Analysis of Thermomagnetic Convection in a Vertical Layer of Magnetic Fluid with Arbitrary Orientation of Magnetic Field

Transcription

1 Analysis of Thermomagnetic Convection in a Vertical Layer of Magnetic Fluid with Arbitrary Orientation of Magnetic Field A dissertation for the confirmation of the degree of Doctor of Philosophy by Md Habibur Rahman Department of Mathematics Faculty of Science, Engineering and Technology Swinburne University of Technology Hawthorn, Victoria, Australia 215

2 Abstract Linear stability of pure magnetic and magneto-gravitational convection in a layer of ferrofluid between two vertical differentially heated non-magnetic plates in a uniform oblique external magnetic field under zero- and nonzero gravity conditions has been investigated. In the first part of the dissertation pure thermomagnetic convection in the gravity-free environment is studied. For each set of physical governing parameters, the equivalent problem is solved for a range of wavenumbers. The arising thermomagnetic convection is caused by the spatial variation of magnetization occurring due to its dependence on the temperature. The critical values of the governing parameters at which the transition between motionless and convective states is observed are determined for various field inclination angles and for fluid s magnetic parameters that are consistently chosen from a realistic experimental range. The obtained results are compared with those available for paramagnetic fluids. In contrast to the linear distribution of magnetic field that exists in a layer of a paramagnetic fluid the magnetic field in a layer of ferrofluid varies non-linearly. It has been shown that unlike in paramagnetic fluids, the stability characteristics magnetoconvection in ferrofluids depend not only on the inclination of the applied magnetic field, but also on its magnitude. This is shown to be due to the nonlinearity of the magnetic field distribution within ferrofluid. It is shown that similar to paramagnetic fluids the most prominent pure thermomagnetic convection patterns in ferrofluid align with the in-layer component of the applied magnetic field but in contrast to paramagnetic fluids the instability patterns detected in ferrofluids can be oscillatory. In the second part of the dissertation both gravitational and magnetic effects acting simultaneously are accounted for and shown to lead to distinct mechanisms, namely, thermogravitational (buoyancy-driven) and thermomagnetic, responsible for the appearance

3 ii of various instability modes. The characteristics of all instability modes are investigated, and three main types of instability patterns corresponding to thermogravitational, magnetic and magneto-gravitational convection are found to exist in a normal magnetic field. The oblique external magnetic field is shown to lead to the preferential shift of instability structures toward the hot wall, which introduces an asymmetry to the problem and the subsequent qualitative change in the stability characteristics. It is found that for each field inclination angle there exists a preferred field orientation angle that promotes the onset of magneto-gravitational instability the most. The variation of fluid magnetisation due to thermal disturbances has been found to always have a destabilizing effect. However depending on the orientation of the applied field the related variation of a magnetic field can to draw the energy from the perturbed flow field thus playing a stabilising role. It is also found that the role of the buoyancy effects changes from destabilising in the gravity-dominated flow to stabilising in flows with strong magnetic effects.

4 Dedication This dissertation is dedicated to my beloved parents and sons, Alif & Adib and to my dear wife, Lipi.

5 Acknowledgements This research, while analytical and computational in its nature, has been inspired by the experimental research that had been conducted in the Magnetic Fluids Laboratory at Perm State National Research University, Russia under the scientific leadership of Professor Gennady F. Putin. Sadly, Professor Putin passed away just a few weeks before this dissertation has been completed. Thus this work is dedicated to his memory. I am immensely grateful to my supervisor, Associate Professor Sergey A. Suslov from the Department of Mathematics at Swinburne University of Technology for his encouragement, guidance and support from the initial to the final stage of this research. His patience and professionalism are greatly appreciated. I consider myself fortunate for having undertaken the research work under his enthusiastic, inspiring and helpful guidance. I extend my gratitude to my co-supervisor, Dr Tonghua Zhang at the Department of Mathematics, Swinburne University of Technology, for his kind help and cooperation. I would like to thank Professor A. A. Bozhko and Mr A. S. Sidorov of the Physics Department at Perm State National Research University, Russia for discussing the relevance of the presented computational results to the experimental studies conducted in their group. I would also like to thank all teachers and research students at the Department of Mathematics at Swinburne University of Technology who contributed so much to my work through sharing their knowledge with me. Special thanks to all of my friends, relatives, and colleagues at the Department of Mathematics, KUET, Khulna, Bangladesh for their encouragement and moral support.

6 v I am extremely thankful to my bosom friend, Associate Professor Ali Akbar from the Department of Applied Mathematics, University of Rajshahi, Bangladesh for his great instructions, inspiration, constant moral support and discussions at all times and in particular during this study. I am grateful to the authority of Swinburne University of Technology for selecting me as a PhD research student and providing scholarship support during my study. I would like to thank the staff of Swinburne University of Technology for their warm assistance and endless help to fulfil my requests. My deep appreciation goes to my beloved parents, sons, brothers, nephews and nieces for their love, motivation and continuous moral support in every aspect. Finally, I would like to thank my wife, Dilruba Khanam (Lipi), for her patience, understanding and precious love during this research. Without her constant moral support and encouragement, this dissertation would not be possible. Md Habibur Rahman, 215

7 Declaration I, hereby declare that the matter embodied in this dissertation is the result of investigations carried out by me under the supervision of Associate Professor Sergey A. Suslov at the Department of Mathematics, Faculty of Science, Engineering and Technology, Swinburne University of Technology, Hawthorn, Victoria, Australia, and to the best of my knowledge, it has not been submitted for the award of any degree, diploma or fellowship at any other University or Institute. Md Habibur Rahman, 215

8 Contents Acknowledgments iv Declaration vi List of Figures xx List of Tables xxiii 1 Introduction Objective and Motivation Magnetic Fluids Literature Review Convection in Ferromagnetic Fluids Rayleigh-Bénard Convection in Ferromagnetic Fluids Marangoni Convection in Ferromagnetic Fluids

9 CONTENTS viii Modulated Convection in Ferromagnetic Fluids Thermomagnetic Convection in Ferromagnetic Fluids in Various Configurations Other Effects in Magnetic Fluids Pure Thermomagnetic Convection Introduction Problem Formulation and Governing Equations Nondimensionalisation and Problem Parameters Basic Flow Linearized Perturbation Equations Squire s Transformation Numerical Results Discretisation Check of Numerical Accuracy Flow Stability Characteristics Perturbation Energy Balance Perturbation Fields Magneto-Gravitational Thermal Convection 9

10 CONTENTS ix 3.1 Introduction Problem Formulation and Governing Equations Nondimensionalisation and Problem Parameters Basic Flow Linearized Perturbation Equations Squire s Transformation Numerical Results Comparison with Selected Previous Numerical Results Flow Stability Characteristics Stability Characteristics of Flows in a Normal Field Wave-like Instabilities in Oblique Fields Optimal Orientation of Wave-like Instabilities Complete Stability Diagrams for an Equivalent Two-dimensional Problem Perturbation Energy Balance and Perturbation Fields Conclusions Pure Thermomagnetic Convection Mixed Magneto-Gravitational Convection Recommendations for the Future Work

11 CONTENTS x References 166 Publications 175

12 List of Figures 1.1 Surfacted ferrofluid (Tynjälä 25). Used with permission Cross-sectional schematic view of cell illustrating convection rolls A schematic view of instability patterns observed experimentally in a differentially heated layer of ferrofluid placed in a magnetic field, see figure 9 in Suslov et al. (212). The magnetic field is applied normally in the center of the layer, but its direction is inevitably distorted near the edges Sketch of the problem geometry. The vector of external magnetic field, H e forms angles δ and γ with the coordinate axes The comparison of magnetic fluid properties defined using Langevin s law and MMF2 model for δ = and T = 293 K (Θ = 7.5 K is chosen in plot (d)) Numerical solution for the magnitude H of the undisturbed magnetic field ((a) and (d)) and its cross-layer component H x ((b) and (e)) for H e = 1 (top row), H e = 1 (bottom row), χ = χ = 3 and various field inclination angles δ. Plots (c) and (f) show the corresponding relative error of the asymptotic solution (2.4.9)

13 LIST OF FIGURES xii 2.5 Same as figure 2.4 but for χ = 1.5, χ = Numerical solution for the magnitude M of the undisturbed fluid magnetization ((a) and (d)), its cross-layer component M x ((b) and (e)) and magnetic pressure P ((c) and (f)) for H e = 1 (top row), H e = 1 (bottom row), χ = χ = 3 and various field inclination angles δ Same as figure 2.6 but for χ = 1.5, χ = Refraction of magnetic lines ((a) and (c)) and the distribution of Kelvin force (solid line, and dashed line for δ = ) ((b) and (d)) in a layer of magnetic fluid heated from the left for the field inclination angle δ = 5, γ =, χ = χ = 3, H e = 1 ((a) and (b)) and H e = 1 ((c) and (d)) Same as figure 2.8 but for δ = Same as figure 2.8 but for δ = Same as figure 2.8 but for χ = 1.5, χ = Same as figure 2.9 but for χ = 1.5, χ = Same as figure 2.1 but for χ = 1.5, χ = (a) Critical magnetic Rayleigh number Ra mc and (b) wavenumber α c as functions of the field inclination angle δ for transverse rolls at H e = 1 and γ =. The respective plots for H e = 1 are indistinguishable within the figure resolution Critical wave speed c = σi α c as the function of the field inclination angle δ for transverse rolls at γ = for (a) H e = 1 and (b) H e = Same as figure 2.14 but for longitudinal rolls at γ =

14 LIST OF FIGURES xiii 2.17 (a) Critical magnetic Rayleigh number Ra mc, (b) wavenumber α c and (c) wave speed c as functions of the azimuthal angle γ for various angles δ for H e = 1 and χ = χ = Same as figure 2.17 but for χ = χ = Same as figure 2.17 but for χ = 1.5 and χ = Disturbance energy integrands at the critical point of magnetoconvection threshold Ra mc = 178.3, α c = at H e = 1, δ = γ = and χ = χ = Same as figure 2.2 but for Ra mc = 232.4, α c = 1.951, δ = Same as figure 2.2 but for Ra mc = 533.7, α c = 2.116, δ = Same as figure 2.2 but for Ra mc = 272.8, α c = 2.381, δ = (Colour online) Perturbation eigenfunctions of the fluid velocity v 1 = (u, v), temperature θ 1, magnetization M 1 and magnetic field H 1 for magnetoconvection at H e = 1, δ = γ = and χ = χ = 3 at the critical point Ra mc = 178.3, α c = Colour scale is arbitrary as the amplitude of perturbations cannot be determined in the framework of a linear analysis Same as figure 2.24 but for Ra mc = 232.4, α c = 1.951, δ = Same as figure 2.24 but for Ra mc = 533.7, α c = 2.116, δ = Same as figure 2.24 but for Ra mc = 272.8, α c = 2.381, δ = Same as figure 2.24 but for longitudinal rolls at Ra mc = 267., α c = 1.916, δ = 1 and γ =

15 LIST OF FIGURES xiv 3.1 Sketch of the problem geometry. The vector of external magnetic field, H e forms angles δ and γ with the coordinate axes Leading disturbance temporal amplification rates σ R (left) and frequencies σ I (right) as functions of the combined wavenumber α for ( Gr m, Gr) = (, ) (onset of thermogravitational convection) at δ = γ =, χ = χ = 5 and Pr = Same as Figure 3.2 but for ( Gr m, Gr) = (3.35, ) (onset of stationary magneto-convection) Same as Figure 3.2 but for ( Gr m, Gr) = (43.3, 11.75). In the left plot the left and right maxima correspond to small- and large-wavenumber waves, respectively, and the middle maximum corresponds to a stationary roll pattern Schematic diagram of main flow patterns in an oblique magnetic field: (a) thermogravitational waves; (c) stationary vertical thermomagnetic rolls; (d) oblique thermomagnetic waves Perturbation eigenfunctions of the fluid velocity v 1, temperature θ 1, magnetization M 1 and magnetic field H 1 for magneto-convection in a normal field (δ = ) for H e = 1, χ = χ = 5 and at the critical point for Gr m =, Gr = , and α = Same as figure 3.6 but for Gr m = 3.35, Gr = and α = Same as figure 3.6 but for Gr m = 43.3, Gr = and α = Same as figure 3.6 but for Gr m = 43.3, Gr = and α = Same as figure 3.6 but for Gr m = 43.3, Gr = and α =

16 LIST OF FIGURES xv 3.11 Comparison among the critical parameter values: (a) Grashof number Gr (the flow is stable under the respective curves), (b) wavenumber α and (c) wave speeds c as functions of the field inclination angles δ and γ for Gr m = 12, H e = 1, Pr = 55 and χ = χ = Same as figure 3.11 but for χ = χ = Same as figure 3.11 but for χ = 1.5 and χ = (a) The value of the field orientation angle γ min at which the instability first occurs and (b d) the corresponding critical parameters as functions of the field inclination angle δ for Gr m = 12, H e = 1, Pr = 55 and χ = χ = Same as figure 3.14 but for χ = χ = Same as figure 3.14 but for χ = 1.5 and χ = Comparison of the critical parameter values for the upward (solid line) and downward (dashed line) propagating waves: (a) Grashof number Gr (the flow is stable under the respective curves), (b) wavenumber α and (c) wave speeds c as functions of the azimuthal angle γ for Gr m = 12, H e = 1, Pr = 55, δ = 5 and χ = χ = Same as figure 3.17 but for χ = χ = Same as figure 3.17 but for χ = 1.5 and χ =

17 LIST OF FIGURES xvi 3.2 Comparison of the critical parameter values for thermo-magnetically less (H e = 1, solid line) and more (H e = 1, dashed line) sensitive fluids: (a) Grashof number Gr (the flow is stable under the respective curves), (b) wavenumber α and (c) wave speeds c as functions of the azimuthal angle γ for Gr m = 12, Pr = 55, δ = 5 and χ = χ = 3. Type I (thermogravitational convection) instability Same as figure 3.2 but for χ = χ = Same as figure 3.2 but for χ = 1.5 and χ = (a) Stability diagram for an equivalent two-dimensional problem; (b) the critical wavenumber α c and (c) the corresponding wave speeds along the stability boundaries shown in plot (a) for H e = 1, Pr = 55 and χ = χ = 5 in a normal magnetic field (δ = ). Parametric curves corresponding to the Type I, II and III instabilities are shown by the solid, dashed and dash-dotted lines, respectively Same as Figure 3.23 but for χ = χ = Same as Figure 3.23 but for χ = 1.5 and χ = (a) Maximum amplification rate for an equivalent two-dimensional problem; (b) the corresponding wave numbers α and (c) wave speeds for Gr m = 12, H e = 1, Pr = 55 and χ = χ = 5 in a normal magnetic field (δ = ). The solid and dashed lines correspond to the Type I and II instabilities, respectively Leading disturbance temporal amplification rates σ R (left) and frequencies σ I (right) as functions of the combined wavenumber α for ( Gr m, Gr) = (31.37, 5.) at δ = γ =, χ = χ = 5 and Pr =

18 LIST OF FIGURES xvii 3.28 Same as figure 3.27 but for ( Gr m, Gr) = (51.3, 21.) Same as figure 3.26 but for Gr m = 35. The solid, dashed and dash-dotted lines correspond to the Type I, II and III instabilities, respectively (a) Stability diagram for an equivalent two-dimensional problem; (b) the critical wavenumber α c and (c) the corresponding wave speeds along the stability boundaries shown in plot (a) for H e = 1, Pr = 55 and χ = χ = 5 in an oblique magnetic field for δ = 5 and γ =. The solid and dash-dotted lines correspond to the Type I/II and III instabilities, respectively Same as figure 3.3 but for χ = χ = Same as figure 3.3 but for χ = 1.5 and χ = Same as Figure 3.27 but for ( Gr m, Gr) = (6.4, ) and δ = 5, γ =. The solid and dashed lines represent the first and second leading eigenvalues σ of the linearised stability problem, respectively Leading disturbance temporal amplification rates σ R (left), frequencies σ I (middle) and the amplification rate difference σ R = σ 1 R σr 2 as functions of the combined wavenumber α for ( Gr m, Gr) = (13.88, 1.7), δ = 5 and γ =. The solid and dashed lines represent the first and second leading eigenvalues σ of the linearised stability problem, respectively (a) Maximum amplification rate for an equivalent two-dimensional problem; (b) the corresponding wave numbers α and (c) wave speeds for Gr m = 12, H e = 1, Pr = 55 and χ = χ = 5 in an oblique magnetic field for δ = 5 and γ =. The solid and dashed lines correspond to the Type I and II instabilities, respectively

19 LIST OF FIGURES xviii 3.36 Same as figure 3.35 but for Gr m = Same as figure 3.35 but for Gr m = Same as Figure 3.3 but for H e = Variation of stability diagrams for H e = 1, Pr = 55 and χ = χ = 5 in an oblique magnetic field for γ = and (top to bottom rows) δ =, 5, 8.5, 1 and 15. The solid, dashed and dash-dotted lines correspond to the Type I, II and III instabilities, respectively Leading disturbance temporal amplification rates σ R (left) and frequencies σ I (right) as functions of the combined wavenumber α for χ = χ = 5, Pr = 55, δ = 8.5, γ = and ( Gr m, Gr) = (197, 1.35). Type I/III instability. The solid and dashed lines represent the first and second leading eigenvalues σ of the linearised stability problem, respectively Same as Figure 3.4 but for ( Gr m, Gr) = (187.3, 8.9). The The Type II instability Same as Figure 3.4 but for ( Gr m, Gr) = (2.3, ). The Type II instability Comparison of the critical values for the upward (solid line) and downward (dashed line) wave modes in an oblique field for H e = 1, Pr = 55, γ = and δ = 5 and χ = χ = 5 (top row), χ = χ = 3 (bottom row)

20 LIST OF FIGURES xix 3.44 Comparison of the critical values for H e = 1, Pr = 55, χ = χ = 3, δ = 5 and various field orientation angles γ: (a) stability diagram for an equivalent two-dimensional problem; (b) critical wavenumber α c and (c) the corresponding wave speeds along the stability boundaries shown in plot (a) Same as figure 3.44 but for δ = (a) The maximum amplification rate of thetype II instability and (b) the corresponding wavenumber α as functions of the field orientation angle γ for δ = 5, Gr =, Gr m = 12, H e = 1, Pr = 55 and χ = χ = The critical parameter values: (a) Grashof number Gr (the flow is stable above the curve), (b) wavenumber α and (c) wave speeds c as functions of the field orientation angle γ for δ = 5, Gr m = 12, H e = 1, Pr = 55 and χ = χ = 5. The Type II instability Same as figure 3.47 but for Gr m = 35. The Type I (solid lines) and II (dashed lines) instabilities Same as figure 3.47 but for Gr m = 4. The Type I (solid lines) and II (dashed lines) instabilities Same as figure 3.47 but for Gr m = 4 and δ = 1. The Type I (solid lines) and II (dashed lines) instabilities

21 LIST OF FIGURES xx 3.51 The perturbation energy integrals entering equation (3.9.1) characterizing the thermomagnetic (Σ m1 + Σ m2, the solid line) and thermogravitational (Σ Gr, the dash-dotted line) mechanisms of convection and the exchange with the basic flow (Σ uv, the dashed line) as functions of the ratio Gr m / Gr along the stability boundaries shown by the solid lines (a) in figure 3.23(a) for the normal field and (b) in figure 3.3(a) for the oblique field Disturbance energy integrands at the critical points corresponding to parameters listed in rows 13 (top) and 14 (bottom) in Table Disturbance fields at the critical points corresponding to parameters listed in rows 13 (top) and 14 (bottom) in Table 3.9. Cross-section along the main periodicity direction Disturbance energy integrands at the critical points corresponding to parameters listed in rows 15 (top) and 16 (bottom) in Table Disturbance fields at the critical points corresponding to parameters listed in rows 15 (top) and 16 (bottom) in Table 3.9. Cross-section along the main periodicity direction Disturbance energy integrands at the critical points corresponding to parameters listed in row 17 in Table Disturbance fields at the critical points corresponding to parameters listed in row 17 in Table 3.9. Cross-section along the main periodicity direction. 16

22 List of Tables 1.1 A brief summary of literature review of research related to ferrofluid flows The typical values of experimental parameters and properties of the ferrofluid manufactured in Scientific Laboratory of Practical Ferromagnetic Fluids, Ivanovo, Russia under Technical Conditions and used in experiments reported in Suslov et al. (212) and Bozhko et al. (213) The critical values of Ra m, α and disturbance wave speed c = σ I / α and the corresponding perturbation energy integrals Σ k and Σ m1 for magnetoconvection at δ =, H e = 1 and various values of χ and χ Same as table 2.2 but for transverse rolls at δ = 1 and γ =, H e = 1 (odd-numbered lines), H e = 1 (even-numbered lines) Same as table 2.3 but for longitudinal rolls at δ = 1 and γ =

23 LIST OF TABLES xxii 3.1 The critical values of Gr m, Gr, α, disturbance wave speed c = σ I / α and the maximum speed of the basic flow ṽ max for mixed convection in a perpendicular (δ = ) external magnetic field H e = 1 at χ = χ = 5 and various values of Prandtl number Pr The critical values of Gr, α and disturbance wave speed c = σ I / α for leading two waves of mixed convection in a normal magnetic field (δ = ) for Gr m = 12, H e = 1, Pr = 55 and various values of χ and χ The critical values of Gr, α and disturbance wave speed c = σ I / α for the upward propagating wave of mixed convection in an oblique magnetic fields for Gr m = 12, γ =, Pr = 55, H e = 1 (odd-numbered lines), H e = 1 (even-numbered lines) and various values of χ and χ Same as Table 3.3 but for the downward propagating wave The representative critical values of the Squire-transformed Grashof number Gr, wavenumber α and disturbance wave speed c = σ I / α for the Type II istability in a normal magnetic field (δ = ) for Gr m = 12, H e = 1, Pr = 55 and various values of χ and χ Values of the perturbation energy integrals Σ k, Σ m1, Σ m2, Σ Gr and Σ uv computed for the Type I instability in a normal magnetic field (δ = ) at Gr m = 12, Pr = 55, H e = 1 and various values of χ and χ and at the corresponding critical values of α, Gr given in Table 3.2 for the upward (odd-numbered lines) and downward (even-numbered lines) propagating waves Same as Table 3.6 but for an oblique magnetic field (δ = 5, γ = ) and for the critical values of α, Gr given in Tables 3.3 and

24 LIST OF TABLES xxiii 3.8 Same as Table 3.7 but for H e = Selected critical values of parameters and perturbation energy integrals for Pr = 55, H e = 1 and χ = χ =

25 Chapter 1 Introduction 1.1 Objective and Motivation This study deals with thermomagnetic convection in ferromagnetic fluids, which are also referred to as ferrofluids or magnetic fluids. The effects of uniform heat source and magnetic field are considered. Over the few past decades, the applications of magnetic and electric fields in fluid flow control gained a considerable attention with prospects in areas such as medicine, chemical engineering, nuclear fusion, high speed noiseless printing etc. The main area relevant to this study is ferrohydrodynamics: the study of nonconducting fluid motions caused by forces created by magnetic fields. To understand the physics of a complex flow behavior of magnetic fluids, and also to obtain a vital information for industrial applications a systematic study through a proper theory is important. Therefore, the goal of the work is to study various convective instabilities in ferromagnetic fluids influenced by buoyancy and ponderomotive force due to magnetic effects. Fluids used in most of the technologically important applications are non-isothermal. That is why the study of convection and heat transfer is needed. Since ferrofluids respond to both thermal and magnetic fields, their physical and mathematical description is a challenging task. Most of the available studies related to heat transfer

