦ)∂j∂iΦ (A.7) Obviously the second term cancels the fifth as well as the third term cancels the sixth. The remaining two contributions cancel by realizing that the magnetic scalar potential has to satisfy the Laplace equation. In total all magnetic contributions cancel in the bulk equations for the magnetic medium. 103 104 Decoupling of the dynamic system Appendix B Magnetic fields Within the scope of our assumptions the hydrodynamic bulk equations completely decouple from the magnetic bulk equations. This enables us to find solutions to the bulk system separately. Furthermore, since the magnetic system is entirely independent of any hydrodynamic variable (except that the distortions of the magnetic field should be proportional to the surface deformation) it is worth determining the magnetic field for a given surface deformation ξ(x, y, t) first and substitute afterwards into the system of hydrodynamic equations. In this section we give a detailed derivation for all the magnetic field expressions used in the main text. B.1 The Heaviside-Lorentz system of electromagnetic units Throughout this thesis the Heaviside-Lorentz or the rationalized Gauss system of electromagnetic units has been used to describe the magnetic phenomena [69]. The choice of this system is set by the fundamental hydrodynamic equations in chapter 2 and in particular by the choice of the energy density (2.7). We will therefore introduce this system of units in this section based on the book of Jackson [69] and although this thesis only considers magnetic fields we will, for completeness, include the electric degrees of freedom in this introductory section as well. The fundamental laws in the electrodynamic theory, Coulomb’s law of electrostatics and the Amp`ere law, only give proportionalities between measured forces, FCand FArespectively, and the distance and the magnitude of two electric charges or electric currents. Coulomb’s law reads FC=k1 q1q2 r2(B.1) where q1and q2are electric charges, ris the distance between them and k1is a proportionality constant and Amp`ere’s law is given by dFA dl = 2k2 I1I2 r(B.2) relating the force per unit length to the electric currents I1and I2that are carried by two parallel, infinitely long conducting wires of negligible cross-section separated by the distance r. 105 106 Magnetic fields Quantity Heaviside-Lorentz SI Speed of light c(µ0ǫ0)−1/2 Magnetic induction B B/√µ0 Magnetic field H√µ0H Magnetic scalar potential Φ √µ0Φ Magnetization M√µ0M Magnetic permeability µ µ/µ0 Table B.1: This table shows the conversion rules between the Heaviside-Lorentz system of units used in this thesis and the SI units (taken from [69]) for the macroscopic variables relevant in this thesis. The magnetic permeability of vacuum is given by µ0= 4π·10−7H/m and the speed of light by c= 2.99792458 ·108m/s. The proportionality constants k1and k2are either given by the eqs. (B.1) and (B.2) if the unit charge has been chosen independently or one can choose them arbitrarily with the consequence of defining unit charge. Due to the common definition of the electric current as the time rate of change of charge, one can give the relative dimension of k1with respect to k2as k1=c2k21, where cdenotes the speed of light. In the Heaviside-Lorentz system of units these proportionality constants are arbitrarily chosen as k1= 1/(4π) and k2= 1/(4πc2) and