26 CHAPTER 1. INTRODUCTION 2 and convection in fluids follow Newtonian description. Typically, a set of constitutive equations is used involving a constant viscosity assumption. The flow of ferromagnetic colloidal suspension of magnetic solid particles such as magnetite in a carrier liquid between two differentially heated plates placed in a uniform external magnetic field is considered. The instability patterns detected when the applied field is normal to the plates are found to consist of stationary magnetoconvection rolls and propagating thermomagnetic or thermogravitational waves (Suslov 28). Two distinct mechanisms, thermogravitational and thermomagnetic, are responsible for the appearance of these instability modes. The objective of this study is to investigate various convective instabilities in ferromagnetic fluids driven by buoyancy and ponderomotive magnetic forces in an obliquely applied magnetic field. The physical nature of the so-induced instability modes and their most prominent features will also be determined to provide guidance for the future experimental investigation. For the current study, it is chosen a three dimensional geometry of a wide and tall vertical fluid layer cooled from one side, heated from the other and placed in a magnetic field which is inclined at an arbitrary angle to the plates. Such a configuration is easy to recreate experimentally and it enables one to focus on investigating the physical mechanisms leading to a nontrivial fluid motion without being distracted by complicated boundary effects. 1.2 Magnetic Fluids Ferromagnetism is the property of cobalt, nickel, iron, their alloys and some minerals that have these metals as compounds. Magnetic properties of such materials weaken as their temperature increases, and they are lost completely above a certain temperature called Curie point. Curie point is below the melting temperature of ferromagnetics so that the melts of ferromagnetic materials are non-magnetic. In contrast to melts, magnetic fluids are multi-phase media containing solid magnetic particles that can be magnetized. Such

27 CHAPTER 1. INTRODUCTION 3 Figure 1.1: Surfacted ferrofluid (Tynjälä 25). Used with permission. suspensions can be used to transfer heat, and heat and mass transport in such liquid magnetics can be controlled by using an external magnetic field. Synthetic magnetic fluids, also known as ferrofluids, are electrically non-conducting stable colloidal suspensions consisting of the carrier liquid (kerosene, water or mineral oil) and magnetic (iron, cobalt, nickel etc.) nanoparticles with the characteristic size d p 1 nm covered by surfactants (oleic acid) to prevent them from forming aggregates. The most widely studied ferrofluids contain colloidal magnetite (Fe 3 O 4 ). A typical ferromagnetic fluid can have up to 1% of magnetic solids and up to 1% of surfactant by volume (Odenbach 22a). Due to the demagnetization and the chemical adsorption impact at the boundary of the magnetic core, there is a layer of demagnetized magnetite of thickness 1 nm near the particle boundary. As seen from Figure 1.1 each particle is coated with an appropriate surfactant and the resulting fluid is known as surfacted ferrofluid. The studies of magnetic properties of such colloids have been conducted since 193s (Elmore 1938) but they intensified noticeably in the 196s and 197s when the industrial production of magnetic fluids became possible (Bashtovoy et al. 1988). Nowadays a large body of literature exists on the properties of ferrofluids, see, for example, (Rosensweig 1979, 1985, Bashtovoy et al. 1988, Blums et al. 1989, 1997) and references therein. In the absence of a magnetic field, the magnetic moments of individual particles in ferrofluids

28 CHAPTER 1. INTRODUCTION 4 are randomly oriented so that the fluids have no net magnetization. Thus they are often categorized as superparamagnets rather than ferromagnets (Albrecht et al. 1997). However, when placed in a magnetic field, they orient along the applied field and the fluid becomes magnetized. The degree of magnetization depends on the applied field strength and the local temperature and concentration of magnetic particles. Led by the arising Kelvin force a magnetized fluid tends to flow toward regions of a stronger magnetic field. In this study it is assumed that the concentration of magnetic phase remains uniform and therefore the influence of only the thermal and magnetic fields on the flow structure will be studied. Such an assumption is reasonable if the characteristic timescale of convection flows of interest is much shorter than that of magnetic fluid segregation due to Soret effect or thermophoresis of solid particles (Shliomis & Smorodin 22). Thus the ferrofluid magnetization will be assumed to depend only on the magnitude of the applied magnetic field as well as on the fluid magnetic susceptibility, which is the ratio of the magnetization in fluid to strength of the applied magnetic field. 1.3 Literature Review Artificially manufactured ferrofluids respond to an external magnetic field similarly to natural paramangnetic and diamagnetic fluids (i.e., water, protein solution, paramagnetic melts) and gases (i.e., oxygen). However the degree of the magnetization which can be achieved in artificial ferrofluids is many orders of magnitude higher than that in natural magnetic fluids. Because of that ferrofluids found many practical applications in electronic devices, energy conversion devices, simulation of zero gravity situations in ground-based experiments, separation of oil from water, analytical instrumentation, tunable optical filters and defect detection sensors and in areas such as medicine, mechanical and aerospace engineering, art, etc. Since the synthesis of ferrofluids begun in the 196s the use of colloidal suspensions of ferromagnetic particles in a carrier fluid has led to new designs for pumps (Park & Seo 23), valves (Goldstein 1977), actuators (Kamiyama 1986) and heat pipes (Ming et al. 29). Ferrofluids are used in many industrial appli-

29 CHAPTER 1. INTRODUCTION 5 cations such as digital data storage, resonance imaging, and to dampen vibrations. In medicine, ferrofluids are used as the contrast agents for magnetic resonance imaging and can be used for cancer detection. In this case ferrofluids contain iron oxide nano-particles and are called SPION which stands for "Superparamagnetic Iron Oxide Nano-particles". There is much experimentation using ferrofluids in cancer treatment called magnetic hyperthermia. It is based on the fact that a ferrofluid placed in an alternating magnetic field releases heat. As a result, if magnetic nano-particles are injected in a tumour and a patient is placed in an alternating magnetic field of the appropriate amplitude and frequency, the tumour temperature is raised and the tumour cells are killed by necrosis if the temperature increases above 45 C. Science museums and some arts have special devices on display that use magnets to make ferrofluids circulate around specially shaped surfaces in a fountain show-like fashion to entertain guests. Sachiko Kodama is known for her ferrofluids art. Ferrofluids were used by an electronic rock band, "Pendulum", the in music video, "Water color". The postmetal band "Isis" is also known for the use of ferrofluids in music-videos. An American art studio, "CZFerro", began using ferrofluids in its productions in 28. As ferrofluids are paramagnetic, they obey Curie s law and thus become less magnetized at higher temperatures. Such temperature sensitive ferrofluids induce especially strong practical interest (Nakatsuka et al. 199). For example, such fluids are used in loudspeakers to remove Ohmic heat generated by the voice coil. In the presence of temperature variation, magnetic buoyancy force is induced in a ferrofluid which leads to fluid motion known as convection. Convection in general is one of the major modes of heat transfer and it has many forms e.g. natural, forced etc. Natural convection or free convection occurs due to temperature differences which affects the density and thus relative buoyancy of the fluid. Denser components fall while less dense ones rise leading to a bulk fluid motion. Natural convection can only occur in a gravitational field. A common example of natural gravitational convection is a pot of boiling water in which the hot and less dense water at the bottom layer moves upward in plumes and the cool and denser water near the

30 CHAPTER 1. INTRODUCTION 6 top of the pot sinks. Gravitational convection is a type of natural convection induced by buoyancy resulting from fluid density variation with temperature. In forced convection, also called advection, fluid movement results from external surface action such as that of a fan or a pump. When temperature sensitive ferrofluid experiences a temperature variation in the presence of an external magnetic field, a thermomagnetic force arises, which drives stronger magnetized colder fluid particles to the regions with a stronger magnetic field. This phenomenon is known as magneto-convection and thermomagnetic convection will be focused in the present study. The review of relevant literature that follows is summarized according to the following classification of topics: Convection in Ferromagnetic Fluids; Rayleigh-Bénard Convection in Ferromagnetic Fluids; Marangoni Convection in Ferromagnetic Fluids; Modulated Convection in Ferromagnetic Fluids; Thermomagnetic Convection in Ferromagnetic Fluids in Various Configurations; Other Effects in Magnetic Fluids Convection in Ferromagnetic Fluids A convection occurs due to temperature differences, which affect the density and thus relative buoyancy of the fluid. Denser components will fall while less dense ones will rise leading to a bulk fluid motion. A natural convection can only occur in a gravitational field.

31 CHAPTER 1. INTRODUCTION 7 Berkovskii & Bashtovoi (1971) discussed the gravitational convection problem for incompressible non-conducting ferromagnetic fluid resulting from the magnetocaloric effect. The natural convection with a vertical temperature gradient is equivalent to this problem. Numerical estimates of the critical magnetic field gradients are given and closed form solutions for both velocity and temperature are obtained in their study. A similar problem was solved both analytically and numerically using a perturbation method by Kamiyama et al. (1988). The relevant magnitudes of the magnetization parameter and thermal Rayleigh number together with the uniform pressure gradient were shown to significantly affect the dynamics of ferromagnetic fluid in a vertical cavity. Heat transfer in natural convection of a magnetic fluid in a cubic enclosure with isothermal hot and cold vertical end walls was investigated experimentally by Kikura et al. (1993). A water-based magnetic fluid was considered. The influence of the magnetic field on the Nusselt numbers were examined and characteristics of the heat transfer of the magnetic fluid were discussed. An interesting type of thermomagnetic convection was observed when a horizontal magnetic field was applied. The cold layer was affected by the magnetic force and the low- temperature fluid was driven along the gradient of the magnetic field. It was found that the heat transfer of magnetic fluid could be controlled by changing the applied magnetic field. Sawada et al. (1993) investigated experimentally the natural convection of a magnetic fluid in concentric annuli. Two concentric cylinders made of copper, kept at different temperature were placed horizontally. The authors carried out several types of experiments to clarify the effects of the intensity and the direction of magnetic fields on the natural convection. An ordinary natural convection was observed when there was no magnetic field. When a magnetic field gradient was applied in the opposite direction to the gravity, convection was found to occur in the direction opposite to that of natural convection. Even the applied magnetic field with small intensity played an important role in the heat transfer in magnetic fluid. Thus it was recognized that the application of a magnetic field can affect the natural convection of a magnetic fluid.

32 CHAPTER 1. INTRODUCTION 8 A convection of a magnetic fluid in a square cavity heated from below has been investigated experimentally and numerically for the vertically imposed external magnetic field by Yamaguchi et al. (1999). The results obtained disclose that the applied magnetic field has a destabilizing influence and the flow mode in the supercritical state appears to be quite different from that seen when the magnetic field is absent. A stronger heat transfer occurs due to the very active circulation motion of flow in the cell, resulting in the higher Nusselt number for larger magnetic Rayleigh number. Yamaguchi et al. (22) examined numerically a convection of a magnetic fluid in a square cavity partitioned in the middle and heated from below in an external applied magnetic field. The adiabatic side walls were considered. The authors focused their attention on investigating the effect of magnetic field on heat transfer by considering temperature and flow fields at representative Rayleigh numbers. It has been found that in the presence of magnetic field the heat transfer rate decreases for small Rayleigh numbers (Ra 5) and increases for Ra > 5. However, in the absence of magnetic field the higher heat transfer rate has been detected even for small Rayleigh numbers. Wang & Wakayama (22) numerically investigated the magnetic effects on the natural convection in non- and low-conducting diamagnetic fluids under non-uniform magnetic fields with various directions. They focused on the effects of the direction of magnetization force and the dependence on Rayleigh number. The magnetic fluid was restrained in an electrically insulating rectangular cavity and the convection was driven by a horizontal temperature gradient across the fluid layer. The authors noticed that when a magnetic field was imposed on a non-conducting fluid horizontally or vertically, depending on the direction of the magnetization force, the natural convection could be damped or promoted, respectively. However, if the field was imposed parallel to the fluid layer, the natural convection was only weakly promoted. A combined natural and magnetic convective heat transfer in a ferrofluid in a cubic enclosure with two differentially heated opposite vertical walls was simulated numerically by

33 CHAPTER 1. INTRODUCTION 9 Snyder et al. (23). In order to validate the theory of magnetoconvection, the isothermal wall and Nusselt numbers were compared to experimental work of Sawada et al. (1994). The results were obtained using standard computational fluid dynamics (CFD) codes, with minor modifications to account for the Langevin factor when needed. The CFD packages FIDAP and FEMLAB were used to predict numerically the effects of magnetic fields on convection in ferrofluids. In the presence of a magnetic field gradient the magnetoconvection was induced and cooler ferrofluid was found to flow along the gradient of the magnetic field dislocating hotter ferrofluid Rayleigh-Bénard Convection in Ferromagnetic Fluids Classical Rayleigh-Bénard convection (RBC) is caused by the instability of a fluid layer which is confined between two horizontal plates and is heated from below to produce a fixed temperature difference. In the standard Rayleigh-Bénard problem, the instability is driven by a density difference caused by a thermal expansion of the fluid. Since fluids typically have positive thermal expansion coefficient, the hot fluid at the bottom of the cell expands and produces destabilizing density gradient in the fluid layer. If the density gradient is sufficiently large, the hot fluid rises causing a convective flow which results in enhanced transport of heat between the two horizontal plates. There are two processes that oppose fluids flow amplification. Firstly, viscous damping in the fluid directly opposes the fluid flow. Secondly, thermal diffusion suppresses the temperature fluctuation by causing the temperature of the rising plume of hot fluid to equilibrate with that of a surrounding fluid, decreasing the buoyancy. If the flow amplifying effect exceeds the dissipative effects of thermal diffusion and buoyancy, the convection starts. This competition of forces is parameterized by the Rayleigh number, which is proportional to the temperature difference appropriately normalized to take into account the geometry of the convection cell and the physical properties of the fluid. The rotation of the cells alternates from counter-clockwise to clock-wise as shown in figure 1.2. If the temperature difference is very large, then the fluid rises very quickly, and a turbulent

34 CHAPTER 1. INTRODUCTION 1 Cold Hot Figure 1.2: Cross-sectional schematic view of cell illustrating convection rolls. flow may occur. If the temperature difference is not far above the onset, the arising flow resembles overturning of cylinders referred to as convection rolls. Finlayson (197) first explained how an external uniform magnetic field applied to a ferrofluid with varying due to a temperature gradient magnetization results in a nonuniform magnetic body force, which leads to thermomagnetic convection. He discussed theoretically the effect of a magnetic field applied vertically to an initially quiescent horizontal ferrofluid layer heated from below. He showed that the magnetic mechanism dominates over the gravitational buoyancy mechanism in thin fluid layers that are about 1 mm thick. For fluid layers contained between two flat and free boundaries, the exact solution is obtained for some specific parameter values and it is shown that oscillatory instability cannot occur. Using Galerkin method approximate solutions for stationary instability are derived for two rigid boundaries (see also (Finlayson 197, pp. 758 and 759)). Under certain conditions his analysis predicted a strong coupling between the buoyancy and magnetic forces and showed that the applied magnetic field can be used to control magnetic convection, which is important in ferrofluid technology. Schwab et al. (1983) conducted an experiment in a ferrofluid layer to test the effects of a homogeneous magnetic field perpendicular to the layer in the case of magnetic Bénard convection. The critical temperature difference was determined by measuring the effective thermal conductivity. The theoretical predictions of the threshold of convection were found to agree with these observations.

35 CHAPTER 1. INTRODUCTION 11 Aniss et al. (1995) investigated Rayleigh-Bénard convection in magnetic fluid confined in a Hele-Shaw cell where a viscous fluid in a narrow slot was bounded by two thermally insulated parallel vertical and two perfectly heat conducting horizontal plates. The fluid was heated from below and cooled from above. The dimensional analysis was performed using the Hele-Shaw approximation and an asymptotic expansion. The authors presented the results of linear and weakly nonlinear analyses of stability near the threshold of convection. The Hele-Shaw approximation (ɛ 2, where ɛ is the aspect ratio of the cell) has been used to obtain a system of simplified partial differential equations. It has been observed that the advection term can be eliminated from the motion equation using the Hele-Shaw approximation so that the only nonlinear contribution comes from the advection term of the energy equation. It has also been found that the linear stability analysis provides the same critical parameters as those reported in Frick & Clever (198). Siddheshwar (1995) studied a linear stability analysis of ferrofluid in Rayleigh-Bénard configuration in a transverse uniform magnetic field between penetrable magnetic boundaries. He used Galerkin method to determine the critical eigenvalue for rigid-rigid and free-free boundaries. The eigenvalue for stationary convection was obtained. Oscillatory convection was ruled out and the principle of exchange of stabilities was found to be valid. The author confirmed the qualitative outcomes of the previous investigation (Siddheshwar 1993) that were the limiting case of his study. Qin & Chadam (1995) performed a nonlinear stability analysis in the case of thermal convection of a ferromagnetic fluid in a porous medium heated from below. They discussed the inertia effect to accommodate the high velocity flow by using the Darcy-Forchheimer model (Nield & Joseph 1985) that includes a quadratic inertia term. To adapt the energy stability analysis to this case, special mathematical considerations were necessary to calculate the additional nonlinear term representing the magnetic body force. It was found that the incorporation of the inertia effect is not only physically significant but also mathematically important.

36 CHAPTER 1. INTRODUCTION 12 Russell et al. (1995) investigated heat transfer in a horizontal layer of strongly magnetized ferromagnetic fluid heated strongly from above under the influence of a uniform vertical applied magnetic field. The authors noticed that convective patterns with a large wavenumber exist at critical conditions and derived simplified equations for the temperature field in a ferromagnetic fluid. It was found that heat transfer depends non-linearly on the temperature difference. Recktenwald & Lücke (1998) studied the influence of boundaries with finite heat conductivity on thermal convection in a Rayleigh-Bénard setup in magnetized ferrofluids. A linear stability analysis of the conductive state was performed using a shooting method. It was shown that the critical wavenumber is strongly affected by the conductivity of the boundaries. The coefficients of Ginzburg-Landau amplitude equation for convection slightly above the threshold were evaluated as functions of fluids Prandtl number, the magnetic Rayleigh number and the boundary conductivities. Pattern formation due to double-diffusive convection in a horizontal layer of a ferromagnetic fluid heated from below and placed in an externally applied normal magnetic field was investigated by Bajaj & Malik (1998). The authors discussed the stability of steady state patterns such as rolls, squares and hexagons in this paper. It was found that rolls are structurally stable on the square lattice, whereas on the hexagon lattice bifurcation can lead to either hexagons or rolls depending on the value of the magnetization parameter. It was also found that the amplitudes of patterns of rolls, squares and hexagons decrease with the increase in the value of the magnetization parameter. Sekar et al. (2) analyzed the effect of ferro thermohaline convection in a rotating medium salted from above and heated from below. The salinity influences the density and magnetization of a ferromagnetic fluid. The authors obtained the conditions for both oscillatory and stationary modes using linear stability analysis and found that the stationary mode is favored over the oscillatory one. It was found that in the case of ordinary fluids rotation tends to prefer oscillatory mode of destabilisation, where as in this case

37 CHAPTER 1. INTRODUCTION 13 of magnetic fluids the system continues to get destabilised through stationary mode. It was also found that for positive salinity Rayleigh number in a larger (which is salted from above and heated from below), the critical thermal Rayleigh number decreases as salinity Rayleigh number increases. Auernhammer & Brand (2) showed that a thermal convection in a horizontal layer of a magnetic fluid in a vertically applied magnetic field can be driven by buoyancy or by magnetic forces (due to the thermomagnetic effect). Depending on the direction of the applied temperature gradient, buoyancy effects can be stabilizing (heating from above) or destabilizing (heating from below), whereas the magnetic forces always play a destabilizing role in magnetic fields perpendicular to the interface. They discussed the effect of rotations in a ferromagnetic fluid using both linear and weakly nonlinear analyses of the governing hydrodynamic equations using Boussinesq approximation. It was found that magnetically dominated convection does not distinguish between the systems heated from above and below. Abraham (22) analyzed the RBC problem in a micropolar ferromagnetic fluid layer in a vertical uniform magnetic field. Physically, micropolar fluid represents a fluid consisting of rigid randomly oriented particles suspended in a viscous medium, where the deformation of fluid particles is ignored (Eringen 1966). The influence of the various micropolar and magnetization parameters on the threshold of stationary convection was discussed. It was found that micropolar fluid heated from below is more stable than ordinary fluids. The nature of the magnetization effects on convection in a micropolar ferromagnetic fluid was found to be similar to that in Newtonian ferromagnetic fluids. Kaloni & Lou (24b) developed linear and weakly nonlinear analyses of thermal instability in a horizontal layer of a ferromagnetic fluid heated from below rotating around a vertical axis in a vertical non-uniform magnetic field. By the use of multi-scale perturbation method, the amplitude equation was derived. It was then concluded that magnetic field the accelerated convection onset and the ratio of the heat transfer by convection to

38 CHAPTER 1. INTRODUCTION 14 that by conduction (Nusselt number) decreased as magnetic field increased. Ramanathan & Muchikel (26) analyzed the effect of temperature-dependent viscosity on the ferroconvective instability in a porous medium. It was observed that the temperaturedependent viscosity has a destabilizing effect on the onset of convection. It was also found that the stationary mode of instability is preferred to the oscillatory one. Kaloni & Lou (24a) analyzed the convective instability problem in a thin horizontal layer of a magnetic fluid heated from below in the presence of a uniform vertically applied magnetic field to the plates. The authors considered the effects of the relaxation time (Brownian relaxation mechanism) and the rotational viscosity. Chebyshev pseudospectral method was used to solve the arising eigenvalue problem. Various types of results under gravity-free conditions were presented and the critical temperature gradient was determined for a variety of situations. It was found that the effect of magnetic field accelerates the convection and reduces the ratio of heat transfer by convection to that by conduction. Barletta & Nield (29) analysed the classical Rayleigh-Bénard problem in an infinitely wide horizontal fluid layer with isothermal boundaries heated from below. The effects of the pressure work and viscous dissipation were taken into account in the energy balance. A linear analysis was performed in order to obtain the conditions of marginal stability and the critical values of the wavenumber and of Rayleigh number for the onset of convective rolls. Mechanical boundary conditions were considered so that the boundaries were both rigid, stress-free, or the upper stress-free and the lower rigid. It was shown that the critical value of Rayleigh number could be significantly affected by the contribution of pressure work, mainly through the functional dependence on the Gebhart number (that characterises the importance of a pressure work) and thermodynamic Rayleigh number. However, while the pressure work term affects the evaluation of the critical wavenumber and the critical Rayleigh number for the threshold of linear instabilities, the viscous dissipation term plays no role in this analysis as it represents a higher order effect. Lalas & Carmi (1971) performed a nonlinear analysis of thermoconvective stability of

39 CHAPTER 1. INTRODUCTION 15 ferrofluids using the energy method. The authors considered ferrofluid layer between two infinite plates heated from below in the presence of gravity and a constant magnetic field. They adopted an assumption that the magnetization is independent of the magnetic field intensity and showed that the linear stability theory and the energy analysis lead to the same results for steady ferromagnetic flow so that no subcritical instabilities occur. It was also found that a typical ferrofluid flow could be unstable only when the magnetic force is in the same direction as gravity. Gupta & Gupta (1979) discussed thermal instability conditions in a horizontal layer of a ferrofluid heated from below under the action of Coriolis force in a vertically applied uniform magnetic field. For oscillatory convection, Rayleigh number was plotted as a function of the wavenumber for several values of the magnetization parameter. Critical Rayleigh number was found to decrease with the increase of the magnetization parameter. It was also noticed that oscillatory convection cannot occur if the Prandtl number is greater than unity (see Gupta & Gupta (1979, pp. 276)). Gotoh & Yamada (1982) presented the linear instability conditions for convection in a horizontal ferrofluid layer between two ferromagnetic boundaries heated from below in the presence of a vertical magnetic field. They used Galerkin method for solving the disturbance equations and took Legendre polynomials as the trial functions. They solved numerically the resultant eigenvalue problem and obtained that the critical Rayleigh number varies inversely with the magnetic number. They found that both the magnetization of the boundaries and the nonlinearity of fluid magnetization reduce the critical Rayleigh number and that the effects of magnetic force and buoyancy compensate each other. The thermoconvective instability in a horizontal layer of ferrofluid in a strong magnetic field applied vertically has been studied by Stiles & Kagan (199). They observed a satisfactory agreement between their experiments and the theoretical model developed by Finlayson (197). Rudraiah & Sekhar (1991) examined the effect of a uniform distribution of heat source

40 CHAPTER 1. INTRODUCTION 16 on the onset of stationary convection in a horizontal magnetic fluid layer restrained by isothermal non-magnetic boundaries with internal heat production in the presence of a transverse applied magnetic field. The solutions were obtained using Galerkin method. The authors considered different isothermal boundary combinations (rigid-rigid, rigidfree, and free-free). It was found that when the internal magnetic number increases, the internal Rayleigh number decreases in analogy with the effect of the external magnetic number. It was also shown that the rigid-rigid boundary combination is the most stable configuration compared to rigid-free and free-free boundary conditions. Vaidyanathan et al. (1991) discussed the magnetoconvective instability in a ferromagnetic fluid using Brinkman model for saturated porous medium of very large permeability in a vertical magnetic field and showed that only stationary convection can occur. A linear convective instability analysis was performed accounting the critical temperature gradient when only the magnetic mechanism was important and when both buoyancy and magnetic mechanisms were present. The weakly nonlinear and linear thermoconvective stability of ferromagnetic fluid enclosed by two horizontal rigid plates maintained at different temperatures in a strong uniform external vertical magneto-static field was examined by Blennerhassett et al. (1991). They noticed that when magnetically saturated ferrofluid is heated from below the critical parameters for the linear stability problem agree with those of Finlayson. If the ferrofluid is heated from above, critical temperature gradients can attain extremely high values and the horizontal separation of roll cells can be less than half of the spacing observed when the lower plate is warmer. When ferrofluid is heated from above, in the non-standard regime, where convection arises due to magnetic forces, the Nusselt numbers for a given supercritical temperature gradient are significantly higher than when the ferrofluid is heated from below. Subsequently, Blennerhassett et al. (1991) & Stiles et al. (1992) investigated a weakly nonlinear and linear thermoconvective stability in weakly magnetized ferromagnetic fluids. When a ferrofluid was heated from above, they observed that the magnitudes of the horizontal critical wavenumbers were noticeably larger

41 CHAPTER 1. INTRODUCTION 17 than those for a ferrofluid heated from below. Bajaj & Malik (1997) performed theoretically linear and nonlinear analyses of a convective instability and in a layer of a magnetic fluid subject to a vertical temperature gradient in the gravitational field. The authors considered an infinite horizontal layer of an incompressible and nonconducting magnetic fluid of finite depth and a uniform magnetic field that is vertically applied across the fluid layer. They determined the stability of steady convection patterns using a bifurcation theory. It was found that the supercritical rolls are stable on both square and hexagonal lattices. It was also found that the critical wavenumber decreases with the increase in Rayleigh number. Huang et al. (1997) performed the thermoconvective instability analysis of a laterally unbounded nonconducting horizontal layer of paramagnetic fluid heated from below subject to a uniform oblique magnetic field by using a linear stability analysis of the Navier- Stokes equations complemented with Maxwell s equations and accounting for the magnetic body force. The two-dimensional convective rolls with the axes parallel to the horizontal component of the magnetic field were shown to lead to the onset of convective instability. In the presence of magnetic field the corresponding critical Rayleigh number and critical wavenumber for the threshold of such rolls are greater than the well-known critical Rayleigh number ( ) and critical wavenumber (3.117), respectively (Chandrasekhar 1981). It was also found that only a stationary instability could occur in this geometry. Huang et al. (1998) studied the thermoconvective instability of unbounded nonconducting paramagnetic fluid layer heated from above or below with the effect of a static and lateral nonuniform magnetic field by using a linear stability analysis of the Navier-Stokes equations complemented by Maxwell s equations. The authors confirmed that a Buoyancydriven convection can be controlled by subjecting the layer to a nonuniform magnetic field. They showed that in the presence of nonuniform magnetic field the effect on convection due to the field gradient depends on the sign of the magnetic Rayleigh number

42 CHAPTER 1. INTRODUCTION 18 Ra m. It was also found that convection is promoted at negative Ra m and is suppressed at positive Ra m. A convective instability in a horizontal magnetic fluid layer placed in a uniform vertical magnetic field with a focused laser beam parallel to the field passing through the layer was discussed by Luo et al. (1999). The authors found that the critical field for the onset of instability in this configuration increases with the increasing thickness of the layer. It was also observed that a new bifurcation caused by the interaction between horizontal field and the magnetic moment of the fluid particles when a horizontal magnetic field was applied in addition to the vertical field. The obtained results suggest that the mass and heat transfers in colloidal fluid could be controlled and monitored by an external magnetic field. Russell et al. (1999) analyzed the structure of two-dimensional vortices in a thin layer of a magnetized ferromagnetic fluid heated from above in the limit of large critical wavenumbers. They presented a nonlinear asymptotic description of the vortex pattern that occurs directly above the critical point in the parametric space where instability first sets in. Sunil et al. (24) examined the thermosolutal convection in a layer of ferromagnetic fluid soluted and heated from below in a uniform magnetic field applied vertically. An exact solution is obtained using a linear stability analysis for the case of two free boundaries. It was found that the fluid magnetization has a destabilizing effect for the case of stationary convection, whereas stable solute gradient postpones instability. Owing to the presence of the stable solutal gradient the oscillatory modes appear that are inexistent in its absence. A sufficient condition for the non-existence of the oscillatory convection was also determined. Shliomis & Smorodin (22) theoretically studied the convective instability in a flat ferromagnetic fluid layer under a uniform magnetic field applied transversely. A temperature gradient applied across the layer induces a concentration gradient of magnetic particles due to Soret effect. Both gradients result in a spatial variation of magnetization that in-

43 CHAPTER 1. INTRODUCTION 19 troduces a gradient of magnetic field intensity within the layer of fluid. The field gradient induces a redistribution of magnetic particles due to magneto-phoresis. The resulting magnetic force attempts to mix the fluid. The linear analysis discussed for the case of realistic boundary conditions at restraining horizontal plates predicts oscillatory instability. It was also concluded that the stationary instability only occurs if particle thermodiffusion is weak. Vaidyanathan et al. (25) made an attempt to obtain the condition for the onset of thermoconvective instability in a horizontal layer of a ferromagnetic fluid heated from below and salted from above due to Soret effect in the presence of a normal magnetic field. The authors used linear analysis and investigated both oscillatory and stationary instabilities. It has been found that the system is destabilized for various physical parameters through stationary instability. The salinity and thermal magnetic gradients compensate each other. It has also been found that the system can be in convective mode even when magnetic and thermal gradients are small. Martinez-Mardones et al. (28) obtained theoretical and numerical results for a binary magnetic mixture with rotation. They focussed on the effect of magneto-phoresis and Kelvin forces. They also discussed the stabilizing effect of rotation on fluid suspensions. Sekar et al. (26) investigated Soret effect in multi-component ferromagnetic fluid saturating a porous medium with large variation in penetrability. It was noticed that stationary instability is preferred regardless of the values of penetrability of pores. Sprenger et al. (213a) analysed the thermomagnetic convection in a horizontal layer of ferrofluid driven by a temperature difference and a magnetic field. Its onset is characterized by Rayleigh number. In binary fluids the temperature gradient also drives thermal diffusion caused by Soret effect. The interaction of two transport phenomena was investigated with the focus on the point of transition from enhanced to hindered or even suppressed convection. The zero-magnetic-field Soret coefficient (S T ) in ferromagnetic fluids is positive but the measurements of the magnetic Soret coefficient in applied mag-

44 CHAPTER 1. INTRODUCTION 2 netic field showed that its sign depends on the strength of the magnetic field. It was found that the convective motion enhanced by S T > K 1, hindered in the interval.11 K 1 < S T < K 1 and even suppressed at S T <.11 K 1. Sprenger et al. (213b) investigated the thermomagnetic convection in a horizontal layer of ferromagnetic fluids with thermodiffusion. The orientation of applied magnetic field was either parallel or perpendicular to the vertical temperature gradient. It was found that depending on a magnetic field Soret coefficient can reach high values and change sign. The corresponding change in the direction of motion of nano-particles strongly affects the threshold of thermomagnetic convection. The linear stability analysis exposes that thermodiffusion with a positive sign of Soret coefficient promotes the onset of convection while negative coefficient suppresses it. Sunil et al. (27) theoretically investigated double-diffusive convection in a layer of micro-polar ferromagnetic fluid heated and soluted from below in a uniform vertical magnetic field. The authors used linear stability theory and normal mode analysis to study the onset of convection. They found that the micro-polar heat conduction and micro-polar viscous effect always have a stabilizing effect. The principle of exchange of stabilities was shown to hold for the micro-polar ferromagnetic fluid heated from below in the absence of a micro-polar viscous effect, micro-inertia and a solutal gradient. However, the oscillatory modes appear in presence of these effects. Shivakumara et al. (29) studied the thermomagnetic convection threshold in a ferromagnetic fluid saturating a horizontal porous layer in a uniform vertical magnetic field for variety of temperature and velocity boundary conditions. To model a flow in porous medium Brinkman-Lapwood extended Darcy equation was used. The lower boundary of the porous layer was assumed to be rigid-ferromagnetic and the upper boundary was either rigid-ferromagnetic or stress-free. The thermal conditions included fixed heat flux at the lower boundary and a convective-radiative exchange at the upper boundary subjected to fixed temperature or heat flux conditions. By using Galerkin method and a regular

45 CHAPTER 1. INTRODUCTION 21 perturbation expansion when both boundaries were insulated the eigenvalue problem was derived and solved. It was found that the increase in the value of the magnetic number and the measure of nonlinearity of magnetization hasten, whereas the increase in Biot number (that characterises a heat transfer through the boundary), the viscosity ratio and the decrease in the magnetic parameter and Darcy number (that represents the effect of the permeability of the medium relative to its cross-sectional area) postponed magnetoconvection. Suresh & Vasanthakumari (29) analysed the effect of magnetic field-dependent viscosity on ferroconvection in an anisotropic porous medium using Darcy model. A linear stability analysis was performed for both oscillatory and stationary modes using Galerkin method. The critical magnetic Rayleigh number was calculated for various values of the parameters which characterize the flow. It was noticed that the increase in magnetoviscosity stabilizes the system with respect to a stationary mode. Zablotsky et al. (29) investigated thermomagnetic convection in ferromagnetic fluids focussing on the study of efficiency of heat transfer. The main goal of this study was to investigate numerically thermomagnetic convection of a special temperature-sensitive ferromagnetic fluid in a rectangular cell with permanent magnets attached to its walls. The volume of ferrofluid was cooled from the top and heated from the bottom. A strong non-uniform magnetic field was applied to drive the thermomagnetic convection. It was observed that the intensification of heat transfer was significantly higher than that in the case of simple thermogravitational convection when the cell was heated from below. It was also found that the efficiency of the thermomagnetic convection depends on the placement of magnets and maximum thermomagnetic intensification was achieved when magnets were placed as close to the warm end of the cell as possible. Banerjee et al. (211) studied a steady state thermomagnetic convection in a square cavity with localized heat sources. The external magnetic field was described by Maxwell s equations. The effect of magnetization saturation of ferrofluid on heat transfer augmenta-

46 CHAPTER 1. INTRODUCTION 22 tion was studied using Langevin s law. Thermal interactions between the heaters and the fluid at convection-dominated regions were visualized through heatline, streamline and isotherm plots. It was found that the fluid magnetization contours follow the layout of the Kelvin force field. Nanjundappa et al. (211) investigated the penetrative convection in a horizontal porous layer saturated with ferromagnetic fluid in a uniform vertical magnetic field via the internal heating model using Brinkman-extended Darcy equation. The authors considered the rigid-isothermal boundaries of the porous layer to be either paramagnetic or ferromagnetic. They solved the eigenvalue problem numerically using Galerkin method with either magnetic or thermal Rayleigh number as the eigenvalue. They found that the internal heating in a porous layer affects the stability of the system significantly. The buoyancy and magnetic forces are complementary to each other. It was also found that the paramagnetic boundaries with large magnetic susceptibility postpone the onset of ferroconvection compared to the case of a very low magnetic susceptibility and ferromagnetic boundaries. Increasing the value of magnetic Rayleigh number, heat source strength and non-linearity of magnetization hastens the threshold of ferroconvection. The stability of the system heated from above in the absence of thermal buoyancy was also discussed and it was found that the system with increasing internal heat source strength is more stable when the ferrofluid saturated a porous layer. Thermal convection in a horizontal layer of ferromagnetic fluid in a vertical magnetic field and an imposed shear flow caused by an external horizontal pressure gradient was analyzed by Bajaj (211). The author used Chebyshev tau method to find the marginal stability of the flow. The instability patterns in the form of rolls inclined to the shear direction were considered. The threshold of rolls aligned with the shear flow was found unaffected by shear of the base flow. However, the external pressure gradient characterized by Reynolds number affects the threshold of rolls inclined to the direction of a shear flow. The effects of the pressure gradient on the temperature and velocity fields and the magnetic potential at the marginal state was also studied. It was shown that the

47 CHAPTER 1. INTRODUCTION 23 critical value of Rayleigh number for the onset of instability increases with the increase of the Reynolds number, whereas the critical wavenumber at the onset of instability decreases with the increase of Reynolds number. It was also found that the vertical velocity disturbance increases with the increase of the Reynolds number most prominently at the mid-plane Marangoni Convection in Ferromagnetic Fluids Marangoni convection occurs when the surface tension of an interface of fluid depends on the concentration of species or on the temperature. Schwab (199) experimentally examined the stability of flat layers of ferromagnetic fluids when a vertical temperature gradient and a vertical magnetic field were applied. He showed that magnetostatic stresses fortify the surface deformation of Marangoni convection but they work against the surface deformation of Rayleigh-Bénard convection. A nonlinear stability analysis based on the energy method has been developed by Qin & Kaloni (1994) to discuss the impact of buoyancy and surface tension on the motion in a ferrofluid layer heated from below. They assumed the free surface to be non-deformable and flat and pointed out the possibility of subcritical instabilities. The influences of rotation on the thermoconvective instability in horizontal layer of ferromagnetic fluid heated from below under a vertical uniform magnetic field has been analyzed by Venkatasubramanian & Kaloni (1994). A linear stability analysis in the layer of a magnetic fluid with deformable free surface heated from below and placed in a vertical uniform magnetic field has been discussed by Weilepp & Brand (1996). The buoyancy, the temperature dependent surface tension and the focusing of the magnetic field work as destabilizing factors due to surface fluctuations. They presented the boundaries of non-oscillatory instability in the parametric space for this problem.

48 CHAPTER 1. INTRODUCTION 24 The effect of various basic temperature gradients on the thresholds of pure magnetic and combined Rayleigh-Bénard and Marangoni convection flows in an infinite horizontal layer of a ferromagnetic fluid in a vertical uniform magnetic field was investigated by Shivakumara et al. (22). By using Galerkin method the resulting eigenvalue problem was solved for various basic temperature gradients. The authors discussed the mechanisms of suppressing and enhancing the magnetoconvection. It was found that the stability of Rayleigh-Bénard-Marangoni magnetoconvection was significantly affected by basic temperature gradients. It was also found that the surface tension effect was negligible while the buoyancy was predominant force driving convection. Shivakumara et al. (21) studied the threshold of Brinkman-Bénard-Marangoni (BBM) convection in a horizontal layer of a very coarse porous medium saturated with an initially quiescent magnetized ferromagnetic fluid in a uniform vertical magnetic field. The upper boundary was free and open to the ambient and was subjected to fixed temperature or fixed heat flux conditions, while the lower rigid boundary was subjected to a fixed heat flux. The resulting eigenvalue problem was solved using Galerkin method supplemented by perturbation technique when both boundaries were insulated. The surface tension, buoyancy and magnetic forces were found to reinforce each other in hastening the threshold of BBM convection. It was noticed that the increase in Biot number and the porosity parameter and decrease in the magnetic Rayleigh number as well as nonlinearity of fluid magnetization had a stabilizing effect on the threshold of BBM convection. The authors also discussed Brinkman-Bénard (BB) and Brinkman-Marangoni (BM) convection separately and made a comparison between them in their paper. It was found that the increase in the magnetic Rayleigh number leads to the decrease in the critical wavenumber in the case of BM convection but the opposite trend was observed in the case of BB convection Modulated Convection in Ferromagnetic Fluids Instabilities in a fluid layer can be controlled by a modulation of external conditions. A few analyses focused on Rayleigh-Bénard and thermomagnetic convection with modu-

49 CHAPTER 1. INTRODUCTION 25 lations are discussed below. Such modulations have been induced either by (i) imposing spatial variation of a magnetic field, or (ii) using time-periodic temperature boundary conditions, or (iii) changing the geometry of a flow domain, or (iv) applying a time dependent magnetic field. Aniss et al. (21) investigated theoretically the effect of a sinusoidal temporal variation of the applied magnetic field on the threshold of convection in a horizontal layer of a magnetic fluid heated from above and bounded by two isothermal non magnetic plates. Floquet theory was used to determine the convective threshold for both free-free and rigid-rigid boundaries. The authors discussed the possibility of generating a competition between the harmonic and sub-harmonic modes at the threshold of convection with an appropriate choice of the ratio (M 1 ) of the magnetic and gravitational forces. In this situation, the magnetic Rayleigh number was modulated periodically in time relative to its mean. It was found that the stable equilibrium configuration corresponding to M 1 1 can be destabilized by a sub-harmonic mode by the magnetic modulation for large frequencies. Siddheshwar & Abraham (23) analyzed theoretically the thermal instability in a horizontal layer of a ferromagnetic fluid confined between two infinite boundaries in a vertical uniform magnetic field subjected to time-periodic variations of boundary temperatures. The authors obtained a perturbation solution in powers of the amplitude of the applied temperature field and considered small disturbances only. The results of this study indicated that the time-periodic boundary temperature variation leads to subcritical motion. It was found that the fluid flow was stabilized for small frequency and destabilized for large frequency. It was also found that low Prandtl number fluids were more vulnerable to destabilization compared to large Prandtl number fluids. The thermosolutal convection in an infinite layer of magnetic fluid heated from below in a vertically applied magnetic field and subject to vertical two-frequency vibrations has been examined by Bajaj (25). Using Floquet theory he determined the regions of instability

50 CHAPTER 1. INTRODUCTION 26 for various values of parameters and found bicritical point, where both harmonic and subharmonic instabilities appear at the same critical value of the modulation amplitude but at different wavenumbers. It was found that the effect of modulation on stability decreases at small wavenumbers. It was also noticed that the decreasing of the particle concentration in magnetic fluids affects the instability behavior promoting its onset. Matura & Lücke (29) investigated thermomagnetic convection in a ferromagnetic fluid layer in a time-periodic magnetic field. They analyzed a linear stability and a nonlinear response of a ferromagnetic fluid layer heated from above and from below. A competition between the stabilizing thermal and viscous dissipation and destabilizing buoyancy and Kelvin forces was discussed. Floquet theory was used to investigate stability boundaries of the motionless conductive state with respect to a harmonic and subharmonic excitation. By using a finite difference method full numerical simulations were performed to obtain nonlinear convective states. The effect of low and high-frequency modulation on the nonlinear oscillations and on the stability boundaries was discussed. Bajaj (21) numerically investigated the thermomagnetic instability in a horizontal layer of ferromagnetic fluid heated from below with the boundaries periodically moving with zero and nonzero average velocity in a vertical magnetic field. Two magnetically penetrable rigid boundaries were maintained at fixed temperatures. Chebyshev tau method and Floquet theory were used to analyze the stability of such flow. The oscillatory shear flow was found to affect the threshold of instability patterns in the form of rolls in the direction transverse to the shear flow. It was observed that the oscillatory solutions were synchronous to the modulation. The amplitude and the oscillation of the rigid boundaries influenced significantly the velocity and temperature fields and the magnetic potential. Lange & Odenbach (211) analyzed thermomagnetic convection in a thin horizontal layer of magnetic fluid confined between two horizontal impenetrable boundaries in a spatially symmetrically modulated magnetic field. The temperature gradient and the magnetic field were oriented vertically. It was found that for any nonzero magnetic field the base state

51 CHAPTER 1. INTRODUCTION 27 was a convective one in contrast to the classical purely thermally driven Rayleigh-Bénard system. The nonzero flow field of the base state was formed by a double vortex which reflects the symmetrical modulation of magnetic field. The linear stability analysis of long wave modulations exposed that the critical Rayleigh number for the stability of the base state increased with the magnitude of a driving magnetic force Thermomagnetic Convection in Ferromagnetic Fluids in Various Configurations Shliomis (1973) clarified the conditions under which instability arises in a ferrofluid in a cavity heated non-uniformly from below in a gravitational field and a non-uniform vertical magnetic field. The author introduced a dimensionless criterion that characterizes the stability of the equilibrium of ferrofluid. Several researchers investigated various aspects of thermomagnetic convection in a vertical layer of magnetic fluid. Berkovsky et al. (1976) studied experimentally and numerically a convective heat transfer across a vertical layer of a ferrofluid between two differentially heated finite parallel plates placed in a horizontal uniform magnetic field in the presence of gravity. The authors used a finite-difference method to solve numerically the boundary-value problem consisting of the dimensionless energy, vorticity and stream function equations with the corresponding boundary conditions. The effect of the direction of the gradient of magnetic field strength relative to that of the temperature gradient on convective motion was discussed. A critical relationship between the aspect ratio of the layer, gravitational and magnetic Rayleigh numbers and heat transfer has been obtained. It was found that heat transfer is more sensitive to the magnetic field variation than to the gravitational buoyancy. However, when the magnetic Rayleigh number is small compared to the gravitational one heat transfer is mainly attributed to gravitational convection. When the magnetic Rayleigh number is almost equal to the gravitational one heat transfer increases if the direction of the gradient of magnetic field coincides with that