the Heaviside-Lorentz system therefore differs from the usual Gaussian system of units by a factor 4π. Measured in the Heaviside-Lorentz units, the magnetic and the electric field variables acquire the same physical units and the constitutive equations read D=E+P(B.3) H=B−M(B.4) where Ddenotes the electric displacement field, Ethe electric field and Pthe electric polarization. The last equation is exactly the relation between the magnetic field H, the magnetic flux density Band the magnetization Mas we obtain by thermodynamic means in eq. (2.9). In the Heaviside-Lorentz system of units the Maxwell equations acquire the form ∇·D=ρel (B.5) ∇×H=J c+∂D c∂t (B.6) ∇×E=−∂B c∂t (B.7) ∇·B= 0 (B.8) where ρel and Jdenote the electrical charge density and its corresponding current, respectively. In the absence of the latter and for time independent magnetic fields, which is 1At this point one can only claim that the relative dimensions of k1and k2are that of velocity squared, whereas the magnitude is still arbitrary. Deriving the wave equation from this, however, fixes the still undetermined magnitude of this velocity to that of the speed of light in vacuum (cf. [69]). B.2 Expansion to higher perturbative orders 107 one of our assumptions, these equations become the static Maxwell equations (2.36) and (2.37) as used in this thesis. To convert the equations in this thesis to the SI system, the rules given in table B.1 can be applied. B.2 Expansion to higher perturbative orders B.2.1 The Maxwell equations For the nonlinear discussion of an instability involving a deformable surface, it is convenient to distinguish the externally applied magnetic field Hext in all orders from the distortion field hdue to the deformed surface and correspondingly for the magnetic flux densities H=Hext +hand B=Bext +b(B.9) The external magnetic field is, in our geometry, always directed parallel to the z−axis (cf. fig. 2.1 on p. 18), however, the magnitude remains tunable. We therefore expand the applied external field as well as the flux density according to Hext =Hc+ǫH(1) +ǫ2H(2) +... (B.10) Bext =Bc+ǫB(1) +ǫ2B(2) +... (B.11) where ǫdenotes the normalized difference between the applied magnetic field and the critical one (this is the expansion we used in eq. (5.1)). In addition, the deformed surface will cause the magnetic field to be distorted. These deviations from the applied magnetic field are taken into account by the field hand b that are also expanded similarly as h=ǫh(1) +ǫ2h(2) +... (B.12) b=ǫb(1) +ǫ2b(2) +... (B.13) The same expansion applies to the corresponding fields in the vacuum. The deviations from the applied field still have to satisfy the linear magnetostatic equations, b=µhand ∇·b= 0 = ∇×h, which allows for the introduction of a magnetic scalar potential [69] h=−∇Φ that is then governed by the Laplace equation ∆Φ = 0 and ∆Φvac = 0 (B.14) The scalar magnetic potentials attain the expansion in terms of ǫfrom the fields and correspondingly are written as Φ = ǫΦ(1) +ǫ2Φ(2) +ǫ3Φ(3) +... (B.15) This one to one correspondence between distortion field hand the scalar potential Φ within the different orders is only true within the scope of our assumptions of chapter 5 where long wavelength variations of the arising pattern are discarded. Since the Laplace equation is linear and homogeneous, its expansion to the higher orders is trivial. The bulk solutions can be found in each order independently and in particular, Fredholm’s theorem will be fulfilled in each order. 