52 CHAPTER 1. INTRODUCTION 28 of the temperature gradient. However, heat transfer decreases when the magnetic field and temperature gradients are in opposite direction. Suslov (28) investigated comprehensively a linear stability of convection flow in a layer of ferromagnetic fluid between two vertical differentially heated plates placed in a uniform external normal magnetic field. The author presented complete stability diagrams for two- and three-dimensional disturbances. It was found that two distinct mechanisms, thermogravitational and magnetic, are responsible for the appearance of three instability modes. The physical nature of all three modes was investigated in detail and the most prominent features were identified. The instability patterns were shown to depend on the governing parameters and to consist of vertical stationary magnetoconvection rolls and/or vertically or obliquely counterpropagating thermogravitational or thermomagnetic waves. It was also found that the growth rate of the stationary magnetoconvective instability is larger than that for the thermogravitational or thermomagnetic waves in a substantial part of a parametric space Blums (1987) investigated numerically and experimentally the convection in a vertical layer of magnetic fluid in the presence of a non-uniform magnetic field. If the magnetic field is not uniform magneto-convection arises in magnetic fluid due to the magnetization dependence on magnetic field. Non-uniform magnetization can also arise due to magneto-diffusion of colloidal particles. Such a non-uniformity can become large enough to produce magneto-convection even in an isothermal magnetic fluid. The distribution of the magnetic particles in nonsteady magneto-diffusive convection was studied. The experimental isoconcentration lines were found to be in good agreement with the numerical results. It was also found that the mean velocity of convection in the boundary layer determined numerically agreed well with that in experiment. Nakatsuka et al. (199) investigated experimentally a thermomagnetic convection that arises when a temperature-sensitive magnetic fluid is heated in a glass beaker in a nonuniform magnetic field. Convection arising along the gradient of a magnetic field en-

53 CHAPTER 1. INTRODUCTION 29 hances the heat transfer from a solid surface into the fluid. The effects of a magnetic field on the heat transfer from a solid surface into a magnetic fluid was also examined by another model experiment involving Magnetic Fluid Heat Siphon. It was found that the results of both experiments were in good agreement. The onset of thermomagnetic convection in a magnetic fluid enclosed between two concentric cylinders heated from inside, cooled from outside and placed in an azimuthal magnetic field with a radial variation has been investigated experimentally by Odenbach (1995). The temperature difference between the inner and outer cylinders gives rise to the magnetization gradient in a fluid. It was found that the gradients of magnetic field strength and magnetization are antiparallel. The periodic temperature distribution along the azimuthal lines around the cylinder produced by hot fluid streaming toward the outer cylinder and cold fluid streaming to the inner one provided the information about the size and the number of vortices. Zebib (1996) conducted a theoretical investigation of the behavior and stability of thermomagnetic flow in micro-gravity conditions. In his study convection was driven in a ferrofluid contained in a cylindrical shell subject to the imposed radial magnetic and temperature gradients. It was shown that the convection starts as a result of a supercritical bifurcation. Siddheshwar & Abraham (1998) studied the convection in ferromagnetic fluids occupying a rectangular vertical cavity subject to a uniform heat flux along the vertical walls. They obtained a closed form solution based on the Oseen linearization technique and the similarity analysis. Magnetisation and buoyancy arising due to different heat fluxes at the walls give rise to convection. It was noticed that Nusselt number increases due to the magnetization effect. The effect of a non-uniform concentration distribution on double diffusive convection in Boussinesq magnetic fluid layer contained between two vertical rigid boundaries has been analyzed using Galerkin method by Rudraiah et al. (1998). They established the

54 CHAPTER 1. INTRODUCTION 3 conditions for the occurrence of stationary and oscillatory modes for various nonlinear basic concentration distributions. It was found that different non-uniform concentration gradients and diffusivity ratios significantly influence the stability of the system when convection was induced by buoyancy force and/or magnetization. Morimoto et al. (1999) investigated the pattern formation in thermomagnetic convection in micro-gravity experiments and performed linear and nonlinear numerical analyses. Thermosensitive magnetic fluid was confined between two concentric cylinders heated from inside and cooled from outside so that a temperature gradient was established in the radial direction. An applied magnetic field with radial variation was created by a solenoid coil. The authors investigated the influence of the aspect ratio of the layer of a magnetic fluid on the pattern formation. They obtained the critical magnetic Rayleigh and wavenumber by solving the linearized eigenvalue problem using a finite-difference method. It was found that the critical magnetic Rayleigh number decreases when the aspect ratio increases. The results of linear stability analysis agree closely with the experimental micro-gravity results. The authors also solved the nonlinear equations by using the control volume method and obtained the flow patterns similar to those observed in experiment. By using the nonlinear analysis they showed that the values of the critical magnetic Rayleigh number agree with those obtained using a linear stability analysis. Therefore, the bifurcation occurring in the considered system is supercritical. Tangthieng et al. (1999) investigated heat transfer augmentation in ferromagnetic fluids between two vertical parallel plates subject to steady spatially non-uniform magnetic fields. They used the finite element method to solve the extended Navier-Stokes equations and reported the enhancement of heat transfer in ferrofluids in magnetic fields. The thermomagnetic convection of magnetic fluids in a cylindrical geometry in a uniform vertical magnetic field has been studied by Lange (22). The model system of co-axial cylinders with inner heating was considered. Using the Kelvin force density a general condition was derived for the existence of a potentially unstable stratification in the mag-

55 CHAPTER 1. INTRODUCTION 31 netic fluid. It was found that the requirement for a potentially unstable stratification was met only in a narrow gap. The author performed a linear stability analysis in order to determine the critical external magnetic induction for the onset of thermomagnetic convection in a magnetic fluid. It was observed that decreasing the temperature difference causes a dramatic increase in the critical induction. Shliomis et al. (23) investigated the onset of a thermomagnetic convection in stratified ferromagnetic fluids. A non-uniform magnetic field was found to promote a nonhomogeneous distribution of magnetic particles in a fluid. It was shown that the concentration equilibrium settlement rate is very slow due to the smallness of the particle diffusion coefficient. If the equilibrium does not have enough time to establish, a ferromagnetic fluid acts as a monocomponent fluid so that stationary convection occurs. However, an oscillatory instability sets when a non-uniform concentration profile is formed. Ganguly et al. (24) simulated the thermomagnetic convection in a layer of magnetic fluid in a differentially heated square cavity that is infinitely long in the third dimension. A two-dimensional magnetic field produced by a line source dipole and described by Maxwell s equations was considered. The lower and upper walls were adiabatic and a magnetic line dipole was placed adjacent to the lower wall halfway along the channel length and at one-fourth of the channel height below its inner surface. The authors neglected gravity in order to focus on the thermomagnetic effects. Fluid motion occurred due to the gradients of both the magnetic field and the temperature. It was observed that the colder fluid with larger magnetic susceptibility was driven toward the regions of stronger magnetic field causing thermomagnetic convection that displaced warmer fluid with a lower magnetic susceptibility. The height-averaged Nusselt number was shown to increase with the increasing magnetic dipole strength and the temperature difference, but decrease with the increasing length scale and fluid viscosity. Bozhko & Tynjälä (25) examined numerically and experimentally the effect of gravitational sedimentation of magnetic particles on ferrofluid convection in a horizontal layer

56 CHAPTER 1. INTRODUCTION 32 heated from below and subject to a uniform external magnetic field normal to the layer. A two-phase mixture model was used in the simulations to take into account the density gradient due to sedimentation. Temperature difference across the layer and concentration of magnetic phase were varied in order to control the driving magnetic forces and damping sedimentation effects. In the absence of a magnetic field, the oscillatory convection was observed over the entire investigated range of temperature differences. In contrast, in presence of magnetic field it was found that the thermal gradient renders an internal magnetic field gradient working as a driving force for convection through the temperature-dependence of magnetic susceptibility. Hennenberg et al. (26) considered convection in a ferrofluid layer bounded by two horizontal rigid plates subject to a lateral temperature gradient taking into account an externally imposed strong oblique magnetic field. This problem combining the heat balance and momentum equations with Maxwell equations contains two Rayleigh numbers (gravitational and magnetic) characterising the buoyancy and Kelvin forces that induce motion opposed by the viscous dissipation and heat diffusion. The steady solution of the problem is independent of the inclination of the magnetic field and is presented as a power series in a small parameter ɛ H measuring the ratio of the magnetization variation across the layer to the magnitude of the external field. It was found that when the strength of the magnetization relative to the applied magnetic field is small i.e. ɛ H <.1 then the ratio of the maximum velocity to Birikh maximum velocity (which is the velocity obtained in same system without magnetic field) is constant for any inclination angle of the applied magnetic field. It was also found that when the ratio of the magnetic Rayleigh number to the gravitational Rayleigh number is large, the roles of buoyancy and magnetic field were reversed. Lian et al. (29) developed a mathematical model for describing flow and heat transport features of the temperature-sensitive magnetic fluid and for designing an automatic energy transport device based on the thermomagnetic effect. This model included the coupling of the three fundamental phenomena (magnetic, thermal and fluid dynamic features) for

57 CHAPTER 1. INTRODUCTION 33 describing thermomagnetic convection of magnetic fluid in a loop. It was found that a stable circulation flow could be maintained in a loop-shaped channel in the presence of the temperature gradient and the external magnetic field. The magnetic field strength and the fluid temperature difference between the heating and cooling sections were found to dominate among several factors influencing the device performance. Saravanan (29) studied theoretically the influence of magnetic field on the threshold of convection excited by a centrifugal acceleration in porous medium filled with magnetic fluid. He considered the layer exhibiting anisotropy in mechanical and thermal sense. He obtained numerical solutions using Galerkin method. He noticed that the magnetic field has a destabilizing effect and depending on the anisotropy parameters it can be suitably adjusted to enhance convection. He also found that the effect of anisotropies of porous media filled with magnetic fluid is significantly different from that filled with an ordinary fluid. Belyaev & Smorodin (21) studied the linear stability of a convective flow in a flat vertical layer of ferromagnetic fluid under a transverse temperature gradient in a uniform magnetic field perpendicular to the plates described by the Langevin law of magnetization. The stability of flow with respect to three-dimensional perturbations was analyzed, and the stability characteristics were obtained. The authors confirmed the existence of the stationary and two types of wave modes previously reported in Suslov (28). It was found that thermomagnetic waves can exist for a wide range of values of the magnetic susceptibility, Prandtl number and the Langevin parameter. The upper and lower boundaries of the interval of Prandtl numbers were determined where thermomagnetic waves with the large wavenumber found in Suslov (28) are most hazardous. Suslov et al. (212) investigated theoretically and experimentally thermomagnetic convection flows in a vertical layer of incompressible nonconducting ferromagnetic fluid bounded by two differentially heated long and wide parallel plates placed in an external uniform perpendicular magnetic field. The authors investigated experimentally various

58 CHAPTER 1. INTRODUCTION 34 flow patterns and confirmed the existence of oblique thermomagnetic waves predicted by Suslov (28). It was found that the wavenumber of the detected convection patterns depends sensitively on the applied magnetic field and on the temperature difference across the layer. They suggested a quantitative criterion for detecting the parametric point where the dominant role in producing a flow instability is transferred between the thermomagnetic and thermogravitational mechanisms based on the disturbance energy balance analysis. It was found that such a transition occurs when Grashof and magnetic Grashof numbers are of comparable sizes. Bozhko et al. (213) investigated experimentally convective flows of a stratified magnetic fluid in differentially heated vertical layers in a normal uniform magnetic field. The authors used two experimental chambers with different aspect ratios (height-to-thickness) to demonstrate the influence of concentration stratification caused by gravitational sedimentation of magnetic particles and aggregates on the structure of a convective motion. It was found that when a density gradient is attained in an isothermal layer over a period of several weeks prior to the experiment, cellular structures appear after the heating starts. It was also found that the fluid mixing caused by thermomagnetic convection homogenises the fluid faster so that the cellular structures get washed out within a shorter time compared to thermo-gravitational convection in the absence of magnetic field Other Effects in Magnetic Fluids While might not be directly relevant to the current study, for the sake of completeness a number of other physical effects that do not exist in non-magnetic fluids but that have been detected in ferrofluids both experimentally and numerically will be mentioned next. Bacri & Salin (1982) examined the evolution of the shape of ferrofluid drops in the presence of a magnetic field. The prolate ellipsoid shape of the fluid drop becomes unstable above a certain magnetic field threshold. The ellipsoid shape of the drop was created due to minimization of the total energy that consists of magnetic and surface tension compo-

59 CHAPTER 1. INTRODUCTION 35 nents. The former is minimised in elongated droplets aligned with the field while the latter is the smallest in spherical droplets. It was found that the drop shape jumped from almost spherical to a much more elongated with the increasing magnetic field and the elongated shape resorbed to near spherical one with the decreasing magnetic field. Pleiner et al. (22) studied the phase transition from an isotropic ferrofluid to a nematic one by considering Landau free energy depending on the nematic order parameter that characterizes the formation of chain structures of fluid particles in a strong external magnetic field and magnetization. The transitions are of the first kind and the ferromagnetic state can be reached either directly from the isotropic one or via the superparamagnetic nematic one. The external magnetic field induces a finite magnetization as well as nematic order in ferrofluid. It was found that in the presence of an external magnetic field the superparamagnetic and ferromagnetic nematic phases have identical symmetry properties and only differ quantitatively in the value of the magnetization. It was also found that the transition under a strong external field was continuous. A number of past studies have dealt with surface effects in ferrofluids that can be traced back to the so-called Rosensweig instability (Rosensweig 1985). Lange et al. (2) used linear stability theory to study the wavenumber selection of a free surface instability arising in a horizontal layer of viscous magnetic fluid of a finite depth in a normally applied magnetic field. The maximum growth rate and the corresponding wavenumber have been computed for various combinations of viscosity and thickness of a fluid layer. It has been found that the increase of magnetic induction leads to a mostly linear increase of a wavenumber, which is consistent with the experimental data. Another noteworthy results reported in Lange et al. (2) is that in thin (film) layers of a fluid the behaviour of the disturbance wavenumber was found to be independent of the fluid viscosity and thus of the potential magnetoviscous effects. Richter & Barashenkov (25) observed experimentally a stable two-dimensional solitonlike structures arising on the free surface of a horizontal layer of ferrofluid in a vertically

60 CHAPTER 1. INTRODUCTION 36 applied stationary magnetic field created by two Helmholtz coils. The authors used a magnetic fluid with a high relative permeability (similar to that used in the current study) and found that a surface deformation of fluid first occurred at the edge of the layer. Only when the magnetic induction was increased soliton-like structures and, subsequently, a fully developed pattern of Rosensweig instability emerged. This behaviour has been traced back to the refraction of magnetic field at the edge of a magnetic fluid layer, which confirms that the edge effects in realistic experiments with magnetic fluids can have a strong influence (see also Suslov et al. (212)). As will be discussed in Chapter 2 accounting for such an influence (at least partially) is one of the main motivating factors for the present study. Yecko (29) investigated the stability of two-layered channel flow of sheared interface separating a ferrofluid and ordinary viscous fluid with respect to disturbances with an arbitrary wavelength using three dimensional linear stability analysis. The unperturbed state corresponds to a two-layer Poiseuille profile in which a uniform magnetic field was applied at an arbitrary angle with respect to the interface. Neutral curves and stability characteristics at low Reynolds number were discussed and found to depend sensitively on the linear and nonlinear magnetic properties of the material. Taking into account the nonlinearity of the magnetic medium and using a fully viscous flow analysis the critical stability characteristics were obtained directly from the interplay of modes of different character corresponding to the most unstable part of the spectrum. The undertaken literature review spans a wide variety of research that has been conducted in the filed of ferrofluid flows to date. For convenience of reference table 1.1 summarises papers discussed here. In conclusion of this review it is noted that every system in nature is subjected to small perturbations. Thus as confirmed by the above summary of research related to ferrofluid flows, most of the related studies use the small perturbation theory. To shed light on the physics of flows this theory will be used here as well and the linear stability of the basic state of flow with respect to infinitesimal disturbances will be investigated. In this study

61 CHAPTER 1. INTRODUCTION 37 Table 1.1: A brief summary of literature review of research related to ferrofluid flows. References Finlayson (197), Schwab et al. (1983), Abraham (22) and Lange (22) Kikura et al. (1993) and Sawada et al. (1993) Odenbach (1995) Huang et al. (1997) Yamaguchi et al. (22) Nakatsuka et al. (22) Snyder et al. (23) Suslov (28) Main features Analytical, numerical, and experimental investigations of convective instability. The effects of a vertical spatially uniform magnetic field to horizontal ferrofluid layer heated from below are analysed. Experimental investigation in a cubical enclosure and concentric horizontal annuli, respectively, under the influence of a magnetic field. The field gradient was measured in a single direction only. The magnetic field orientation was not described. Experimental determination of the threshold of thermomagnetic instability in a cylindrical geometry in microgravity. The investigation has considerable fundamental relevance, but the heat transfer enhancement through thermomagnetic convection is not quantified. Linear stability study of a stationary instability pattern occuring in a paramagnetic fluid placed in a uniform oblique magnetic field in the absence of gravity. The oscillatory instability is not detected. Numerical investigation of a flow in a partitioned rectangular enclosure. The spatial description of the imposed magnetic field is absent. Experimental investigation to characterize the boiling heat transfer in the heat pipes using water-based and ionic magnetic fluids. The magnetic field and its gradient are specified only in one direction. The effect of variations in the field strength is not included, and therefore quantitative relationships between the heat transfer augmentation and field strength are not provided. Three-dimensional numerical simulation using a commercial CFD code. The magnetic field is considered but its full description is not given. A comprehensive linear stability analysis of convection flow in a layer of ferromagnetic fluid between two vertical differentially heated plates placed in a uniform external normal magnetic field. A complete stability diagram for two- and threedimensional disturbances is presented. Two distinct mechanisms, thermogravitational and magnetic, are found to lead to the appearance of three instability modes.

62 CHAPTER 1. INTRODUCTION 38 Belyaev & Smorodin (21) Suslov et al. (212) A linear stability of a convective flow in a flat vertical layer of ferromagnetic fluid in a transverse temperature gradient in a uniform magnetic field perpendicular to the plates assuming the Langevin law of magnetization. The stability of flow with is investigated for fluids with various Prandtl numbers. Energy analysis and experimental investigation of thermomagnetic convection flows in a vertical layer of nonconducting ferromagnetic fluid in a thin and wide vertical rectangular cavity placed in an external uniform normal magnetic field. Various flow patterns are investigated and the existence of oblique thermomagnetic waves predicted in Suslov (28) is confirmed experimentally. the major steps of the analyses reported in (Finlayson 197, Suslov 28, Belyaev & Smorodin 21, Suslov et al. 212) will be followed focusing on the investigation of the influences of the orientation of the applied magnetic field on the fluid behaviour. Physical mechanisms driving the observed ferrofluid motions will be clarified theoretically. Comparative results for fluids with weak (paramagnetic fluids) and strong (ferrofluids) degrees of magnetisation will be obtained and the patterns resulting from the competition between thermomagnetic and thermograviational mechanisms of convection will be discussed. This comprehensive investigation will also provide a guidance for further experimental studies.

63 Chapter 2 Pure Thermomagnetic Convection 2.1 Introduction As ferrofluids are stable colloidal suspensions of magnetic nanoparticles in ordinary nonmagnetic carrier liquids, their flows are sensitive to the influence of magnetic field, which can be used to control fluid flow dynamics and heat transfer in various applications. Various mechanisms leading to convective instability are present in ferrofluids. Fluid motion in ferrofluid does not require gravity to be initiated in a non-uniformly heated fluid. It can be conveniently controlled by varying the applied external magnetic field (Schwab et al. 1983, Tangthieng et al. 1999, Bozhko & Putin 23, 29, Zablotsky et al. 29). Therefore so-induced convection is considered to be an important alternative to gravitational convection in heat exchange systems where natural convection cannot arise due to the lack of gravity (orbital stations) (Odenbach 1995, Bozhko & Putin 29) or extreme confinement (microelectronics) (Matsuki & Murakami 1987, Koji et al. 27, Mukhopadhyay et al. 25, Lian et al. 29). The current chapter is focused on features of a pure magnetic convection and the gravitational acceleration is set to zero in the governing equations. The results obtained for mixed magneto-gravitational convection regimes will be reported in Chapter 3 of this dissertation.

64 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 4 Consistent with the previous studies (Suslov 28, Suslov et al. 212) a planar geometry is considered here: a fluid layer is confined by long and wide non-magnetic plates. The choice of such a classical geometry is dictated by a number of factors. Firstly, it is a common prototype configuration for realistic heat exchangers. Secondly, it allows one to make a significant analytical and computational progress without being overwhelmed by geometrical details. Thirdly, this geometry is relatively easy to realize in experiments. The alternative setup that has been extensively studied numerically consists of two coaxial co-rotating or differentially rotating cylinders (Zebib 1996, Tagg & Weidman 27, Altmeyer et al. 21, e.g.). However, as mentioned in Suslov et al. (21) experimental implementation of such a geometry is somewhat more complicated and magnetic field applied in this case is necessarily non-uniform, which complicates the understanding of basic thermomagnetic mechanisms that are of interest here. There are a number of motivating factors for this study stemming from experimental observations reported in Bozhko & Putin (1991), Bozhko et al. (1998), Bozhko & Putin (23), Suslov et al. (212). In particular, it has been observed that when a normal magnetic field is applied to a sufficiently wide and long ferrofluid layer the convection patterns arising near the edges of the experimental layer differ drastically in both orientation and behaviour from their counterparts seen in the central part of the layer. Namely, while the most prominent pattern in the middle part of the layer is stationary, the propagating structures have been detected near the edges that form some angle with the boundary, see figure 2.1. The exact reasons of such a different behaviour of a ferrofluid near the edges of the flow domain remain unclear to date and the present study explores one of the plausible explanations. At the boundary between magnetic (ferrofluid) and non-magnetic (container wall) media the magnetic field lines inevitably refract, which is the consequence of Maxwell s boundary conditions for a magnetic field, see Section 2.2. As a result even if the applied magnetic field is assumed to be normal to the layer, which was the case in the majority of previous studies, e.g. Finlayson (197), Suslov (28), the field lines necessarily curve near the layer edges so that they are effectively sucked in the magnetic medium. Such a behaviour of magnetic field lines near the borders is very sensitive to

65 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 41 g Inclined rolls Vertical rolls Figure 2.1: A schematic view of instability patterns observed experimentally in a differentially heated layer of ferrofluid placed in a magnetic field, see figure 9 in Suslov et al. (212). The magnetic field is applied normally in the center of the layer, but its direction is inevitably distorted near the edges. the minor details of the border geometry and its defects so that it is virtually impossible to know what the local inclination angle of the magnetic field is. To render the problem tractable to the analysis the following compromise is suggested: an infinitely wide and long layer of fluid is still considered and assumed that the applied field is uniform, however its arbitrary inclination with respect to the plane of the layer is allowed. Effectively, this adds two field direction angles γ and δ to the standard problem s parameter list, see figure 2.2 and equation (2.2.1). Since ferrofluids do not retain magnetization in the absence of an externally applied magnetic field they are sometimes classified as superparamagnets rather than ferromagnets (Albrecht et al. 1997). The next motivating factor for this current study is to demonstrate that such a terminology might be misleading if used blindly. It will be shown that even though the governing equations used to describe the behaviour of ferrofluids (see equations (2.2.2) (2.2.5)) are indeed similar to those typically used for natural paramagnetic fluids such as oxygen (Ageikin 195, Huang et al. 1997, e.g.) flow instability patterns predicted using these equations in ferrofluids are qualitatively different from those found in paramagnetic fluids. It will be shown in Section 2.4 that this distinction is brought

66 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 42 about by a nonlinearity of magnetic field inside the layer of a ferrofluid, which is caused by a field inclination and a much stronger degree of ferrofluid magnetisation compared to that of paramagnetic fluids. This in turn leads to another methodological motivation of this current study: in contrast to paramagnetic fluids the flow stability characteristics of ferrofluids depend sensitively on the actual values of their magnetic susceptibilities that vary widely with the magnitude of the applied field. Therefore a special care should be taken when comparing computational and experimental results for ferrofluids to make sure that the analysis has been performed for fluid properties that vary in a way consistent with experimental conditions. This aspect of theoretical ferrofluid research has received an insufficient attention in the previously reported studies and will be discussed in detail in Section 2.3. Remarkably, in contrast to the previous studies of magnetoconvection arising in a normal field (Finlayson 197, Suslov 28) it will be shown that the magnitude of the applied inclined magnetic field becomes an additional governing parameter of the problem. Prior to proceeding with presenting the current results it should be stressed that ferrofluids are complex multi-component colloidal systems and treating them as monofluids with spatially uniform properties cannot be justified in general. There is a growing body of evidence that effects such as gravitational sedimentation (Bozhko & Tynjälä 25, Bozhko et al. 213) and thermophoresis (Völker & Odenbach 23, Lange 24, Ryskin & Pleiner 24, 27) of magnetic nano-particles as well as magnetoviscosity (Bashtovoy et al. 1988, Odenbach & Raj 2, Odenbach 22a,b, Pop & Odenbach 26, Engler et al. 29, e.g.) can significantly affect experimentally observed ferrofluid flows. The detailed set of specific assumptions and physical conditions under which the governing equations considered in Section 2.2 are expected to be accurate has been discussed in Suslov et al. (212) and will be justified in Sections In the current study it will be assumed that the concentration of magnetic phase remains uniform and therefore the influence of the thermal and magnetic fields on the flow structure will only be investigated. Such a simplified approach appears to be reasonable, at least in the initial study, since the time scale for establishing thermosolutal gradients is significantly longer than that of the de-

67 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 43 velopment of a thermo-magnetic instability and the sedimentation effects are not present due to the lack of gravity. The magnetoviscous viscous effects, in particular, the increase of ferrofluid s effective viscosity in the applied magnetic field, while not modeled explicitly, is partially taken into consideration by allowing the governing parameters that include fluid viscosity, namely, the magnetic Rayleigh number (see Section 2.3) or magnetic and gravitational Grashof numbers and Prandtl number (see Section 3.3), to vary widely so that their true (but not known exactly) experimental values are included in the range investigated here. 2.2 Problem Formulation and Governing Equations The current study will follow the major steps of the analysis presented in (Finlayson 197), and (Suslov 28). Namely, a linear flow stability analysis will be used to shed light on the physical mechanisms of the observed convection and to provide parametric guidance for further quantitative experimental study of fully nonlinear convection states driven by complex mechanisms in thin layers of ferro-colloids. Consider a layer of a ferromagnetic fluid that fills a gap between two infinitely long and wide parallel nonmagnetic plates as shown in figure 2.2. The plates are separated by the distance 2d and are maintained at constant different temperatures T ± Θ. An external uniform magnetic field, H e = (Hx, e Hy, e Hz) e such that H e = H e, where H e x = H e cos δ, H e y = H e sin δ cos γ, H e z = H e sin δ sin γ, (2.2.1) is applied at an arbitrary inclination to the layer. This field causes an internal magnetic field H such that H = H within the layer. The external field induces fluid magnetization M such that M = M, which is assumed to be co-directed with the internal magnetic field: M = χ H, where χ is the integral magnetic susceptibility of the fluid. As discussed, for example, in Odenbach (24) and references therein this is true if the magnetic particle size does not exceed d p 13 nm (the estimations based on the exper-

68 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 44 y T + Θ T Θ γ H e z δ x 2d Figure 2.2: Sketch of the problem geometry. The vector of external magnetic field, H e forms angles δ and γ with the coordinate axes. imental data, see Section 2.3, show that for the fluid used in the cited experiments the average value of d p satisfies this condition). In this case the ratio of the Brownian particle magnetization relaxation time τ B = (4πd 3 pη )/(kt), where k B = J/K is Boltzman constant, to the viscous time τ v = ρ d 2 p/η characterizing the macroflow development is τ B /τ v 1 5. Thus it is safe to assume that the orientation of the magnetic moments of individual particles and thus of the fluid magnetization follows the direction of a local magnetic field. The orientation of magnetic particle aggregates however can in principle be affected by the mechanical torque due to the local shear of the flow so that they can misalign with the local magnetic field. Yet the experiments reported in Odenbach & Müller (25) show that such a misalignment only becomes noticeable for shear rates exceeding 15 s 1 while the shear rate for typical convection flows that are of interest here are of the order of.1 s 1 or even smaller. Therefore the misalignment of the magnetization and magnetic field vectors can be safely neglected. We also assume that the fluid magnetization depends only on magnetic field and temperature, which is the case when the concentration of magnetic particles remains uniform in experiments (gravita- 1

69 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 45 tional sedimentation, thermo- or magneto-phoresis of solid particles are not considered). The right-hand system of coordinates (x, y, z) with the origin in the midplane of the layer is chosen in such a way that the plates are located at x = ±d and the y and z axes are parallel to the plates. Assuming that the temperature difference 2Θ between the walls is sufficiently small the Boussinesq approximation of the continuity, Navier-Stokes and thermal energy equations that are complemented with Maxwell s equations for the magnetic field are adopted: v =, (2.2.2) v ρ t + ρ v v = p + η 2 v + µ M H, (2.2.3) T t + v T = κ 2 T, (2.2.4) H =, B =, (2.2.5) where B = µ ( M + H), M M(H, T) = H. (2.2.6) H In the above equations v is the velocity vector with the respective components (u, v, w) in the x, y and z directions, t is time, T is the temperature, p is the pressure, B is the magnetic flux density, ρ is the fluid density, η is the dynamic viscosity, κ is the thermal diffusivity of the fluid, and µ = 4π 1 7 H/m is the magnetic constant. The subscript denotes the values of the fluid properties evaluated at the reference temperature T and reference internal magnetic field H (to be defined in Section 2.4). In writing equation (2.2.3) it is assumed that the fluid remains Newtonian. It has been found in experiments of Bogatyrev & Gilev (1984) that this is a reasonable approximation for fluids with the concentration of solid phase not exceeding f =.1. The more recent measurements (however with different fluids) reviewed in Odenbach (22b, Ch. 4) have indicated that ferrofluids placed in the magnetic field can also behave as Bingham fluids with a nonzero yield-stress that increases approximately quadratically with the applied magnetic

70 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 46 field. However the yield-stress magnitude remains very small for the field strength range relevant to the current study so that the Newtonian fluid approximation is well justified here. As evidenced by numerous studies cited in Section 2.1 the viscosity of concentrated ferrofluids depends on the applied magnetic field and the local flow shear that influence the concentration of aggregates formed as a result of a dipole interaction between magnetised particles. In general, both the average and local values of viscosity can vary. Even though multiple experiments aiming at quantifying such a dependence have been reported in literature Odenbach (22a), Odenbach & Müller (25), Pop & Odenbach (26, e.g.) the data collected in these measurements cannot be used directly to model flows in geometries and conditions that are significantly different from those of rheological experiments. Fortunately, in this study the reference layer-average fluid viscosity only enters the non-dimensional governing equations in combination with other fluid properties forming magnetic Rayleigh number (see Section 2.3). Its value has been allowed to vary over a wide range, which effectively includes all experimental conditions even though the exact value of magnetoviscosity remains unknown. The unknown variation of the local viscosity and other fluid properties subject to the action of the locally varying magnetic field and shear presents a more daunting problem. It is well known (Suslov & Paolucci 1995a,b, e.g.) that if sufficiently large such a variation can strongly influence the structure of the flow and its stability. Yet to make analytical progress in absence of a quantitative rheological model it has been forced to neglect these spatial variations of fluid properties in equation (2.2.3). This is consistent with a widely used Boussinesq thermal approximation adapted for magnetic fluids (Bashtovoy et al. 1988) and is expected to be reasonable if the temperature and magnetic field variation across the layer remain small. The qualitative agreement between the computational results and the experimental observations reported in the previous work (Suslov et al. 212) indicates that indeed such a simplification preserves sufficient accuracy of the model and makes it tractable. A further discussion and a quantitative justification of this simplification have been provided in Section 2.3 and Section 2.4.

71 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 47 The last term in equation (2.2.3) represents a ponderomotive (Kelvin) force that acts on a magnetized fluid in a nonuniform magnetic field driving it toward regions with a stronger magnetic field as discussed in Bashtovoy et al. (1988). In order to close the problem, a magnetic equation of state is required which is assumed to be in the simplest linear form valid for small temperature and field variations within the layer, M = M + χ H K T, H H H, T T T. (2.2.7) Here H and M = χ H are the magnitude of the magnetic field and the magnetisation at the location with temperature T, χ = M/ H (H,T ) is the differential magnetic susceptibility and K = M/ T (H,T ) is the pyromagnetic coefficient. Using equation (2.2.7), rewrite equation (2.2.6) as M = χh + (χ χ)h K T H H. (2.2.8) Equation (2.2.8) was used for computations reported in Rahman & Suslov (215). However it leads to expressions that are algebraically quite involved. Therefore it is desirable to use a simplified linearized version of equation (2.2.8), namely, its linearization M = [M + (χ χ ) H K T] e + χ H, (2.2.9) where e H /H = (e 1, e 2, e 3 ), χ M /H, H = H + H and H = H + H so that H = H is the constant vector representing the major direction of the magnetic field and denotes small increments. This was done previously in Finlayson (197) and Suslov (28). Here the simplified equation (2.2.9) will also be used and the results will be compared with those reported in Rahman & Suslov (215) for full equation (2.2.8). Eliminating the magnetization in favor of the magnetic field one then obtains from the

72 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 48 second of equations (2.2.5) (1 + χ ) H + (χ χ ) H e K T e =. (2.2.1) This equation shows that thermomagnetic coupling occurs mostly when the magnetic field and the temperature gradient have components in the same direction. It is convenient to redefine pressure p entering the momentum equation (2.2.3) so that it includes both a hydrostatic component and a potential of Kelvin force (see also Odenbach (22a, pp. 86, 87)). In order to do this equation (2.2.7) is used to write µ M H = µ [M + χ H K T] H = µ [M H χ H2 ] µ K T H. It will be demonstrated in Section that only the non-potential component F K = µ K T H of Kelvin force can lead to the destabilization of a static mechanical equilibrium and result in magnetoconvection. Upon introducing the modified pressure P = p µ [ M H χ H2 ], (2.2.11) equation (2.2.3) is written as ρ v t + ρ v v = P + η 2 v µ K T H, (2.2.12) The standard no-slip/no-penetration and thermal boundary conditions are imposed. v =, T = ±Θ at x = d (2.2.13)

73 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 49 for velocity and temperature, respectively. The magnetic boundary conditions are ( H e H) n =, ( B e B) n = at x = ±d, (2.2.14) where superscript e denotes fields outside the layer and n = (1,, ) is the normal vector to the walls. Using equation (2.2.1) the second of the conditions in equation (2.2.14) is rewritten as [ He {(1 + χ )H + (χ χ ) H ± KΘ} e (1 + χ ) H] n = at x = ±d. (2.2.15) 2.3 Nondimensionalisation and Problem Parameters The governing equations and boundary conditions are nondimensionalised using (x, y, z) = d(x, y, z ), v = κ d v, t = d2 κ t, P = ρ κ 2 d 2 P, T = Θθ, H = KΘ 1+χ H, H = KΘ 1+χ H, M = KΘ 1+χ M, M = KΘ 1+χ M. Then after omitting primes for simplicity of notation they become v =, (2.3.1) v t + v v = P + Pr 2 v Ra m Prθ H, (2.3.2) θ t + v θ = 2 θ, (2.3.3) H =, (2.3.4) (1 + χ ) H + (χ χ ) H e (1 + χ) θ e =, (2.3.5) M = [(χ χ )(H N) (1 + χ)θ] e + χ H (2.3.6)

74 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 5 Table 2.1: The typical values of experimental parameters and properties of the ferrofluid manufactured in Scientific Laboratory of Practical Ferromagnetic Fluids, Ivanovo, Russia under Technical Conditions and used in experiments reported in Suslov et al. (212) and Bozhko et al. (213). Notation Parameter Typical value f Volume concentration of magnetic phase.1 ρ Density kg/m 3 β Coefficient of thermal expansion K 1 κ Thermal diffusivity m 2 /s η Dynamic viscosity in the absence of magnetic field kg/m s H e External magnetic field A/m T Average (reference) temperature in the layer 293 K 2Θ Temperature difference between the walls 1 3 K 2d Distance between the walls 6 mm with the boundary conditions [ H e {(χ χ )(H N) (1 + χ)} e (1 + χ ) H] n =, (2.3.7) v =, θ = 1 at x = ±1. (2.3.8) The dimensionless parameters appearing in the problem are Ra m = µ K 2 Θ 2 d 2 η κ (1 + χ), Pr = η, N = H (1 + χ). (2.3.9) ρ κ KΘ The magnetic Rayleigh number Ra m characterizes the importance of magnetic forces compared to viscous forces, Prandtl number Pr is the ratio of viscous and thermal diffusion transports, and parameter N represents the nondimensional magnetic field at the reference location. The typical parameter values that are reported for the recent experiments (Suslov et al. 212, Bozhko et al. 213) are listed in Table 2.1. Based on physical quantities listed in this table the estimated value of Prandtl number Pr is around 55 and it is used for current computations. In the case of pure magnetoconvection (in the absence of gravity) considered here the critical values of Ra m are found to be invariant with respect to the

75 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 51 values of Prandtl number. To compare the previous results with the results of Chapter 3 reported in terms of magnetic Grashof number, Gr m (Pr) = Ra m Pr, (2.3.1) where Ra m is the invariant value reported in the current chapter and Pr is the value of Prandtl number used elsewhere. Among other important physical quantities characterizing the field dependent magnetic properties of the fluid are differential and integral magnetic susceptibilities χ and χ and pyromagnetic coefficient K that depend on the applied magnetic field and the temperature. The paramagnetic coefficient K only enters the governing equations as a component of the nondimensional groups (2.3.9) so that its exact value is not required for the current analysis, see a similar discussion of fluid viscosity in Section2.2. However it will be seen in Section2.4 that the magnitude of K (and thus of parameter N) can be conveniently used to distinguish between paramagnetic and ferromagnetic fluids. At the same time the values of magnetic susceptibilities χ and χ are important problem parameters entering the governing equations directly. It is a common practice to derive them from Langevin magnetization law that reads M L (H) = M L(ξ), L(ξ) = coth ξ 1 ξ, ξ = µ π M sd 3 ph 6k B T, (2.3.11) where M is the experimentally measured saturation magnetization of the fluid, L(ξ) is Langevin s function and ξ is Langevin s parameter. Here M s is the saturation magnetization of a magnetic phase at a given temperature, k B = J/K is Boltzmann constant and d 3 p is the average cube of the diameter of a magnetized core of solid particles. Due to Curie effect (demagnetization of a ferromagnetic with the increasing temperature) and thermal expansion of a carrier fluid the saturation magnetization of magnetic material

76 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 52 and ferrofluid vary as M s = M s 1 β 2 T 2 1 β 2 T 2, (2.3.12) 1 β M = 2 T 2 M 1 β 2 T 2 (1 β (1 f )(T T )), (2.3.13) where β is the coefficient of thermal expansion of a carrier fluid, β 2 is Curie coefficient and f is the volume fraction of the magnetic phase. In this study, magnetite particles with M s = 48 ka/m and β 2 = K 2 at reference temperature T = 293 K are considered, which correspond to the experimental fluids referred to in the previous publications (Suslov 28, Suslov et al. 212, Bozhko et al. 213). The fluid saturation magnetization measured at T = 293 K was M = 43 ka/m. However both experimental measurements and molecular dynamics simulations show that the magnetization law of a realistic ferrofluid deviates significantly from the Langevin dependence. The main reason for this is that Langevin s law assumes no inter-particle interactions, which is not the case for experimental fluids with magnetic phase concentration as high as f =.1. A comprehensive review of this issue is given in Ivanov et al. (27). There the authors showed that the significant improvement of the accuracy of the magnetization law for a ferrofluid is obtained via the use of the so-called second order Modified Mean Field (MMF2) model that is essentially a two-term expansion of Weiss Mean Field model (Weiss 197, Tsebers 1982). It is obtained by replacing the Langevin parameter ξ with ξ = µ π M sd 3 p H 6k B T, H = H + 1 ( 3 M L(H) ) dm L (H). (2.3.14) dh Here H is the effective magnetic field that takes into account mean magnetic interactions between particles in a concentrated magnetic fluid. The only physical quantity that remains unknown in the above formulae is the average cube of the diameter of magnetized particle cores. This quantity depends strongly on the (unknown) size dispersion of nano-particles and their aggregates present in the fluid.

77 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 53 Thus d 3 p was determined by matching the predictions of the initial differential magnetic susceptibility lim H χ(h) of the fluid with its experimentally determined value χ = 3 (Pshenichnikov 27, Lebedev & Lysenko 211). It turned out that the MMF2 model produces this value if d 3 p ( ) 3 m 3. Even though the MMF2 model is shown to produce accurate values for magnetic properties of a ferrofluid, it requires the value of a local magnetic field H as an input. However this quantity depends on the geometry of the considered problem. For example, it was shown in Suslov (28) that when an external magnetic field H e is applied perpendicularly to an infinite differentially heated layer of a ferrofluid bounded by parallel non-magnetic plates the internal magnetic field in the midplane of the layer is H = H e /(1 + χ ). The values of H and the corresponding fluid magnetization M are shown in figure 2.3(c) for the experimental range of external magnetic field H e 35 ka/m. If the magnetic field is applied obliquely then the relationship between the external and internal fields becomes much more complicated, see Section 2.4. However in this study it is mostly interested in a qualitative behaviour of the system for relatively small field inclination angles δ 15. Therefore the data presented in figure 2.3 for δ = will suffice for the parameter estimation purposes. The comparison of relevant quantities obtained using Langevin and MMF2 models given in figure 2.3 demonstrates that Langevin s law systematically underestimates the values of the magnetic susceptibilities and the pyromagnetic coefficient. Thus MMF2 model is used to choose the range of parameters for which the numerical results will be presented. As seen from figure 2.3(a) the values of magnetic susceptibilities change over the range of the applied magnetic field from (χ, χ ) = (3, 3) to just below (χ, χ ) = (1.5, 2.5). Thus the numerical results will be reported for these two pairs of values. It also is noted that the previous numerical results are available (see Suslov (28)) for (χ, χ ) = (5, 5). Therefore for comparison purposes the computations will also be performed for this pair of parameters as well as for (χ, χ ) = (3, 5). Note that as discussed in Finlayson (197), and Suslov (28), χ = χ along the linear segment of the magnetization curve. How-

78 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION (a) 12 1 (b) χ, χ χ, MMF2 χ, MMF2 χ, Langevin χ, Langevin K,A/(m K) MMF2 Langevin H e, ka/m H e, ka/m 25 2 H, MMF2 M, MMF2 H, Langevin M, Langevin 12 1 H, M, ka/m H e, N H e, MMF2 N, MMF2 H e, Langevin N, Langevin (c) H e, ka/m 2 (d) H e, ka/m Figure 2.3: The comparison of magnetic fluid properties defined using Langevin s law and MMF2 model for δ = and T = 293 K (Θ = 7.5 K is chosen in plot (d)).

79 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 55 ever, χ < χ when the fluid s magnetization approaches saturation. Therefore by choosing different values of the differential and integral magnetic susceptibilities, the effect of nonlinearity of the magnetization law on flow stability is investigated. The value of the pyromagnetic coefficient K enters the definition of magnetic Rayleigh number. In this study the data from figure 2.3(b) is not used to limit the computational range of Ra m, but it is rather presented here to enable the design of chambers for future experiments where the thickness of the gap between the walls will need to be chosen so that it results in the values of parameters close to critical ones reported in this study. Figure 2.3(d) shows the values of the normally applied nondimensional external magnetic field H e and the corresponding reference internal magnetic field N = N δ=. These parameters are not independent. It has been shown in Suslov (28) that H e = (1 + χ )N (2.3.15) when δ =. It is also known that neither of these two parameters affect the stability characteristics of the flow when the magnetic field is perpendicular to the layer if the magnetisation law is linear (Finlayson 197, Suslov 28). 1. However, when the nonlinearity of magnetization is taken into account and when the field is applied obliquely the role of these parameters becomes nontrivial. It will be demonstrated in Section 2.4 that whenever δ = the magnetic field lines within the layer curve. This curvature will be shown to be responsible for a qualitative change in the behaviour of the perturbation fields compared to that observed when δ =. The relationship between parameters N and H e for a general oblique field cannot be given explicitly (apart from some limiting cases discussed in Section 2.4) yet it is still well approximated by (2.3.15). It is convenient to use N for expressing solutions for magnetic field within the layer symbolically and H e for parameterizing the numerical stability results. Figure 2.3(d) shows that for all experimentally relevant conditions the value of H e (and N ) remain large and close to constant for fixed 1 Here prime is used to denote a non-dimensional external field in figure 2.3(d) and to distinguish it from dimensional external field H e, however for simplicity of notation H e will be used to denote non-dimensional field in the subsequent text.