108 Magnetic fields B.2.2 The boundary conditions Since the surface normal nis not constant but depends on the surface deflection (as do the distorted field contributions), a higher harmonic coupling to previous orders is possible (in contrast to the system of bulk equations). For the upcoming calculation it is useful to determine first the fields at the boundary z=ξ H=Hc+ǫH(1) −(∇Φ(1))z=0+ǫ2H(2) −(∇Φ(2))−ξ(1)(∂z∇Φ(1))z=0 +ǫ3H(3) −(∇Φ(3))z=0−ξ(1)(∂z∇Φ(2))z=0−1 2[(ξ(1)2∂2 z+2ξ(2)∂z)∇Φ(1)]z=0(B.16) and accordingly for the magnetic field Hvac and the magnetic flux densities Band Bvac. The contributions in (B.16) that are explicitly proportional to ξ(1) or ξ(2) are due to the deformable surface. As mentioned, the surface normal n, initially directed parallel to the z−axis, changes its orientation in the course of time as the surface perturbation grows (cf. fig. 2.1 on p. 18). To give a proper expansion of the boundary conditions, we additionally have to expand the surface normal as a function of the surface deflection ξ(x, y, t) n=n0+ǫn(1) +ǫ2n(2) +ǫ3n(3) (B.17) with the different perturbative contributions given by n(1) =   −∂xξ(1) −∂yξ(1) 0   ,n(2) =   −∂xξ(2) −∂yξ(2) 1 2(∂xξ(1))2+1 2(∂yξ(1))2   (B.18) and n(3) =   −∂xξ(3) −1 2(∂yξ(1))2(∂xξ(1))−1 2(∂xξ(1))3 −∂yξ(3) −1 2(∂xξ(1))2(∂yξ(1))−1 2(∂yξ(1))3 (∂yξ(1))(∂yξ(2)) + (∂xξ(1))(∂xξ(2))   (B.19) With the previous considerations on hand, we are able to expand the boundary conditions in terms of ǫ. The fact that the normal component of the magnetic flux density is continuous at the boundary gives the following condition n·Hvac −H=n·Bvac −B+M =n·M(B.20) Consider the linear perturbative order of the last equation n(1) ·Hvac c−Hc+n(0) ·H(1)vac −H(1)=n(1) ·M(0) +n(0) ·M(1) (B.21) For the constant contributions (constant with respect to xand y), we find H(1)vac z−H(1) z=M(1) z(B.22) while the contributions proportional to n(1) cancel identically. The corresponding expression for the second order contribution to the applied field, H(2)vac z−H(2) z=M(2) z, can be obtained straightforwardly. B.2 Expansion to higher perturbative orders 109 The boundary condition for the tangential components of the magnetic field (2.38) is given in linear order n(1) ×(Hvac c−Hc) + n(0) ×H(1)vac −H(1)= 0 (B.23) which can be simplified substituting eq. (B.22) to (with a∈ {x, y}) h(1)vac a−h(1) a=−(∂aξ(1))M0(B.24) In the second perturbative order we find n(2) ×Hvac c−Hc+n(1) ×H(1)vac −H(1) −∇Φ(1)vac +∇Φ(1) +n(0) ×H(2)vac −∇Φ(2)vac +kcξ(1)∇Φ(1)vac −H(2) +∇Φ(2) +kcξ(1)∇Φ(1)= 0 (B.25) which is simplified in the same manner (by exploiting the results of the previous order) to ∂aΦ(2)vac −∂aΦ(2)−(∂aξ(2))Mc−(∂aξ(1))M(1) +(∂aξ(1))∂zΦ(1)vac −∂zΦ(1)−kcξ(1)∂aΦ(1)vac +∂aΦ(1)= 0 (B.26) with a∈ {x, y}. Upon substituting the linear solutions (B.34) and (B.35) this immediately leads to expressions (B.39) and (B.40) used in section B.4 to find the magnetic eigenvectors in the second perturbative order. Finally we deduce for the tangential boundary condition in the third perturbative order n(3) ×(Hvac c−Hc) + n(2) ×H(1)vac −H(1) −(∇Φ(1)vac) + (∇Φ(1)) +n(1) ×H(2)vac −H(2) −(∇Φ(2)vac) + (∇Φ(2)) + kcξ(1)(∇Φ(1)vac) + kcξ(1)(∇Φ(1)) +n(0) ×H(3)vac −H(3) −(∇Φ(3)vac) + (∇Φ(3))−ξ(1)(∂z∇Φ(2)vac) + ξ(1)(∂z∇Φ(2)) −1 2(k2 cξ(1)2 −2kcξ(2))(∇Φ(1)vac) + 1 2(k2 cξ(1)2 + 2kcξ(2))(∇Φ(1)) = 0 (B.27) where it will be sufficient for our discussion to consider only the contributions proportional to the main characteristic modes ξ(1) as discussed in section 5.3.2. Along the same lines the boundary condition that guarantees the continuity of the normal component (2.39) of the magnetic flux density is derived. In first perturbative order we get n(1) ·(Bvac c−Bc) + n(0) ·B(1)vac −B(1)= 0 (B.28) which is straightforwardly simplified to b(1)vac z−b(1) z= 0 (B.29) For the corresponding condition in the second perturbative order we obtain n(2) ·Bvac c−Bc+n(1) ·B(1)vac −B(1) −∇Φ(1)vac +µ∇Φ(1) +n(0) ·B(2)vac −B(2) −∇Φ(2)vac +µ∇Φ(2) +kcξ(1)∇Φ(1)vac +µkcξ(1)∇Φ(1)= 0 (B.30) 116 Magnetic fields Appendix C The hydrodynamic boundary conditions C.1 Expansion of the boundary conditions In this section we discuss the expansion of the hydrodynamic boundary conditions in terms of ǫ. Recall first, that we require the tangential stress at the free surface to vanish whereas the normal stress is balanced by surface tension and gravity n×T·n=n×Tvac ·n(C.1) n·T·n−n·Tvac ·n=σT∇·n−ρGξ (C.2) The contributions of the stress tensor to the different perturbative orders are defined by the expansions of the macroscopic variables, eqs. (5.1,5.2), and by the expansion of the surface normal n, eq. (B.17). The linear eigenvectors of the hydrodynamic set of equations are either proportional to ekz or eqz (cf. section 3.5). For the boundary conditions one has to evaluate them at z=ξand therefore an expansion similar to (B.16) is needed that explicitly accounts for the deformability of the surface. C.2 The linear perturbative order The boundary conditions in linear order are straightforwardly calculated, but are repeated here for completeness. For the tangential stress boundary condition we obtain 2µ2ǫ(1) yz +ν2∂zv(1) y+∂yv(1) z= 0 (C.3) 2µ2ǫ(1) xz +ν2∂zv(1) x+∂xv(1) z= 0 (C.4) whereas the normal stress boundary condition reads 2µ2ǫ(1) zz + 2ν2∂zv(1) z−p(1) +Gρξ(1) −µH0∂zΦ(1) −Hvac 0∂zΦ(1)vac=σT∇·n(1) (C.5) In section 3.2 we introduced potentials for the irrotational and the rotational flow contributions. To solve the system of equations for the potentials we have to translate the boundary conditions above to the corresponding ones valid for the potentials. We 117 118 The hydrodynamic boundary conditions start with the tangential boundary conditions. To be able to express the strain field in terms of the gradient of the velocity field (3.9), we first have to take the derivative of eqs. (C.3,C.4) with respect to time. Upon substitution of eqs. (3.11) we end up with ˜µ2(∂2 z−∂2 y)Ψ(1) x+ ˜µ2(∂y∂x)Ψ(1) y+ 2˜µ2∂y∂zϕ(1) = 0 (C.6) ˜µ2(∂x∂y)Ψ(1) x+ ˜µ2(∂2 z−∂2 x)Ψ(1) y−2˜µ2∂x∂zϕ(1) = 0 (C.7) where the coefficient (µ2+ν2∂(0) t) has been abbreviated by ˜µ2. For the normal stress boundary condition we additionally have to discuss the contributions that explicitly contain the surface deflection ξ(1). At a first glance these contributions may be considered as inhomogeneities. However, the deflection ξ(1) is related to the local velocity. Upon taking the time derivative of eq. (C.5) we can substitute the linearized kinematic boundary condition ∂tξ(1) =v(1) z. Following the same lines as in the case of the tangential boundary conditions, we find −(2˜µ2∂z∂y+Gρ∂y+σTk2∂y)Ψ(1) x+ (2˜µ2∂z∂x+Gρ∂x+σTk2∂x)Ψ(1) y +(2˜µ2∂2 z+Gρ∂z+σTk2∂z)ϕ(1) −∂tp(1) −∂tH0µ∂zΦ(1) −Hvac 0∂zΦ(1)vac= 0 (C.8) Additionally, we can substitute the solution for the pressure p(1), obtained in section 3.2, and the solutions for the magnetic fields, obtained in appendix B, to finally arrive at −(2˜µ2∂z∂y+Gρ∂y+σTk2∂y−µ 1 + µM2 0∂z∂y)Ψ(1) x +(2˜µ2∂z∂x+Gρ∂x+σTk2∂x−µ 1 + µM2 0∂z∂x)Ψ(1) y +(2˜µ2∂2 z+Gρ∂z+σTk2∂z−ρω2−µ 1 + µM2 0∂2 z)ϕ(1) = 0 (C.9) where again the explicit surface deflection ξ(1), arising from the magnetic solutions, has been replaced by the local velocity. The steps leading to eq. (C.8), where we first took the time derivative of eq. (C.5) to implement afterwards the kinematic boundary condition, are very crucial. By inspection of eq. (C.5) we realize that it is finite in the stationary limit revealing the force balance between elastic, gravitation, magnetic and surface tension forces. Eq. (C.8), however, is at least linear in the time derivative and so are the eigenvectors derived from it. The latter property is in accordance with the understanding that there is no motion in the medium, if the surface pattern is fully developed. These considerations show that, due to the kinematic boundary condition, the bulk equations inherently scale one order higher with respect to the time derivative, when compared with the normal stress boundary condition. This different scaling behavior has to be taken into account, if we combine the solvability conditions from the bulk with those from the normal stress boundary condition (chapter 5). C.3 The second perturbative order 119 C.3 The second perturbative order In the second order we find the tangential boundary conditions involving the hydrodynamic fields as 2µ2ǫ(2) xz +ν2∂zv(2) x+∂xv(2) z =−ξ(1)∂z2µ2ǫ(1) xz +ν2(∂xv(1) z+∂zv(1) x)+ (∂yξ(1))2µ2ǫ(1) yz +ν2(∂yv(1) x+∂xv(1) y) −2(∂xξ(1))µ2(ǫ(1) zz +ǫ(1) xx ) + ν2(∂zv(1) z+∂xv(1) x)+ρv(1) xv(1) z≡Ω(2) xz (C.10) 2µ2ǫ(2) yz +ν2∂zv(2) y+∂yv(2) z =−ξ(1)∂z2µ2ǫ(1) yz +ν2(∂yv(1) z+∂zv(1) y)+ (∂xξ(1))2µ2ǫ(1) xy +ν2(∂xv(1) y+∂yv(1) x) −2(∂yξ(1))µ2(ǫ(1) zz +ǫ(1) yy ) + ν2(∂zv(1) z+∂yv(1) y)+ρv(1) yv(1) z≡Ω(2) yz (C.11) In eqs. (C.11) and (C.10) the inhomogeneities on the right hand side have been abbreviated by Ω(2) xz and Ω(2) yz , respectively. In particular these inhomogeneities are proportional to [ξ(1)]2. The normal stress boundary condition (C.2) reads in second order 2µ2ǫ(2) zz + 2ν2∂zv(2) z−p(2) +Gρξ(2) −µHc∂zΦ(2) +Hvac c∂zΦ(2)vac =−2µ2[ǫ(1) zz ]2+ρ[v(1) z]2−McBc(∂yξ(1))2+ (∂xξ(1))2−1 2µ∂zΦ(1)2 +1 2∂zΦ(1)vac2−2µHc(∂yξ(1))(∂yΦ(1)) + 2Hvac c(∂yξ(1))(∂yΦ(1)vac) +1 2µ(∂yΦ(1))2−1 2(∂yΦ(1)vac)2+1 2µ(∂xΦ(1))2−1 2(∂xΦ(1)vac)2 −2µHc(∂xξ(1))(∂xΦ(1)) + 2Hvac c(∂xξ(1))(∂xΦ(1)vac) +ξ(1)∂z2µ2ǫ(1) zz + 2ν2∂zv(1) z−p(1) −µ 1 + µM2 ckξ(1)+µ 1 + µM(1)Mckcξ(1) −σT∆ξ(2) (C.12) Furthermore, we obtain for the kinematic boundary condition in second order ∂(0) tξ(2) +∂(1) tξ(1) + (v(1) ·∇)ξ(1) =v(2) z+ξ(1)∂zv(1) z(C.13) The physical boundary is at z=ξ, giving rise to an additional dependence on ξ. In eqs. (C.11-C.13) such terms have already been made explicit (e.g. the last one of (C.13)). Thus these boundary conditions are effective ones that have to be taken at z= 0. Inspecting the expressions (C.12) and (C.13) one immediately realizes that two qualitatively different contributions are present. On the one hand we obtain contributions proportional to the higher harmonic