80 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 56 temperature difference 2Θ between the walls. Therefore H e = 1 is chosen in the current computations, which is close to the values achieved in experiments. The calculations show that the numerical stability results become almost independent of H e when its value exceeds 4 (this is the case when the temperature difference between the walls does not exceed about 3 K, which is the experimental maximum). The selected results are also presented for H e = 1. This would correspond to the temperature difference T between the walls of more than 1 degrees, which is experimentally unachievable with the fluids used in the referenced experiments. However such a value of H e could be realistic for a stronger magnetiseable fluid (e.g. that with a higher concentration of magnetic particles and thus with a larger pyromagnetic coefficient). Therefore the numerical results are presented for H e = 1 as well, which allows to see more clearly the peculiar qualitative effects brought about by the inclination of a magnetic field and the resulting curvature of magnetic field lines (see in plots (a) and (c) in figures ). 2.4 Basic Flow The steady motionless solutions of equations (2.3.1) (2.3.8) in the form v =, θ = θ (x), P = P (x), H = (H x (x), H y (x), H z (x)). They should satisfy DP = Ra m Prθ e 1 DH x, D 2 θ =, (2.4.1) (1 + χ )DH x + (χ χ )e 1 e 1 DH x (1 + χ)e 1 Dθ =, (2.4.2) DH y =, DH z =, (2.4.3) and the boundary conditions H e x (χ χ )(H N)e 1 ± (1 + χ)e 1 (1 + χ )H x =, (2.4.4) θ = ±1, H y = H e y, H z = H e z at x = 1, (2.4.5)

81 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 57 where H Hx 2 + H2 y + H2 z, M M 2 x + M2 y + M2 z introducing the unit vector e (x) (e 1 (x), e 2 (x), e 3 (x)) = and D d/dx. Upon ( Hx H, H ) y H, H z H in the direction of the magnetic field the basic flow solutions of equations (2.4.1) are written as x θ = x and P = Ra m Pr xe 1 DH x d x + C, (2.4.6) where C is an arbitrary constant. Equations (2.4.3) along with boundary conditions (2.4.5) result in the expressions for tangential components of the magnetic field that are constant inside the fluid layer H y (x) = Hy e and H z (x) = Hz. e Therefore the tangential components of the applied magnetic field do not change across the layer. It is known (e.g. Suslov (28)) that for a perpendicular field when e = e = (1,, ) the basic flow component of magnetic field in the x direction across the layer is given by H x = N x, where N is defined by (2.3.15). However when the external field is applied obliquely a nonlinear equation (2.4.2) does not have a closed form solution and has to be solved numerically. When the field inclination angle is small, that is when sin δ = H 2 y +H 2 z H e, this solution can also be written asymptotically as H x = N x + x(1 + χ ) 2 sin 2 δ 2(1 + χ)(n x) [(N x)(1 + χ ) + N (χ χ)] x(1 + χ ) 4 sin 4 δ 8(1 + χ) 2 (N x) 3 [(N x) 3 (1 + χ )(3 + χ + 2χ ) (2.4.7) +(N x)n (χ χ)(3 + χ + 2χ )(2N x) +N 3 (χ χ)(1 χ + 2χ )] + o((1 + χ ) 4 sin 4 δ), It can be also shown that for small non-zero inclination angles δ the nondimensional magnetic field at the center-plane of a fluid layer is N N 1 + (1 + χ ) 2 sin 2 δ. (2.4.8) This approximation remains accurate as long as (1 + χ s ) 2 sin 2 δ 1. For example, if χ = 5 the above expression indicates that a significant change in the value of the internal

82 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 58 magnetic field is expected to occur for the field inclination angles as small as δ = 1. This is also evidenced by the numerical stability results that, as will be shown below, experience a qualitative change at similar inclinations. Further asymptotic can be developed by taking into account that N x as discussed in x Section 2.3. Then H x can be given in terms of as N H x = 1 x ( 1 1 χ + 2χ (1 + χ ) 2 sin 2 δ N N 2(1 + χ) χ2 3 + (3 + 2χ )(2χ 4χ ) + x2 N 2 + x3 N 3 χ χ 2(1 + χ) (1 + χ ) 2 sin 2 δ χ χ 2(1 + χ) (1 + χ ) 2 sin 2 δ ( χ + 1χ 2(1 + χ) ) 8(1 + χ) 2 (1 + χ ) 4 sin 4 δ (1 + 3 (1 + χ ) 3 ) 1 + χ sin2 δ (2.4.9) ) (1 + χ ) 2 sin 2 δ) +. A similar asymptotic approach was used in Hennenberg et al. (26) to determine the approximate expression for the magnetic field in a layer of magnetic fluid subject to longitudinal temperature gradient (in contrast to the transverse gradient considered here). If the magnetisation law is linear, that is if χ = χ (for small values of the applied magnetic field, see Figure 2.3(a)) equations (2.4.7) and (2.4.9) reduce to H x = 1 x + x [ (1 + χ ) 2 sin 2 δ 1 3 ] N N 2N 4 (1 + χ ) 2 sin 2 δ + (2.4.1) The first two terms in the asymptotic solution (2.4.1) are equivalent to expression (9) given in Huang et al. (1997) for a magnetic field in a layer of paramagnetic fluid with χ = χ 1. However, the nonlinearity of the magnetic field in the layer was fully neglected in Huang et al. (1997). As seen from expressions (2.4.9) and (2.4.1) this is only true when N, that is when pyromagnetic coefficient K, see definition (2.3.9). This is shown to be a good approximation in the case of paramagnetic fluids (Huang et al. 1997); however, K is large in the current problem, see figure 2.3(b). Therefore N is finite

83 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 59 and the nonlinearity of the magnetic field inside the layer of ferromagnetic fluid cannot be ignored. This is confirmed by fully nonlinear numerical solutions for the magnetic field shown for the finite values of χ = χ = 3 and χ = 1.5, χ = 2.5 in figures 2.4 and 2.5, respectively. The degree of nonlinearity increases with the decreasing value of N that is if the applied magnetic field decreases while the value of the pyromagnetic coefficient remains close to constant. Yet the three term asymptotic solution (2.4.9) remains robust providing the accuracy within 1 2% of the numerically computed values in all considered regimes. Importantly, figures 2.4 and 2.5 demonstrate that the relative deviation of the magnetic field within the layer from its average value cannot exceed 1/N. Using the data presented in figure 2.3(d) one then concludes that the field varies within the layer by less than 4%. This is the natural measure of the error that is introduced in the considered model by assuming that the field-dependent fluid properties remain constant in equation (2.2.3). Once the magnetic field within the layer is determined the unperturbed fluid magnetization is computed using M = {[((χ χ )(H N) (1 + χ)θ )e 1 + χ H x } 2 +{((χ χ )(H N) (1 + χ)θ )e 2 + χ H y } 2 (2.4.11) +{((χ χ )(H N) (1 + χ)θ )e 3 + χ H z ] 2 } 1 2. The typical distributions of the magnetization and magnetic pressure across the layer are shown in figures 2.6 and 2.7. Figures demonstrate that the field inclination leads to a noticeable asymmetry of the distribution of the cross-layer components of magnetic and magnetization fields and their full magnitudes. This asymmetry in basic flow fields will be shown to influence the stability results qualitatively. The change in the values of the differential and integral magnetic susceptibilities leads to a significant change in the magnitude of the magnetic field inside the layer as follows from equation , but it does not lead to qualitative changes in the variation of the magnetic field and its

84 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 6 H(x) N δ = δ =5 δ =1 δ =15 Hx(x) Hx() H a x/hx (a) x (b) x (c) x H(x) N (d) x 1 Hx(x) Hx() (e) x 1 1 H a x/hx (f) x Figure 2.4: Numerical solution for the magnitude H of the undisturbed magnetic field ((a) and (d)) and its cross-layer component H x ((b) and (e)) for H e = 1 (top row), H e = 1 (bottom row), χ = χ = 3 and various field inclination angles δ. Plots (c) and (f) show the corresponding relative error of the asymptotic solution (2.4.9). cross-layer component as is seen Figures 2.4 and 2.5. However, a noticeable change in both the magnitude of the cross-layer component of the magnetization field and its full magnitude occurs (see figures 2.6 and 2.7). The other observation is that the magnitude of the external magnetic field influences the fields inside the layer: weaker oblique external fields lead to a stronger nonlinearity of internal fields (compare the top and bottom rows in figures ). The symmetry-breaking effect of the field inclination is also evident in the behaviour of the magnetic pressure P shown in plots (c) and (f) in figures 2.6 and 2.7. As the field inclination angle increases the pressure near the cold wall grows with respect to that near the hot wall. For example, P (1) P ( 1).5Ra m Pr for δ = 15 (the dotted line in figure 2.6 (f)). As will be discussed later this leads to the preferential shift of instability structures toward the hot wall, which introduces a further asymmetry and qualitative change in stability characteristics compared to the normal field

85 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 61 H(x) N δ = δ =5 δ =1 δ =15 Hx(x) Hx() H a x/hx (a) x (b) x (c) x H(x) N.5.5 Hx(x) Hx() H a x/hx (d) x (e) x.1 (f) 1 1 x Figure 2.5: Same as figure 2.4 but for χ = 1.5, χ = 2.5. case considered in Finlayson (197), and Suslov (28). The behaviour of magnetic field lines inside the layer of a ferrofluid is shown in plots (a) and (c) in figures The increasing field inclination angle leads to a quantitative change in the variation of the curvature of magnetic lines inside the layer and to the normal non-potential component F K = θ dh dx of the nondimensional Kelvin force decrease near the hot wall and its increase near the cold wall. The change in the values of the differential and integral magnetic susceptibilities at the same field inclination angles does not affect qualitatively the magnetic field lines or Kelvin force behaviour (compare plots (b) and (d) in figures 2.8, 2.9, 2.1 with those in figures 2.11, 2.12, 2.13, respectively). In contrast to the case of a normal field considered in Finlayson (197) and Suslov (28) the nondimensional magnitude (relative to the fluid magnetisation) of the obliquely applied field strongly affects the geometry of magnetic lines. The curvature of magnetic lines is especially pronounced in stronger magnetiseable fluids (plot (c) as contrasted to plot (a)).

86 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 62 M(x) χ N (a) δ = δ =5 δ =1 δ = x Mx(x) χ Hx() (b) x P/(RamPr) (c) 1 1 x M(x) χ N (d) Mx(x) χ Hx() (e) P/(RamPr) (f) x x 1 1 x Figure 2.6: Numerical solution for the magnitude M of the undisturbed fluid magnetization ((a) and (d)), its cross-layer component M x ((b) and (e)) and magnetic pressure P ((c) and (f)) for H e = 1 (top row), H e = 1 (bottom row), χ = χ = 3 and various field inclination angles δ. This has a profound effect on the distribution of the normal non-potential component of the nondimensional Kelvin force that can be viewed as a magnetic buoyancy force. It is shown in plots (b) and (d) in figures Such a force is positive near the left wall and negative near the right wall, which corresponds to an inherently unstable situation when hot fluid near x = 1 is forced to flow toward the cold wall at x = 1 and vice versa. This situation is similar to an unstably stratified layer of a regular fluid heated from below in the downward gravitational field. Yet such similarity is complete only if the external magnetic field is normal to the layer. In this case, similar to its gravitational counterpart, the magnetic buoyancy is a linear function of the cross-layer coordinate x and its nondimensional value is independent of the strength of the applied field, see dashed lines in plots (b) and (d) in figures However, when the oblique field of the same

87 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 63 M(x) χ N (a) δ = δ =5 δ =1 δ = x Mx(x) χ Hx() (b) x P/(RamPr) (c) 1 1 x 2 (d) 1 (e).5 (f) M(x) χ N 1 1 Mx(x) χ Hx().5.5 P/(RamPr) x x 1 1 x Figure 2.7: Same as figure 2.6 but for χ = 1.5, χ = 2.5. magnitude is applied to the layer at least three qualitative differences arise due to the nonlinearity of the induced internal field. Firstly, the magnetic buoyancy force becomes more uniform across the layer so that the onset of thermo-magnetic instability is expected to be delayed compared to the normal field situation. Secondly, the magnetic buoyancy force becomes a function of the magnitude of the applied magnetic field. Thirdly, and most importantly, the nonlinearity of the internal magnetic field leads to the situation when the unstable stratification df K < with respect to magnetic buoyancy is noticeably dx stronger in the vicinity of the hot wall, see plots (d) in figures Therefore in contrast to the case of a normal field the cross-layer symmetry of the arising instability structures is broken. In the following sections the physical features of instability patterns will be established and quantified that are brought about by the inclination of an external field and were not found in the previous studies reported in Finlayson (197) and Suslov

88 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 64 Figure 2.8: Refraction of magnetic lines ((a) and (c)) and the distribution of Kelvin force (solid line, and dashed line for δ = ) ((b) and (d)) in a layer of magnetic fluid heated from the left for the field inclination angle δ = 5, γ =, χ = χ = 3, H e = 1 ((a) and (b)) and H e = 1 ((c) and (d)). (28). 2.5 Linearized Perturbation Equations In order to investigate linear stability of the basic state discussed in Section 2.4 with respect to infinitesimal disturbances that are assumed to be periodic in the y and z directions, a standard normal form representation of the perturbed quantities is considered and

89 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 65 Figure 2.9: Same as figure 2.8 but for δ = 1. written as ( v, P, θ, H, H, M, M) = ( v, P, θ, H, H, M, M ) (2.5.1) [ ] + ( v 1 (x), P 1 (x), θ 1 (x), H1 (x), H 1 (x), M1 (x), M 1 (x))e σt+i(αy+βz) + c.c., where σ = σ R + iσ I is the complex amplification rate, α and β are real wavenumbers in the y and z directions, respectively, and c.c. denotes the complex conjugate of the expression in brackets. To satisfy equation (2.3.4) identically, it is convenient to introduce

90 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 66 Figure 2.1: Same as figure 2.8 but for δ = 15. perturbation φ 1 (x)e σt+i(αy+βz) of a magnetic potential so that H 1 = [Dφ 1, iαφ 1, iβφ 1 ] T, H 1 = H1 e = [e 1 D + i(αe 2 + βe 3 )]φ 1, M 1 = [χ Dφ 1 + e 1 (χ χ )H 1 e 1 (1 + χ)θ 1, iαχ φ 1 + e 2 (χ χ )H 1 e 2 (1 + χ)θ 1, iβχ φ 1 + e 3 (χ χ )H 1 e 3 (1 + χ)θ 1 ] T, M 1 = M1 e = [χ + (χ χ )(e 1 e 1 + e 2 e 2 + e 3 e 3 )]H 1 (1 + χ)(e 1 e 1 + e 2 e 2 + e 3 e 3 )θ 1.

91 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 67 Figure 2.11: Same as figure 2.8 but for χ = 1.5, χ = 2.5. The linearization of equations (2.3.1) (2.3.8) about the basic state leads to = Du 1 + i (αv 1 + βw 1 ), (2.5.2) σu 1 + Pr(α 2 + β 2 D 2 )u 1 + DP 1 + e 1 Ra m PrDH x θ 1 ( + Ra m Prθ e 1 D 2 φ 1 + Ra m Prθ i(αe 2 + βe 3 ) + (1 e1 2 ) DH ) x Dφ 1 H ira m Prθ e 1 (αe 2 + βe 3 ) DH x H φ 1 =, (2.5.3)

92 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 68 Figure 2.12: Same as figure 2.9 but for χ = 1.5, χ = 2.5. σv 1 + Pr(α 2 + β 2 D 2 )v 1 + iαp 1 + iαra m Prθ e 1 Dφ 1 αra m Prθ (αe 2 + βe 3 )φ 1 =, (2.5.4) σw 1 + Pr(α 2 + β 2 D 2 )w 1 + iβp 1 + iβra m Prθ e 1 Dφ 1 βra m Prθ (αe 2 + βe 3 )φ 1 =, (2.5.5) σθ 1 + Dθ u 1 + (α 2 + β 2 D 2 )θ 1 =, (2.5.6) = (D 2 α 2 β 2 )φ χ [i(αe βe 3 ) + e 1 D]θ 1 χ χ χ [ ( ) DH (αe βe 3 ) αe 2 + βe 3 + x ie 1 e 1 φ 1 (2.5.7) χ H ( ie 1 (αe 2 + βe 3 ) + ie 1 (αe 2 + βe 3 ) + e 1 (1 e1 2 ) DH ) x Dφ 1 H ] e 1 e 1 D 2 φ 1.

93 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 69 Figure 2.13: Same as figure 2.1 but for χ = 1.5, χ = 2.5. The disturbance velocity and temperature fields are subject to standard homogeneous boundary conditions u 1 = v 1 = w 1 = θ 1 = at x = ±1. (2.5.8) A perturbation of a magnetic field within a fluid layer causes perturbation of the external magnetic field for non-magnetic boundaries as discussed in Finlayson (197). If there are no induced currents outside the layer and a non-magnetic medium (air) fills the surrounding space, then the external magnetic field has a potential φ1 e (x) exp(σt + iαy + iβz), which, as follows from equations (2.2.5) and (2.2.6), satisfies Laplace s equation (D 2 α 2 β 2 )φ e 1 =, (2.5.9)

94 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 7 in the regions x < 1 and x > 1. A physically relevant bounded solution can be written as Ae α 2 +β 2x φ1 e(x) =, x < 1 Be α 2 +β 2x, x > 1. (2.5.1) Upon taking into account (2.2.6), the linearization of the magnetic boundary conditions (2.2.14) leads to Dφ e 1 = (1 + χ )Dφ 1 + e 1 (χ χ )[i(αe 2 + βe 3 ) + e 1 D]φ 1, (2.5.11) φ e 1 = φ 1 at x = ±1. (2.5.12) After eliminating A and B from equations (2.5.1) and (2.5.12), the boundary conditions for φ 1 at x = ±1 become (1 + χ )Dφ 1 ± α 2 + β 2 φ 1 + e 1 (χ χ ) [i(αe 2 + βe 3 ) + e 1 D] φ 1 =. (2.5.13) 2.6 Squire s Transformation Upon using the generalized Squire s transformations (x, y, z) = ( x, ỹ, z), θ = θ, H x = H x, H = H, σ = σ, α 2 + β 2 = α 2, β = β, u 1 = ũ, αv 1 + βw 1 = αṽ, w 1 = w, θ 1 = θ, P 1 = P, φ 1 = φ, Ra m = Ra m, Pr = Pr, N = Ñ, χ = χ, χ = χ, (2.6.1) e 1 = ẽ 1, αe 2 + βe 3 = αẽ 2, e 1 = ẽ 1, αe 2 + βe 3 = αẽ 2,

95 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 71 equations (2.5.2) (2.5.7) become = Dũ + i αṽ, (2.6.2) σũ + Pr( α 2 D 2 )ũ + D P + ẽ 1 Ra m PrD H x θ + Ra m Pr θ ẽ 1 D 2 φ (2.6.3) [ ] + Ra m Pr θ i αẽ 2 + (1 ẽ1 2 ) D H x D H D φ x i α Ra m Pr θ ẽ 1 ẽ 2 φ =, H H σṽ + Pr( α 2 D 2 )ṽ + i α P + α Ra m Pr θ (iẽ 1 D φ αẽ 2 φ) =, (2.6.4) σ w + Pr( α 2 D 2 ) w + i β P + β Ra m Pr θ (iẽ 1 D φ αẽ 2 φ) =, (2.6.5) σ θ + D θ ũ + ( α 2 D 2 ) θ =, (2.6.6) = (D 2 α 2 ) φ χ χ αẽ χ [ + χ χ 1 + χ with the boundary conditions [ αẽ 2 + iẽ 1 ẽ 1 D H x H i α(ẽ 1 ẽ 2 + ẽ 1 ẽ 2 ) + ẽ 1 (1 ẽ 2 1 ) D H x H + χ χ 1 + χ ẽ 1 ẽ 1 D 2 φ 1 + χ 1 + χ [i αẽ 2 + ẽ 1 D] θ ] φ ] D φ (2.6.7) (1 + χ )D φ ± α φ + ẽ 1 ( χ χ )(i αẽ 2 + ẽ 1 D) φ =, (2.6.8) ũ = ṽ = w = θ = at x = ±1. (2.6.9) Equation (2.6.4) is obtained by multiplying equation (2.5.4) by α, equation (2.5.5) by β, adding them together, and dividing the result by α. Note that only equation (2.6.5) contains w and β and thus it can be solved for any particular value of β after σ, P and φ are found from equations (2.6.2) (2.6.4), (2.6.6) and (2.6.7), which form an equivalent two-dimensional problem obtained by formally setting w = β =. It is important to note though that the notion of an equivalent two-dimensional problem in the current context is somewhat different from that arising in problems dealing with non-magnetic fluids. The reason is that even if w and β are set to zero in the above Squire-transformed equations the external magnetic field remains three-dimensional. In general it still has three non-zero components in the x, y and z directions and thus it needs to be described

96 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 72 using two coordinate angles δ and γ that act as independent control parameters of the problem. The above transformations simply mean that the y direction as the periodicity direction of the arising perturbation structures is conveniently viewed while the vector of the applied magnetic field can be arbitrarily oriented. More specifically, the axes of the instability rolls are considered to be always parallel to the z axis in figure 2.2 so that γ = means that the magnetic field has a component in the plane of the fluid layer that is perpendicular to the roll axes while γ = 9 means that this component is parallel to the roll axes. The intermediate values of γ are interpreted accordingly. For the sake of brevity in the following sections it will conveniently be referred to instability patterns computed using the above transformed equations for γ = and γ = 9 as transverse and longitudinal rolls, respectively, while patterns obtained for all other values of γ will be referred to as oblique rolls. It will also be referred to angle δ as the field inclination angle and angle γ as the angle between the axes of the instability rolls and the in-layer component of the applied magnetic field. 2.7 Numerical Results Discretisation Equations (2.6.2) (2.6.7) are discretised using the pseudo-spectral Chebyshev collocation method as introduced in Ku & Hatziavramidis (1984) and Hatziavramidis & Ku (1985) and implemented in Suslov & Paolucci (1995b) and Suslov & Paolucci (1995a). This spatial approximation converges exponentially quickly so that 71 collocation points used in the current computations guarantee that all digits in the reported numerical results are significant and accurate. Upon discretization and exclusion of equation (2.6.5), which is the only equation containing β, the system of equations (2.6.2) (2.6.7) results in a generalised algebraic eigenvalue problem for the complex amplification rate σ ( σx + Y) q =, (2.7.1)

97 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 73 where X and Y = Y( σ, Ra m, Pr, χ, χ ; δ, γ) are matrices and eigenvector q contains the discretised components of (ũ, ṽ, P, θ, φ) T. The eigenvalue problem is solved using the MATLAB (Mat 213) function eig. Once both σ and q are found equation (2.6.5) is written as [ ] [ ] Pr(D 2 α 2 ) σ w = i β P + Ra m Pr θ (ẽ 1 D φ + i αẽ 2 φ) (2.7.2) and solved for w. The inverse Squire s transformation (2.6.1) then recovers full threedimensional solutions for perturbations Check of Numerical Accuracy In order to test the numerical code, the critical values for the magnetic convection threshold in a perpendicular (δ = ) external magnetic field with magnitude H e =1 for the case of Pr = 13 and χ = χ = 4 have been computed using relation (2.3.1) and the critical values Gr mc = and α c = are obtained, which agree well with the values of Gr mc = and α c = 1.95 computed from the corresponding data reported in Finlayson (197). The magnetic convection threshold for Pr = 13 and χ = χ = 5 is also determined and the critical values Gr mc = and α c = are obtained, which are identical to those reported in Suslov (28). As an additional check the critical values of Ra mc = and α c = were computed for the normal field H e = 1 in the limit of a paramagnetic fluid with χ = χ = 1 3. These values are in excellent agreement with the respective values of /16 = and 3.69/2 = reported in Huang et al. (1997) (the coefficients of 1/16 and 1/2 are due to the differences between the used nondimensionalisation scales) Flow Stability Characteristics The numerical values of critical parameters for thermomagnetic convection arising in magnetic fields of various orientations and intensities are given in tables The