coupling [ξ(1)]2of the main characteristic mode. On the other hand, there are still contributions proportional to the main characteristic mode ξ(1) itself. The latter will allow us to find the linear contributions in an amplitude equation even though the control parameter is not present in the bulk equations. To solve the corresponding hydrodynamic bulk equations, we introduced a scalar, ϕ(2), and a vector potential, Ψ(2), in section 5.3, to discuss potential and rotational flow contributions separately. Following the same lines as done in the linear order (section C.2), we can translate the boundary conditions into a corresponding set of equations for 120 The hydrodynamic boundary conditions the amplitudes of the second order potentials ϕ(2) and Ψ(2). We obtain for the tangential contributions (C.11) and (C.10) ˜µ2(∂2 z−∂2 y)Ψ(2) x+˜µ2(∂y∂x)Ψ(2) y+2˜µ2∂y∂zϕ(2) = 2µ2v(1) k∂kǫ(1) yz +∂(0) tΩ(2) yz (C.14) −˜µ2(∂x∂y)Ψ(2) x−˜µ2(∂2 z−∂2 x)Ψ(2) y+2˜µ2∂x∂zϕ(2) = 2µ2v(1) k∂kǫ(1) xz +∂(0) tΩ(2) xz (C.15) using ǫ(1) xz =ǫ(1) yz ≡0 at the boundary (cf. eqs. (C.3,C.4)). The normal stress boundary condition (C.12) translates into −(2˜µ2∂y∂z+ρG∂y)Ψ(2) x+ (2˜µ2∂z∂x+ρG∂x)Ψ(2) y+ (2˜µ2∂2 z+ρG∂z)ϕ(2) −∂(0) tp(2) =∂(0) tHcµ∂zΦ(2) −Hvac c∂zΦ(2)vac+M(1)Mckc µ 1 + µ∂(0) tξ(1) +∂(0) tΩ(2) zz +2µ2v(1) k∂kǫ(1) zz −2ρGξ(1)∂zv(1) z+ρG∂(1) tξ(1) + 2µ2∂(1) tǫ(1) zz −σT∂(0) t∆ξ(2) (C.16) Eqs. (C.14-C.16) follow from (C.11-C.12) by taking the time derivative with respect to t(0) without loss of generality. This is why in eq. (C.16) only the contribution ∂(0) tp(2) and no contribution ∂(1) tp(1) arises, while ∂(0) tǫ(2) ij gives rise to contributions ∼v(2) iand ∼∂(1) tǫ(1) ij (cf. eq. (5.58)). C.4 The third perturbative order We take over the procedure of the previous section to the third order. If we use the solutions (5.81,5.85,5.87) of the hydrodynamic bulk equations in second order, the kinematic boundary condition reads ∂(0) tξ(3) +∂(1) tξ(2) +∂(2) tξ(1) + (v(1) ·∇)ξ(2) + (v(2) ·∇)ξ(1) (C.17) =v(3) z+ξ(1)∂zv(2,2) z+ξ(1)∂zv(2,1)hom z−µ2+˜µ2 q˜µ2 k2 cξ(1)∂(1) tξ(1) +ξ(2)∂zv(1) z+1 2ξ(1)2∂2 zv(1) z The tangential boundary conditions are of the usual structure and given as 2µ2ǫ(3) yz +ν2∂zv(3) y+∂yv(3) z= Ω(3) yx (C.18) 2µ2ǫ(3) yz +ν2∂zv(3) y+∂yv(3) z= Ω(3) xz (C.19) where inhomogeneous contributions, which are at least proportional to the higher harmonic couplings, are collected in the abbreviation Ω(3) ij , as done similarly in second order. The only reason why we have to consider the third order boundary conditions is to obtain the linear contributions to the amplitude equation. The general solution involving the higher harmonic couplings is not needed. Since Ω(3) ij is at least proportional to the higher harmonic couplings it is therefore unimportant in this discussion and is not shown here. Taking the time derivative of eqs. (C.18,C.19) together with (5.118) we find ˜µ2(∂2 z−∂2 y)Ψ(3) x+ ˜µ2(∂y∂x)Ψ(3) y+ 2˜µ2∂y∂zϕ(3) =∂(0) tΩ(3) yx + 2µ2(∂(1) tǫ(2) yz +∂(2) tǫ(1) yz ) +2µ2(v(1) k∂kǫ(2) yz +v(2) k∂kǫ(1) yz )(C.20) −˜µ2(∂x∂y)Ψ(3) x−˜µ2(∂2 z−∂2 x)Ψ(3) y+ 2˜µ2∂x∂zϕ(3) =∂(0) tΩ(3) xz + 2µ2(∂(1) tǫ(2) xz +∂(2) tǫ(1) xz ) +2µ2(v(1) k∂kǫ(2) xz +v(2) k∂kǫ(1) xz )(C.21) C.4 The third perturbative order 121 For the normal stress boundary condition we obtain from eq. (C.2) 2µ2ǫ(3) zz + 2ν2(∂zv(3) z)−p(3) +ρGξ(3) −(µHc∂zΦ(3) −Hvac