98 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 74 Table 2.2: The critical values of Ra m, α and disturbance wave speed c = σ I / α and the corresponding perturbation energy integrals Σ k and Σ m1 for magnetoconvection at δ =, H e = 1 and various values of χ and χ. χ χ α c Ra mc c c Σ k Re(Σ m1 ) Im(Σ m1 ) data in the tables warrants a number of general conclusions. The magneto-convective instability arising in a normal field remains stationary regardless of the specific magnetic properties of the fluid and the magnitude of the applied magnetic field. This is in agreement with the findings previously reported, for example, in (Finlayson 197, Huang et al. 1997, Suslov 28). However, in contrast to all previous studies the instability threshold Ra mc is found to depend not only on the values of the magnetic susceptibilities χ and χ but also on the magnitude of the applied magnetic field, namely, the decrease of the characteristic non-dimensional field parameter N promotes instability and increases the wavelength of the arising patterns. This dependence, however, remains relatively weak: the largest difference between the critical values of magnetic Rayleigh and wavenumbers is found to be under 3.5% and 1.5%, respectively, for a fluid with the highest degree of magnetisation investigated (χ = χ = 5) when the external magnetic field is changed by a factor of 1. The comparison of the current results with the previous study (Suslov 28) shows that the dependence of the critical parameters on the magnitude of the magnetic field is traced back to the form of the constitutive magnetisation equation (2.2.8). Its linearisation used in all previous studies cited above eliminates the dependence of the threshold values on the amplitude of the normal magnetic field. However, as shown in Section 2.4 such an idealization is only robust for the case of paramagnetic fluids with small magnetic susceptibilities but it should not be

99 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 75 expected to be uniformly valid for realistic ferromagnetic fluids. If the dependence of the fluid magnetisation on the magnitude of the applied magnetic field remains linear that is if χ χ, see figure 2.3, then the magnetoconvection threshold parameters decrease monotonically with the values of magnetic susceptibilities to their limiting values ( Ra mc, α c ) (16.5, 1.85) that are independent of the magnitude of the applied magnetic field. However, when the fluid magnetisation approaches saturation so that χ < χ the variation of the differential and integral susceptibilities have opposite influences on the threshold: the decrease of χ at fixed χ promotes instability while the decrease of χ at fixed χ delays it. In realistic ferrofluids, however, the values of both χ and χ decrease with the increasing magnetic field, but at different rates, see figure 2.3 (a). Therefore it is not straightforward to anticipate what the overall effect of a changing magnetic field on the convection onset could be and one needs to rely on the specific computational results. In particular, the data in table 2.2 shows that the critical value of magnetic Rayleigh number decreases by more than 1% when progressively stronger magnetic field is applied to a layer of experimental ferrofluid with the initial susceptibilities χ = χ 3 that are reduced to χ 1.5 and χ 2.5 during a typical experimental run. It is remarkable that as seen from tables 2.3 and 2.4 when an oblique magnetic field is applied to the layer the trends described above are reversed even for small field inclination angles δ: now the decrease of χ at fixed χ delays instability while the decrease of χ at fixed χ promotes it. This indicates the qualitative difference between the instability mechanisms present in normal and oblique fields that will be discussed in more detail in the following sections. The numerical data given in tables also demonstrates a very strong stabilization effect of the field inclination compared to the normal field situation that is further illustrated in figures 2.14(a) and 2.16(a). Such a stabilization is observed regardless of the specific magnetic properties of the fluid for all investigated values of χ and χ.

100 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 76 Table 2.3: Same as table 2.2 but for transverse rolls at δ = 1 and γ =, H e = 1 (odd-numbered lines), H e = 1 (even-numbered lines). χ χ α c Ra mc c c Σ k Re(Σ m1 ) Im(Σ m1 ) Even more striking effect of the field inclination is evident from the data presented in table 2.3: the transverse instability rolls computed for γ = become oscillatory resulting in waves propagating along the direction of the field component that is tangential to the plane of the fluid layer. This is a somewhat unexpected result given that the unperturbed problem possesses a full planar symmetry with no preferred direction. Moreover Huang et al. (1997) even argued that the instability in this problem can only be stationary. The resolution of this apparent paradox is prompted by the comparative computational data presented in table 2.3 for H e = 1 and H e = 1 and by figure They show that the magnitude of the disturbance wave speed c = α σi c is approximately inversely proportional to the magnitude of the applied magnetic field H e, which in turn is proportional to the field parameter N characterizing the nonlinearity of the magnetic field distribution within a layer. It is assumed in Huang et al. (1997) that N and effectively postulated that the magnetic field within the layer varies linearly. No unsteady patterns were found there. Therefore it is concluded that the main reason for the appearance of oscillatory instability in the current problem is the nonlinearity of a magnetic field within the ferrofluid layer as has been discussed in Section 2.4.

101 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 77 Table 2.4: Same as table 2.3 but for longitudinal rolls at δ = 1 and γ = 9. χ χ α c Ra mc c c Σ k Re(Σ m1 ) Im(Σ m1 ) The values of the threshold parameters for longitudinal rolls computed for γ = 9 are given in table 2.4. Remarkably, they remain strictly stationary for all values of the governing parameters. Figure 2.16(a) shows that similar to the critical magnetic Rayleigh number for transverse rolls the one for longitudinal rolls increases quickly with the field inclination angle δ. However for all non-zero angles it remains smaller than that of transverse rolls. This is consistent with findings of Huang et al. (1997) for paramagnetic fluids and confirms an experimental fact that the axes of thermomagnetic rolls appearing away from the boundaries always align with the tangential component of the magnetic field since this configuration is found to be less stable than a transverse one. Having said this it is emphasized that even though longitudinal rolls are always expected to dominate the observed instability patterns the possibility of the existence of transverse rolls should not be ignored for at least two reasons. Firstly, unlike in paramagnetic fluids, in ferrofluids they are qualitatively different from their longitudinal counterparts as they are unsteady. They are also characterised by a wavenumber that depends sensitively on the field inclination angle, see figure 2.14(b), while the wavenumber of longitudinal rolls remains almost constant as the field inclination is increased, see figure 2.16(b). Secondly, near the boundaries of a layer the longitudinal rolls may be suppressed due to the geometry of the boundary or

102 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 78 Ra mc 2 x χ =1.5, χ =2.5 χ =3, χ =3 χ =3, χ =5 χ =5, χ =5 (a) αc (b) δ, δ, Figure 2.14: (a) Critical magnetic Rayleigh number Ra mc and (b) wavenumber α c as functions of the field inclination angle δ for transverse rolls at H e = 1 and γ =. The respective plots for H e = 1 are indistinguishable within the figure resolution. cc χ =1.5, χ =2.5 χ =3, χ =3 χ =3, χ =5 χ =5, χ =5 (a) cc.2.1 (b) δ, δ, Figure 2.15: Critical wave speed c = σi α c as the function of the field inclination angle δ for transverse rolls at γ = for (a) H e = 1 and (b) H e = 1. other influences that are not present in unbounded domains so that oscillatory transverse rolls might be preferred. The experimental observations reported in the previous work Suslov et al. (212) (see figure 9 and 11 there) indeed indicate that this might have been the case in the near-boundary regions of a finite experimental enclosure. Given that the two limiting cases of transverse and longitudinal rolls have qualitatively different characteristics it is of interest to investigate how and at what value of the intermediate angle the transition between stationary and oscillatory patterns occurs. Thus the stability characteristics of oblique rolls have been computed for various values of magnetic susceptibilities and field inclination angles. These are presented in figures They confirm that both the critical magnetic Rayleigh number and wavenumber increase continuously and monotonically from longitudinal to transverse rolls and the

103 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 79 Ra mc χ =1.5, χ =2.5 χ =3, χ =3 χ =3, χ =5 χ =5, χ =5 (a) αc (b) δ, δ, Figure 2.16: Same as figure 2.14 but for longitudinal rolls at γ = 9. Ra mc (a) δ =5 δ =1 δ = γ, αc (b) γ, cc (c) Figure 2.17: (a) Critical magnetic Rayleigh number Ra mc, (b) wavenumber α c and (c) wave speed c as functions of the azimuthal angle γ for various angles δ for H e = 1 and χ = χ = 5. γ,

104 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 8 Ra mc (a) δ =5 δ =1 δ = γ, αc (b) γ, cc Figure 2.18: Same as figure 2.17 but for χ = χ = 3. (c) γ, Ra mc (a) δ =5 δ =1 δ = γ, αc (b) γ, cc Figure 2.19: Same as figure 2.17 but for χ = 1.5 and χ = 2.5. (c) γ, rate at which they do grows quickly with the field inclination angle. The only exception is the behaviour of the wavenumber for relatively large field inclination angles when it reaches its maximum value for oblique rolls forming the angle of about 45 with the tangential field component and then starts decreasing. Of particular interest is the behaviour of the disturbance wave speed. It grows continuously from zero for longitudinal rolls to its maximum for transverse rolls, however, the most rapid growth is observed for γ 5 and γ 13. This suggests that if the value of magnetic Rayleigh number is gradually increased in an experiment then the stationary rolls aligned with the tangential component of the field will appear first. Subsequently, they would be unsteadily modulated by a periodic pattern forming the angle of about 4 45 with the axes of the stationary rolls. A further increase of magnetic Rayleigh number would lead to the increase of the modulation frequency and wavenumber and to the re-orientation of the modulating pattern so that it would become closer to orthogonal with respect to the original stationary rolls.

105 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 81 It is also noteworthy that the Ra mc and α c curves are symmetric with respect to the γ = 9 line while the c c line is centro-symmetric with respect to ( c c, γ) = (, 9 ). To shed light on why this is so refer to plots (a) and (c) in figures where the concave south-west/north-east magnetic field lines are shown for γ =. If γ is changed to 18 the magnetic field lines re-orient to become north-west/south-east and convex. As has been discussed above the appearance of oscillatory disturbances is a consequence of the non-linearity of a magnetic field. Therefore it is concluded that it is this change of the curvature of magnetic lines in the plane perpendicular to the roll axes that is responsible for the change of the sign of the disturbance wave speed Perturbation Energy Balance To confirm the physical nature of the observed instabilities it is instructive to consider the mechanical energy balance in a way similar to that used for example in Hart (1971), and Suslov & Paolucci (1995a). The momentum equations (2.6.3) and (2.6.4) are multiplied by the complex conjugate velocity components ū and v, respectively, add them together and integrate by parts across the layer using the boundary conditions (2.6.9) and the continuity equation (2.6.2) to obtain σσ k = Σ vis + Σ m1 + Σ m2, (2.7.3) where Σ k = Σ vis = Σ m1 = Σ m2 = ( ũ 2 + ṽ 2 ) }{{} E k d x >, Pr( α 2 ( ũ 2 + ṽ 2 ) + Dũ 2 + Dṽ 2 ) }{{} d x = 1, (2.7.4) E vis Ra m PrD H x ẽ }{{ 1 θ } ũ d x, E m1 E m2 d x

106 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 82 and θ E m2 = Ra m PrD H x ũ ((1 ẽ1 2 H )D φ i αẽ 1 ẽ 2 φ) ) 2 Ra m Pr θ (ẽ 1 ũd φ + i α(ẽ 1 ṽ + ẽ2 ũ)d φ α 2 ẽ 2 ṽ φ. Given that Σ k is positive definite the perturbation energy balance equation (2.7.3) determines the complex growth rate σ of linear instability. It does not contain the modified pressure P as it integrates to zero identically. This confirms that the potential component of Kelvin force included in P indeed has no effect on the stability of a layer of a ferromagnetic fluid as has been stated in Section 2.2. The viscous dissipation contribution to the energy perturbation balance is always negative and, given that the eigenfunctions of the linearized problem are defined up to a multiplicative constant, scale them in such a way that Σ vis 1. The remaining two terms are of magnetic nature. As discussed in Suslov et al. (212), Σ m1 represents the variation of fluid magnetization (and thus of the local Kelvin force) due to the thermal perturbations while Σ m2 describes the energy contribution associated with the induced magnetic field variations. Separating the real (Re) and imaginary (Im) parts of (2.7.3) obtained at the critical point = Re(Σ m1 + Σ m2 ) 1, σ I Σ k = Im(Σ m1 + Σ m2 ) (2.7.5) The energy terms with positive real parts promote instability, while the ones with negative suppress it. The second of equations (2.7.5) demonstrates that the nature of the detected oscillatory instabilities is purely magnetic. Tables contain numerical data for various perturbation energy terms that enable us to draw a number of general conclusions. Firstly, the magnitude of the kinetic energy term Σ k never exceeds the value of about 1% of the viscous dissipation while the magnitude of the magnetic contribution Σ m1 always exceeds the dissipation value. This confirms that the instability is of magnetic rather than hydrodynamic or thermal nature and that the visible fluid motion triggered by the instability is not the main recipient of

107 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 83 the energy supplied to the system (in experiments such as those described in Suslov et al. (212) the energy is supplied by heat exchangers attached to the layer walls). Secondly, since Re(Σ m1 ) is always positive, it is concluded that the specific mechanism triggering the instability is the thermally induced variation of fluid magnetisation. Thirdly, since Re(Σ m1 ) > 1 then, according to the first of equations (2.7.5), Re(Σ m2 ) <. This means that the variation of the applied magnetic field caused by perturbations always plays a stabilizing role. In summary, the analysis of mechanical energy balance shows that the energy received by the system through a thermal exchange with the ambient is mostly spent on varying the local magnetisation of the fluid. In turn the latter triggers fluid motion, which is an observable signature of instability. The remaining part of the received energy is spent on modifying the magnetic field. Since the variation of magnetic field is not limited to the interior of the layer this energy largely leaves it and thus cannot be used for supporting a mechanical instability within the system. Typical distributions of the perturbation energy integrands for instability patterns arising in normal and oblique fields are shown in figures , respectively. Since the Ek,Evis.5 1 E k Re(Em1,Em2) E m1.5 E vis E m2 E m x x x Im(Em1,Em2) 1.5 E m1 Figure 2.2: Disturbance energy integrands at the critical point of magnetoconvection threshold Ra mc = 178.3, α c = at H e = 1, δ = γ = and χ = χ = 3.

108 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 84 Ek,Evis.5 1 Re(Em1,Em2) Im(Em1,Em2) E k 1 E m1.6 E m1 1.5 E vis E m2 E m x x x Figure 2.21: Same as figure 2.2 but for Ra mc = 232.4, α c = 1.951, δ = 5. Ek,Evis E k E m1 E m1 1.2 E vis E m2 E m x x x Re(Em1,Em2) Im(Em1,Em2) Figure 2.22: Same as figure 2.2 but for Ra mc = 533.7, α c = 2.116, δ = 1.

109 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 85 Ek,Evis E k 2 E m1 E m1 E 1.6 vis E m2 E m x x x Re(Em1,Em2) Im(Em1,Em2) Figure 2.23: Same as figure 2.2 but for Ra mc = 272.8, α c = 2.381, δ = 15. integrand behaviour for longitudinal rolls is found to be qualitatively similar to that for stationary rolls arising in a normal field, only the results for transverse rolls are presented here. As expected, the viscous dissipation E vis occurs mostly near the solid boundaries and the kinetic energy E k of perturbations is maximised near the center of the layer. The middle panels in all figures show that the magnetisation variation effect E m1 plays a destabilizing role uniformly across the complete width of the layer and with the maximum near its center. On the other hand, the stabilizing effect of magnetic field modification (Re(E m2 ) < ) is maximized in the regions located closer to the layer walls. This is intuitively expected since the internal magnetic field near the walls defines the external field via the field-matching boundary conditions (2.5.12) and (2.5.13). The overall role of the E m1 and E m2 effects does not change in the oblique field, however the field inclination introduces a noticeable asymmetry. The maximum of the destabilizing influence shifts toward the hot wall, compare the locations of the maxima of the dash-dotted lines in the middle panels of figures This is because the unstable magnetic buoyancy stratification in oblique fields is more pronounced near the hot wall, see in plots (b) and (d) in figures and the discussion in Section 2.4. While the magnitude of Re(E m1 ) determines whether the instability is present the right panel in figures shows

110 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 86 that it is the magnitude of Im(E m2 ) that predominantly defines the sign of σ I and thus the propagation direction of transverse and oblique rolls Perturbation Fields To complete this report the plots of typical perturbation fields arising in normal and inclined magnetic fields are presented. The mechanism driving convection is straightforward to see from figure 2.24 for a normal field. Consider, for example, the region near y = 3. There the thermal perturbation θ 1 leads to local cooling. As a result this region becomes stronger magnetised, see the plot for M 1, and the fluid there is driven toward the hot wall where the basic magnetic field is stronger, see in plots (a) and (d) in figures 2.4 and 2.5. This is reflected in the plot of the velocity field showing that indeed cool fluid Figure 2.24: (Colour online) Perturbation eigenfunctions of the fluid velocity v 1 = (u, v), temperature θ 1, magnetization M 1 and magnetic field H 1 for magnetoconvection at H e = 1, δ = γ = and χ = χ = 3 at the critical point Ra mc = 178.3, α c = Colour scale is arbitrary as the amplitude of perturbations cannot be determined in the framework of a linear analysis.

111 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 87 flows toward the hot wall (from right to left) there. This situation is similar to gravitational convection arising in a fluid heated from below. When the applied magnetic field is inclined the mechanism driving convection remains the same even though it is less straightforward to recognize it from figures The thermal and magnetisation perturbation cells align with the applied magnetic field and so does the main fluid flow direction. It is noteworthy that the perturbation cells for magnetic field H 1 corresponding to transverse rolls do not align with the rest of the perturbation field, see the right panel in figures They also become asymmetric. At the same time the structure of the perturbation fields for longitudinal rolls in an inclined field shown in figure 2.28 remains very similar to that seen in figure 2.24 for a normal field. Therefore it is logical to conclude that the phase shift between the magnetic field H 1 and the rest of the perturbation fields is responsible for the change of the instability character to oscillatory for transverse and oblique rolls.

112 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 88 Figure 2.25: Same as figure 2.24 but for Ra mc = 232.4, α c = 1.951, δ = 5. Figure 2.26: Same as figure 2.24 but for Ra mc = 533.7, α c = 2.116, δ = 1.

113 CHAPTER 2. PURE THERMOMAGNETIC CONVECTION 89 Figure 2.27: Same as figure 2.24 but for Ra mc = 272.8, α c = 2.381, δ = 15. Figure 2.28: Same as figure 2.24 but for longitudinal rolls at Ra mc = 267., α c = 1.916, δ = 1 and γ = 9.

114 Chapter 3 Magneto-Gravitational Thermal Convection 3.1 Introduction In Chapter 2 pure magneto-convective flow that arises in a ferrofluid placed in an oblique magnetic field was considered. The prototype applications justifying such an investigation deal with flows that arise in situations where the gravitational buoyancy-driven convection is impossible (Bashtovoy et al. 1988, Odenbach 1995). It is known that the onset of gravitational convection is controlled by the Rayleigh number, the nondimensional parameter, which is proportional to the product of the gravity and the cube of the characteristic size of the domain filled with a fluid. Therefore the value of the Rayleigh number tends to in outer space and in applications such as microelectronic devices. At the same time the magnetic Rayleigh number controlling magnetoconvection and defined in (2.3.9) is independent of the gravity and is proportional to only a square of the characteristic domain size. Therefore magnetoconvection can be induced in a gravity-free environment and it is easier to initiate in congested spaces of microelectronic devices that buoyancy-driven convection thus enabling the use of magneto-convective flows for heat removal in these applications (Polevikov & Fertman 1977, Yamaguchi et al. 22). How-

115 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 91 ever owing to the complex composition of ferrofluids (Charles 22) their flows have to be extensively investigated experimentally before they can receive a wide practical use. Unfortunately, it is virtually impossible to conduct such experiments in environments directly corresponding to the potential working conditions of ferrofluids: direct flow observations in microelectronic devices is impossible due to the microscopic size of flow domains and the experimental studies on-board space stations are prohibitively expensive and remain rare to date. Therefore the bulk of experimental ferrofluid flow investigations is conducted in ground-based experiments within finite-size (of the order of a few centimeters) containers (Bozhko & Putin 23, Zablotsky et al. 29). Therefore the influence of gravitational convection on the flows of non-isothermal ferrofluid cannot be avoided. Thus one of the goals of this chapter is to provide parametric guidance for such mixed magneto-gravitational experiments. Introduction of gravity into the problem expands the set of the governing parameters and as will be shown by the results reported in this chapter leads to qualitatively different instabilities to those observed in a gravity-free setting. Therefore another major goal that will be pursued here is developing the understanding of the physical features that are brought about by the competition between magnetic and gravitational mechanisms of convection. To unambiguously identify the influences brought about by the introduction of the gravity the same basic problem geometry as the one described in Chapter 2 will be considered here with the only addition being the gravity acting along one of the unbounded directions of the fluid layer. The analysis of various types of instabilities caused by the interaction of magnetic Kelvin and gravitational buoyancy forces is the major goal of the study reported in this chapter. The thermogravitational convection mechanism considered here is a consequence of the buoyancy force that occurs due to the thermal expansion of a non-uniformly heated fluid. It is an example of a thermally induced hydrodynamic instability (Chandrasekhar 1981). The stability of a non-isothermal flow in a vertical layer of fluid is one of the classical problems of natural convection (Gershuni & Zhukhovitsky 1953, Batchelor 1954). It is

116 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 92 known that for a class of large-prandtl-number fluids to which kerosene and transformer oil based ferrofluids belong to the instability in this configuration occurs in the form of two waves counterpropagating along the direction of gravity (Gershuni et al. 1989). The most dangerous instability mode detected in a normal magnetic field in the small gravity limit consists of stationary rolls with axes parallel to the direction of gravity (Suslov 28). At the same time it has been shown in Chapter 2 that the oblique magnetic field tends to align the axes of the rolls with its in-layer component. Therefore it remains to be seen what exactly pattern orientation will result when oblique magnetic field and gravity act simultaneously and compete in pattern formation. The study of such combined influences is what distinguishes the current investigations from previous research reported, for example, in Finlayson (197), Bozhko & Putin (1991), Odenbach (1995), Shliomis & Smorodin (22), Hennenberg et al. (26), Suslov et al. (28), Suslov (28), Suslov et al. (212) and references therein. To enable a direct comparison the same specific parameter values prompted by the experimental studies reported in literature to date (Suslov et al. 212, Bozhko et al. 213, Rahman & Suslov 215, e.g.) will be considered as in Chapter 2, but flow instabilities for wider parametric ranges will also be investigated to provide guidance for future experiments. More specifically, the comparative results will be presented for the following situations. A wide class of kerosene and transformer oil based fluids that are characterised by a wide range of Prandtl number values. Various concentrations of magnetic particles in ferrofluids that are characterised by different values of the differential magnetic susceptibility χ. The regimes of linear and nonlinear dependence of fluid magnetisation on the magnitude of the applied magnetic field. The former is typically observed when the amplitude of the applied magnetic field is sufficiently small so that M H = M T H that is the differential and integral magnetic susceptibilities are equal, χ = χ, see figure 2.3(a), (c). The latter case corresponds to strong applied magnetic fields when the fluid magnetisation is close to saturation and χ < χ.