c∂Φ(3)vac) = (µH(2)∂zΦ(1) −H(2)vac∂zΦ(1)vac) + (µH(1)∂zΦ(2) −H(1)vac∂zΦ(2)vac) +Ω(3) zz +σT∇·n(3) (C.22) which by a similar procedure can be written as −(2˜µ2∂z∂y+ρG∂y)Ψ(3) x+ (2˜µ2∂z∂x+ρG∂x)Ψ(3) y+ (2˜µ2∂2 z+ρG∂z)ϕ(3) −∂(0) tp(3) = (µHc∂zΦ(3) −Hvac c∂zΦ(3)vac) + µ 1 + µ(McM(2) +M(1)2)kc∂(0) tξ(1) +∂(0) tΩ(3) zz +ρG(∂(2) tξ(1) +∂(1) tξ(2) +v(2) k∂kξ(1) −ξ(1)∂zv(2) z−ξ(2)∂zv(1) z−1 2ξ(1)2∂2 zv(1) z) +2µ2(∂(2) tǫ(1) zz +∂(1) tǫ(2) zz + (v(1) k∂k)ǫ(2) zz + (v(2) k∂k)ǫ(1) zz ) + σT∂(0) t∇·n(3) (C.23) 122 The hydrodynamic boundary conditions Appendix D Eigenvectors in the second order In this appendix we give the contributions to the eigenvectors in the second perturbative order that are proportional to the higher harmonic couplings ξ(2). Due to the fact that we have to treat the system dynamically throughout all orders, the expressions become tedious and have therefore been calculated with Mathematica. In the following the solutions for the hydrodynamic potentials are represented for the patterns under consideration, hexagons (θij = 2π/3), squares (θij =π/2) and stripes (θij = 0) as well as for the interaction between hexagons and squares (θij =π/6) (cf. fig. 5.2 on p. 59). The inhomogeneous contributions to the vector potential, cf. eq. (5.98), separate into a contribution ∼e(kc+q)zand ∼e2qz. For the hexagonal case (ij =ji = 12 = 23 = 31) we obtain Ψinhom NMij(z) = (k2 c+q2)(2µ2q2−5µ2qkc+ρ[D(0) t]2) 2q(k2 c−q2)(2kcq˜µ2+q2˜µ2−ρ[D(0) t]2)e(kc+q)zD(0) t +3k2 cq(µ2k2 c+ 2µ2q2−ρ[D(0) t]2) (k4 c−5k2 cq2+ 4q4)(˜µ2k2 c−4˜µ2q2+ρ[D(0) t]2)e2qzD(0) t(D.1) with {N, M} ∈ {R, L}. The abbreviation D(0) tstands for 2iω(0) + 2σ(0), 2σ(0), and −2iω(0) + 2σ(0) for N=M=R,N6=M, and N=M=L, respectively. The second coefficient ˜ Ψinhom NMij reads ˜ Ψinhom NMij(z) = (k2 c+q2)(6k3 cµ2−10k2 cqµ2+kcq2µ2+2q3µ2+qρ[D(0) t]2) 2(2k4 c−2k3 cq−3k2 cq2+2kcq3+q4)(2k2 c˜µ2−2kcq˜µ2−q2˜µ2+ρ[D(0) t]2)e(kc+q)zD(0) t −k2 cq(k2 cµ2−2q2µ2+ρ[D(0) t]2) (3k4 c−7k2 cq2+ 4q4)(3k2 c˜µ2−4q2˜µ2+ 2ρ[D(0) t]2)e2qzD(0) t(D.2) For the square pattern we get (ij =ji = 15) Ψinhom NMij(z) = (k2 c+q2)(4µ2k3 c−10µ2qk2 c+2µ2q3+(kc+q)ρ[D(0) t]2) 2(k4 c−2qk3 c−2k2 cq2+2kcq3+q4)(˜µ2k2 c−2˜µ2qkc−˜µ2q2+ρ[D(0) t]2)e(q+kc)zD(0) t +qk2 c(2µ2q2−ρ[D(0) t]2) 4(k4 c−3q2k2 c+2q4)(2˜µ2k2 c−4˜µ2q2+ρ[D(0) t]2)e2qzD(0) t(D.3) =˜ Ψinhom NMij(z) (D.4) 123 124 Eigenvectors in the second order For the stripe geometry, i=j, we obtain Ψinhom NMij(z) = (k2 c+q2)6k2 cµ2−4kcqµ2−2q2µ2−ρ[D(0) t]2 2(kc−q)(3k2 c+4kcq+q2)3k2 c˜µ2−2kcq˜µ2−q2˜µ2+2ρ[D(0) t]2e(kc+q)zD(0) t(D.5) and ˜ Ψinhom NMij(z) = k2 c4k4 cµ2−4q2(2q2µ2−ρ[D(0) t]2)−k2 c(20q2µ2+ 3ρ[D(0) t]2) 4(k2 c−q2)2(4q3˜µ2+qρ[D(0) t]2)e2qzD(0) t +Z1h4(kc−q)2(kc+q)3k2 c˜µ2+2kcq˜µ2+q2˜µ2−2ρ[D(0) t]2i−1e(kc+q)zD(0) t(D.6) with the numerator Z1being given by Z1= (k2 c+q2)24k3 cqµ2−4q4µ2−2q2ρ[D(0) t]2+20k2 cq2µ2+3k2 cρ[D(0) t]2 +8kcq3µ2−4kcqρ[D(0) t]2(D.7) In addition, to describe the interaction between the square and the hexagonal pattern, we need to consider also the case θij =π/6, (ij =ji = 14 = 36 = 25) Ψinhom NMij(z) = N−1 1n(k2 c+q2)(2q3µ2−10k2 cqµ2+k2 cµ2(1+3√3)+qρ[D(0) t]2(D.8) + √3kcq2µ2+kc(1−√3)ρ[D(0) t]2)oe(kc+q)zD(0) t −k2 cq(k2 cµ2(2√3−3) −2(2 −√3)(q2µ2−ρ[D(0) t]2)) ((2 + √3)k2 c−4q2)(k2 c−q2)((2 + √3)k2 c˜µ2−(4q2˜µ2−2ρ[D(0) t]2))e2qzD(0) t where the denominator N1is given by N1= 2(k2 c−q2)n2(2+√3)k4 c˜µ2−4(1+√3)k3 cq˜µ2+q4˜µ2−q2ρ(D(0) t)2 +k2 c(1+√3)ρ[D(0) t]2+2(1−√3)q2˜µ2)+4kcq3˜µ2−2kcqρ[D(0) t]2o(D.9) and ˜ Ψinhom NMij(z) = (3 + 2√3)k2 cµ2+ (2 + √3)(2q2µ2−ρ[D(0) t]2) (k2 c−q2)(√3−2)k2 c+ 4q2(√3−2)k2 c˜µ2+ 4q2˜µ2−ρ[D(0) t]2qk2 ce2qzD(0) t −N−1 2n(k2 c+q2)(3√3−1)k3 cµ2+10k2 cqµ2−2q3µ2 −2qρ[D(0) t]2+kc√3q2µ2−kc[1+√3]ρ[D(0) t]2oe(kc+q)zD(0) t(D.10) with N2= 2(k2 c−q2)h2(2−√3)k4 c˜µ2+4(√3−1)k3 cq˜µ2+q4˜µ2−q2ρ[D(0) t]2 +2k2 c[1+√3]q2˜µ2+k2 c[1−√3]ρ[D(0) t]2+4kcq3˜µ2−qkcρ[D(0) t]2i(D.11) 125 For the scalar potential we obtain from eq. (5.109) in the geometry of hexagons (ij = ji = 12 = 23 = 31) ˆϕNMij =h8ρ[D(0) t]2kcq(q+kc)(k2 c−4q2)i−1n2k5 cq(12˜µ2−14µ2−3ν2D(0) t)−12q4ρ[D(0) t]2 +2k4 c4˜µ2q2−16µ2q2+ν2q2D(0) t+2ρ[D(0) t]2 +k2 c19q2ρ[D(0) t]2+8q4[4˜µ2+4µ2+ν2]D(0) t −4kc9q3ρ[D(0) t]2−4q5[3µ2+ν2D(0) t] +k3 c17qρ[D(0) t]2−96˜µ2q3+52µ2q3+20q3ν2D(0) toD(0) t(D.12) and for the case of squares (ij =ji = 15) ˆϕNMij =h4√2ρ[D(0) t]2kc(kc+q)(k2 c−2q2)(k2 c−2kcq−q2)i−1n4k7 c(6˜µ2−7µ2−2ν2D(0) t) +2q5ρ[D(0) t]2−4k6 cq(10˜µ2−14µ2−5ν2D(0) t) +k2 c16q5˜µ2−16q5µ2−8q5ν2D(0) t+9q3ρ[D(0) t]2 +2kc5q4ρ[D(0) t]2−4q6(3µ2+ν2D(0) t)+k5 c3ρ[D(0) t]2−4q2[22˜µ2−14µ2−3ν2D(0) t] −k3 c35q2ρ[D(0) t]2−4q4[20˜µ2+15µ2+3ν2D(0) t] −k4 c13qρ[D(0) t]2−4q3[18˜µ2−22µ2−9ν2D(0) t]oD(0) t(D.13) For stripes (i=j) we obtain ˆϕNMij =h(kc−q)(kc+q)(3kc+q)ρ[D(0) t]2i−1n2kc(2kc−q)ρ[D(0) t]2 +(kc−q)(3kc+q)k2 c(6˜µ2−7µ2)−3q2µ2+2kcq(˜µ2+2µ2) −ν2(kc−q)(3kc+q)(3k2 c−2kcq+q2)D(0) toD(0) t(D.14) For θij =π/6, (ij =ji = 14 = 36 = 25) we obtain 132 Usual ferrofluids shown here. With the solutions of the second order, the third order Fredholm’s theorem can be fulfilled. The latter reads in the case of ferrofluids h¯vi|ρ∂(2) tv(1) ii+h¯vi|ρ∂(1) tv(2,1) ii=−h¯vi|ρ∂(1) tv(2,2)i−ρh¯vi|∂j(v(1) iv(2,1) j+v(2,1) iv(1) j)i −ρh¯vi|∂j(v(1) iv(2,2) j+v(2,2) iv(1) j)i(E.5) Since the analytical expressions for the eigenvectors v(2,2) iare bulky, the explicit calculation of the cubic coefficients has been performed with Mathematica and the results are shown below. We should mention, however, that also in the third order the right hand side of eq. (E.5) is proportional to [∂(0) t]3and the global factor ([ω(0)]2−[σ(0)]2) can be canceled. The discussion of the normal stress boundary condition in the case of ferrofluids can be taken from the sections 5.3.4 and 5.4. In the second order, the additional condition to the amplitudes (5.115) is valid for ferrogels and ferrofluids, alike, and in the corresponding third order condition (5.133) we have to substitute µ2→0 with the consequence that there is no second order time derivative. The typical time scale in the case of ferrofluids is then given by τ0=ν2kc/(ρG) which is in accordance with previous theoretical discussions [39]. The final amplitude equation is derived in the same way as in section 5.5 for magnetic gels and finally results for the hexagonal pattern in ∂Tξ1=1 2˜ǫflξ1−2 3√Aξ∗ 2ξ∗ 3− |ξ1|2ξ1−Bfl 120 Afl(|ξ2|2+|ξ3|2)ξ1(E.6) For the square pattern the quadratic coefficient is absent and we obtain ∂Tξ1= ˜ǫflξ1− |ξ1|2ξ1−Bfl 90 Afl|ξ5|2ξ1(E.7) where the cubic coefficients are given by Afl≈8.625 (E.8) Bfl 120 ≈3.150 (E.9) Bfl 90 ≈4.266 (E.10) The discussion for the different stable patterns follows the same lines as in section 5.5. 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E. 26, 275 (2008) S. Bohlius, H. Pleiner and H. R. Brand, The amplitude equation for the Rosensweig instability, submitted to Physica D 141