117 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 93 Ferrofluids with different types of magnetic particles, or same fluids placed in significantly different magnetic fields. Both situations are characterised by the variation of the fluid s pyromagnetic coefficient K = M T (see figure 2.3(b)) and H thus the variation of their thermo-magnetic sensitivity. Quantitatively, this is accounted for by the values of the nondimensional parameter N defined in (2.3.9) and the magnitude of the nondimensional applied magnetic field H e, see figure 2.3(d). Since both N and H e are inversely proportional to K more thermo-magnetically sensitive fluids are characterised by the smaller values of these parameters. 3.2 Problem Formulation and Governing Equations In this chapter the same problem geometry is considered as in figure 2.2 in Chapter 2. However the system is placed in a gravitational field as shown in figure 3.1. A non-zero gravity is an important additional factor in the governing equations and the features of mixed magneto-gravitational convection will be discussed in detail below. The gravity vector g has constant components (, g, ) acting in the opposite direction of the y axis parallel to the plates. An oblique external uniform magnetic field is assumed to be the same as in Chapter 2. The governing equations (2.2.2) (2.2.5) still describe the problem. However equation (2.2.3) now reads ρ v t + ρ v v = p + η 2 v + ρ g + µ M H. (3.2.1) Consistent with the Boussinesq approximation valid for small temperature differences between the walls the fluid density variation with temperature T is accounted for only in the buoyancy term in equation (3.2.1) as ρ = ρ [1 β (T T )], (3.2.2) where β is the coefficient of thermal expansion.

118 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 94 y T + Θ T Θ g γ H e z δ x 2d Figure 3.1: Sketch of the problem geometry. The vector of external magnetic field, H e forms angles δ and γ with the coordinate axes. Pressure p entering the momentum equation (3.2.1) is redefined using the equation of state (3.2.2) and the magnetisation equation (2.2.7) and noting that ρ g + µ M H = ρ [1 β T)] g + µ [M + χ H K T] H = {ρ ( r g) + µ [M H χ H2 ]} (ρ β g + µ K H) T, where r = (x, y, z) is the coordinate vector. Equation (3.2.1) then becomes ρ v t + ρ v v = P + η 2 v ρ β T g µ K T H, (3.2.3) where the modified pressure P = p ρ ( r g) µ [M H χ H2] Nondimensionalisation and Problem Parameters The governing equations and boundary conditions are nondimensionalised by using thermal velocity in Chapter2. However, in this chapter the nondimensionalisation is changed

119 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 95 by introducing the viscous velocity, which is representative of non-zero base flow velocity. The other scales are changed consistently so that the governing equations and boundary conditions are nondimensionalised with the reference quantities for length, velocity, temperature and thermodynamic pressure using (x, y, z) = d(x, y, z ), v = η ρ d v, t = ρ d 2 t, P = η2 P, ρ d 2 T = Θθ, g = g e g, H = KΘ 1+χ H, H = KΘ 1+χ H, η M = KΘ 1+χ M, M = KΘ 1+χ M, where ρ is the density and η is the dynamic viscosity at the reference temperature T, e g = (, 1, ) and d is the half-distance between the vertical walls. Then omitting primes for simplicity of notation, the equations (3.2.3) and (2.3.3), respectively, become v t + v v = P + 2 v Grθ e g Gr m θ H, (3.3.1) θ t + v θ = 1 Pr 2 θ, (3.3.2) The rest of the governing equations and boundary conditions (2.3.1) (2.3.8) remain unchanged. The new dimensionless parameters appearing in the problem are Gr = ρ2 β Θgd 3 η 2, Gr m = ρ µ K 2 Θ 2 d 2 η 2 (1 + χ). (3.3.3) The magnetic Grashof number Gr m replaces the magnetic Rayleigh number Ra m, where Ra m = Pr Gr m. The thermal and magnetic Grashof numbers Gr and Gr m characterise the importance of buoyancy and magnetic forces, respectively. The values of Grashof numbers are chosen below to produce the complete parametric instability regions.

120 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION Basic Flow The steady solutions of equations (2.3.1), (3.3.1), (3.3.2) and (2.3.4) (2.3.8) are in the form v = (, v (x), ), θ = θ (x), P = P (x), H = (H x (x), H y, H z ). They should satisfy DP = Gr m θ e 1 DH x, D 2 v = Grθ, D 2 θ =. (3.4.1) The basic flow solutions of equations (3.4.1) are written as v = Gr x 6 (x3 x), P = Gr m xe 1 DH x d x + C, (3.4.2) where C is an arbitrary constant. As will be discussed later the oblique external magnetic field leads to preferential shift of instability structures towards the hot wall, which introduces further asymmetry and qualitative change in stability characteristics compared to the normal field case considered in Finlayson (197) and Suslov (28). In the following sections the physical features of instability patterns and the corresponding critical parameters will be presented for an obliquely applied external magnetic field.

121 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION Linearized Perturbation Equations Using a standard normal form presented by (2.5.1) in Chapter 2 the linearization of equations (2.3.1), (3.3.1), (3.3.2) and (2.3.4) (2.3.8) about the basic state leads to Du 1 + i (αv 1 + βw 1 ) =, (3.5.1) ( σu 1 + α 2 + β 2 + iαv D 2) u 1 + DP 1 + e 1 Gr m DH x θ 1 ( + Gr m θ e 1 D 2 φ 1 + Gr m θ i(αe 2 + βe 3 ) + (1 e1 2 ) DH ) x Dφ 1 H igr m θ e 1 (αe 2 + βe 3 ) DH x H φ 1 =, (3.5.2) σv 1 + Dv u 1 + (α 2 + β 2 + iαv D 2 )v 1 + iαp 1 Grθ 1 + iαgr m θ e 1 Dφ 1 αgr m θ (αe 2 + βe 3 )φ 1 =, (3.5.3) σw 1 + (α 2 + β 2 + iαv D 2 )w 1 + iβp 1 + iβgr m θ e 1 Dφ 1 βgr m θ (αe 2 + βe 3 )φ 1 =, (3.5.4) ( α σθ 1 +Dθ u β 2 D 2 ) + iαv θ 1 =, (3.5.5) Pr (D 2 α 2 β 2 )φ χ [i(αe βe 3 ) + e 1 D]θ 1 χ χ χ [ ( ) DH (αe βe 3 ) αe 2 + βe 3 + x ie 1 e 1 φ 1 (3.5.6) χ H ( ie 1 (αe 2 + βe 3 ) + ie 1 (αe 2 + βe 3 ) + e 1 (1 e1 2 ) DH ) x Dφ 1 H e 1 e 1 D 2 φ 1 ] =. The disturbance velocity and temperature fields are subject to standard homogeneous boundary conditions u 1 = v 1 = w 1 = θ 1 = at x = ±1. (3.5.7) The boundary conditions for φ 1 at x = ±1 is the same as reported by (2.5.13).

122 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION Squire s Transformation Upon using the generalized Squire s transformations (x, y, z) = ( x, ỹ, z), θ = θ, H x = H x, H = H, σ = σ, α 2 + β 2 = α 2, β = β, u 1 = ũ, αv 1 + βw 1 = αṽ, w 1 = w, θ 1 = θ, P 1 = P, φ 1 = φ, αgr = α Gr, Gr m = Gr m, Pr = Pr, χ = χ, χ = χ, (3.6.1) e 1 = ẽ 1, e 1 = ẽ 1, αe 2 + βe 3 = αẽ 2, αe 2 + βe 3 = αẽ 2, and noting that αv = αṽ, where ṽ = Gr( x 3 x)/6, equations (3.5.1) (3.5.6) become Dũ + i αṽ =, (3.6.2) σũ + ( α 2 + i αṽ D 2 )ũ + D P + ẽ 1 Gr m D H x θ + Gr m θ ẽ 1 D 2 φ (3.6.3) [ ] + Gr m θ i αẽ 2 + (1 ẽ1 2 ) D H x D H D φ x i α Gr m θ ẽ 1 ẽ 2 φ =, H H σṽ + Dṽ ũ + ( α 2 + i αṽ D 2 )ṽ + i α P Gr θ (3.6.4) + α Gr m θ (iẽ 1 D φ αẽ 2 φ) =, σ w + ( α 2 + i αṽ D 2 ) w + i β P + β Gr m θ (iẽ 1 D φ αẽ 2 φ) =, (3.6.5) ( α σ θ + D θ ũ + 2 D 2 ) + i αṽ θ =, (3.6.6) Pr [ ] (D 2 α 2 ) φ χ χ αẽ χ [ + χ χ 1 + χ αẽ 2 + iẽ 1 ẽ 1 D H x H i α(ẽ 1 ẽ 2 + ẽ 1 ẽ 2 ) + ẽ 1 (1 ẽ 2 1 ) D H x H + χ χ 1 + χ ẽ 1 ẽ 1 D 2 φ 1 + χ 1 + χ [i αẽ 2 + ẽ 1 D] θ = with the boundary conditions φ ] D φ (3.6.7) (1 + χ )D φ ± α φ + ẽ 1 ( χ χ )(i αẽ 2 + ẽ 1 D) φ =, (3.6.8) ũ = ṽ = w = θ = at x = ±1. (3.6.9)

123 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 99 Only equation (3.6.5) contains w and β explicitly. Therefore this equation can be split from the rest of the transformed system and, if necessary, solved afterwards. The remaining equations form an equivalent two-dimensional problem that can formally be obtained by setting w = β = that is by assuming the two-dimensionality of the perturbation field and its periodicity in the y direction. This enables one to significantly reduce the computational cost of stability calculations. However, even if β and w are set to the external applied magnetic field (2.2.1) still remains three-dimensional in the above Squiretransformed linearized equations and thus the two coordinate angles δ and γ still act as independent control parameters of the problem. The δ angle parameterizes the deviation of the field from the normal direction to the layer, while γ measures the azimuthal angle from the positive y direction that is from the direction opposite to that of the gravity. In this chapter we will refer to the angle δ as the field inclination angle and the angle γ as the field orientation angle. The main reason for this is that in contrast to the analysis of Chapter 2, where the direction of the y axis was arbitrary, here it is rigidly linked to the direction of the gravity rather than to the axial direction of the arising instability rolls. In general the orientation of instability patterns in flows considered here is not known beforehand and will have to be determined by applying the inverse Squire s transformation after the equivalent two-dimensional problem formulated above is solved. However, if as the result of solving the equivalent two-dimensional problem it is established that the instability occurs as Gr is gradually increased while Gr m remains fixed than a definite conclusion can be made about the orientation of the perturbation patterns without the necessity of inverting Squire s transformation. Indeed according to (3.6.1) while the value of the disturbance amplification rate is invariant with respect to Squire s transformation (σ = σ) the value of Gr for the original three-dimensional problem is always larger than that of Gr for the equivalent two-dimensional problem: Gr = α α Gr = α2 + β 2 Gr Gr. α This means that in this case instability indeed first sets in the form of two-dimensional y periodic patterns that correspond to β = and the full original and reduced Squire-

124 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 1 transformed problems have identical solutions. This fact will be used extensively in the discussion of the obtained numerical results in Section 3.7. Note that it follows from the problem geometry and Squire s transformation (3.6.1) that αh e y + βh e z H e = α sin δ cos γ + β sin δ sin γ = α H y = α sin δ cos γ, He or, for an oblique field with δ =, α cos γ + β sin γ = α 2 + β 2 cos γ. so that γ = tan 1 β α ± γ. (3.6.1) In particular, if β = then γ = ± γ. However when α = then γ = 9 ± γ. It is convenient to choose γ as an independent problem parameter characterising the magnetic field orientation keeping in mind its meaning given by (3.6.1). 3.7 Numerical Results Comparison with Selected Previous Numerical Results For each set of physical governing parameters, the problem is solved for a range of wavenumber α to locate the maximum of the disturbance amplification rate σ R as seen, for example, in the left plot in figure 3.2. Then the values of Gr are iteratively modified until the maximum value of σ R becomes smaller than the given threshold value of 1 5. Full stability diagrams discussed in Section are obtained by repeating the computational process for different values of Gr m. To compare with previous results the critical values of the governing parameters for the convection threshold in a perpendicular (δ = ) external magnetic field with magnitude H e =1 have been computed. The gravitational convection threshold ( Gr m = ) is com-

125 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION σ R 2 σ I α α Figure 3.2: Leading disturbance temporal amplification rates σ R (left) and frequencies σ I (right) as functions of the combined wavenumber α for ( Gr m, Gr) = (, ) (onset of thermogravitational convection) at δ = γ =, χ = χ = 5 and Pr = 55. puted for Prandtl number Pr =.71. The critical values Gr = and α = 1.45 are obtained, which after multiplying by the corresponding factors of 16 and 2 (appearing due to the use of different temperature and length scales in non-dimensionalisation), respectively, agree with the previously known accurate result reported in Suslov & Paolucci (1995b). The onset of gravitational convection is also computed for Pr = 7 and the set of critical values ( Gr m =, Gr = , α = 1.38) is obtained, which agree closely with those critical values presented in Belyaev & Smorodin (21). The gravitational convection threshold is also computed for Pr = 13. The obtained critical values Gr = and α = are identical to those reported in Suslov (28). The critical values for the magnetic convection threshold ( Gr = ) for the case of Pr = 13 and χ = χ = 4 are computed and the critical values Gr m = and α = are obtained, which agree well with the values of Gr m = and α = 1.95 computed from the corresponding data reported in Finlayson (197). The magnetic convection threshold for Pr = 13 and χ = χ = 5 is also determined and the critical values Gr m = and α = are obtained, which are identical to those reported in Suslov (28). In addition, the critical values for mixed convection for the case of Pr = 13 and χ =

126 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION σ R 1.5 σ I α α Figure 3.3: Same as Figure 3.2 but for ( Gr m, Gr) = (3.35, ) (onset of stationary magneto-convection). χ = 5 are computed. Two sets of critical values ( Gr = 16.69, Gr m = , α = ) and ( Gr = , Gr m = 1.4, α = ) are obtained, which agree closely with those reported in Suslov (28). 3.8 Flow Stability Characteristics Stability Characteristics of Flows in a Normal Field The stability characteristics of a ferromagnetic fluid flow in a normal magnetic field have already been investigated by Suslov (28). The author discussed only flow stability in linearly magnetized fluid with the specific Prandtl number of Pr = 13. In the current study, comprehensive stability characteristics of convection flow are presented for different values of Prandtl number, arbitrary field inclination angles and both linear and non-linear magnetization of fluids. Three main types of instability patterns corresponding to thermogravitational, magnetic and magneto-gravitational convection are found to exist in normal magnetic field. The

127 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION σ R 1.5 σ I α α Figure 3.4: Same as Figure 3.2 but for ( Gr m, Gr) = (43.3, 11.75). In the left plot the left and right maxima correspond to small- and large-wavenumber waves, respectively, and the middle maximum corresponds to a stationary roll pattern. H e H e (a) (b) (c) Figure 3.5: Schematic diagram of main flow patterns in an oblique magnetic field: (a) thermogravitational waves; (c) stationary vertical thermomagnetic rolls; (d) oblique thermomagnetic waves.

128 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 14 respective typical eigenvalue curves are shown in figures 3.2, 3.3 and 3.4. In the first case ( Gr m ) one maximum of the disturbance amplification rate σ R exists with complex conjugate eigenvalues (see figure 3.2) that indicate the existence of two counterpropagating waves corresponding to the thermogravitational convection instability (Suslov 28). The corresponding perturbation eigenfunctions are shown in figure 3.6. In the second case ( Gr ) one maximum of the disturbance amplification rate σ R is also found but now eigenvalues are real (see figure 3.3). This situation corresponds to a stationary magneto-convection pattern. Figure 3.7 demonstrates the perturbation eigenfunctions corresponding to this stationary magneto-convection pattern. In the third case ( Gr =, Gr m = ) up to three maxima of the disturbance amplification rate σ R (see figure 3.4) can exist, of which the left-and right-most maxima correspond to small- and large-wavenumber waves while the middle one corresponds to a stationary magneto-convection pattern. These instability modes will be referred to as Type I, III and II, respectively. The typical spatial patterns corresponding to these three instability types are shown schematically in figure 3.5 and their cross-layer structures are presented in figures 3.8, 3.9 and 3.1, respectively. The comparison of figures 3.6 and 3.8 shows that the wave patterns corresponding to Type I instability detected when both Gr and Gr m are nonzero are qualitatively similar to the thermogravitational waves even though magnetic effects are always present when Gr m =. The analysis of the influence of such magnetic effects will be given in Section 3.9. The comparison of figures 3.7 and 3.9 also reveals the similarity between Type II instability detected for nonzero Gr and magnetoconvection rolls. However the centrally located Type II instability rolls are skewed compared to the pure magneto-convective rolls observed when Gr =. Such a deformation of rolls is caused by the shearing influence of the basic flow velocity field with the cubic profile given by (3.4.2) where fluid rises near the hot left wall and sinks near the cold right wall. The exact mechanisms causing the appearance of the additional Type III instability mode

129 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 15 Figure 3.6: Perturbation eigenfunctions of the fluid velocity v 1, temperature θ 1, magnetization M 1 and magnetic field H 1 for magneto-convection in a normal field (δ = ) for H e = 1, χ = χ = 5 and at the critical point for Gr m =, Gr = , and α = are not clear, even though the corresponding perturbation fields also appear to be similar to thermogravitational waves, compare figures 3.6 and 3.1. A detailed analysis of this instability mode will also be given in Section 3.9. Numerical values of the critical parameters for these three types of convection in a perpendicular field for various values of Prandtl number Pr are obtained by solving equations (3.6.2) (3.6.9) and are given in Table 3.1. As follows from Table 3.1 the increase in Prandtl number leads to the increase of a wavenumber when the magnetic effect is smaller. As a result the distance between two instability rolls decreases. However, when the magnetic effect is stronger, primarily the increase in Prandtl number leads to the increase of a wavenumber and again decreases it. The set of critical values ( Gr m =, Gr = 15.86, α =.82) for Pr = 2 presented in Table 3.1 is identical to that reported by Belyaev & Smorodin (21). When χ = χ that is if the fluid magnetization depends linearly on the magnitude of the applied magnetic field, then the critical parameter values for external normal magnetic field H e = 1 are identical to those for H e = 1 presented in Table 3.1. Thus, the conclusion made in Finlayson (197) and Suslov (28) that

130 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 16 Figure 3.7: Same as figure 3.6 but for Gr m = 3.35, Gr = and α = Figure 3.8: Same as figure 3.6 but for Gr m = 43.3, Gr = and α =

131 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 17 Figure 3.9: Same as figure 3.6 but for Gr m = 43.3, Gr = and α = Figure 3.1: Same as figure 3.6 but for Gr m = 43.3, Gr = and α =

132 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 18 Table 3.1: The critical values of Gr m, Gr, α, disturbance wave speed c = σ I / α and the maximum speed of the basic flow ṽ max for mixed convection in a perpendicular (δ = ) external magnetic field H e = 1 at χ = χ = 5 and various values of Prandtl number Pr. Pr Gr mc Gr c α c c c ṽ max ± ± ± ± ± ± ± ± ± ± ± ± in this case the stability characteristics of the flow are independent of the magnitude of the applied normal magnetic field is confirmed. As follows from the data in Table 3.1 the onset of a stationary magneto-convection (for Gr = ) occurs at wavenumber that is independent of the values of Prandtl number Pr. The data in the table also confirms that the stationary magneto-convection pattern is characterized by the Grashof number Gr m that is inversely proportional to Pr. Therefore it is concluded that consistent with the analysis given in Chapter 2 the governing parameter responsible for the appearance of pure magnetic convection is magnetic Rayleigh number Ra m = Pr Gr m rather than the magnetic Grashof number Gr m. Note that the disturbance wave speed of gravitational instability modes becomes larger than the maximum basic flow velocity in a large Prandtl-number fluid meaning that the

133 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 19 physical nature of instabilities has nothing to do with the basic flow velocity field. It is also observed that the basic flow becomes less stable when Prandtl number increases. Since Prandtl number is the ratio of fluids viscosity and thermal diffusivity, the large Prandtl number corresponds to small thermal diffusion. Therefore thermal disturbances dissipate slowly in large Prandtl-number fluids. Thus based on the data presented in Table 3.1, it is concluded that the physical nature of instabilities observed for Gr m = is thermally dominated and because of that these waves are called thermal (Gershuni et al. 1989). Thermal waves propagate upward near the hot wall and downward near the cold wall. This conclusion remains mostly true for magneto-gravitational convection when both Gr and Gr m are non-zero. More details will be given in Section 3.9. The representative critical values of Gr, α and disturbance wave speed c for two thermomagnetic waves in a normal magnetic field at Gr m = 12 and Pr = 55 are given in Table 3.2 for various values of magnetic susceptibilities χ and χ. In the case of linear magnetization law i.e. when χ = χ the two waves propagate with equal speeds in the opposite directions. The disturbance waves are characterized by the same wavenumber and the basic flow becomes unstable with respect to the upward and downward propagating waves simultaneously. However, when the values of χ and χ differ i.e. in the case of non-linear magnetization law close to the magnetic saturation the symmetry of wave propagation is broken and the upward propagating wave near the hot wall becomes more dangerous compared to the downward propagating wave near the cold wall (see Table 3.2). The upward propagating wave is characterized by a slightly larger wavenumber than that of a downward propagating wave Wave-like Instabilities in Oblique Fields The representative critical values similar to those given in Table 3.2 for the two counterpropagating waves corresponding to the Type I instability in an oblique magnetic field for various inclination angles are presented in Tables 3.3 and 3.4. As follows from the data in these tables the basic flow becomes more stable when the field inclination angle increases.

134 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 11 Table 3.2: The critical values of Gr, α and disturbance wave speed c = σ I / α for leading two waves of mixed convection in a normal magnetic field (δ = ) for Gr m = 12, H e = 1, Pr = 55 and various values of χ and χ. Upward propagating wave δ = Downward propagating wave χ χ α c Gr c c c α c Gr c c c The wavenumber of the disturbance waves decreases and as a result the distance between two instability rolls increases. The disturbance waves also propagate quicker with the increase of the field inclination angle. In contrast to the normal field, the oblique magnetic field leads to the asymmetry in wave propagation regardless of whether the fluid magnetization is linear or not in the flow domain. Tables 3.3 and 3.4 confirm that the upward propagating waves in various oblique magnetic fields are always characterized by larger wave numbers than those of the downward propagating waves. In the case of a linear magnetization law the basic flow is less stable with respect to the upward propagating wave compared to the downward propagating wave in an oblique magnetic field and thus the upward propagating wave remains the most dangerous as follows from the data presented in Tables 3.3 and 3.4. The wave speed of the upward propagating wave is always smaller than that of the downward one. However, in the case of a non-linear magnetization law the destabilizing roles of the two waves swap near the magnetization saturation. Thus it is concluded that in general the stability characteristics can change qualitatively when the strength of the applied inclined magnetic field increases, which is not the case with normal magnetic field. Note from equation (2.3.9) that the magnitude of the non-dimensional magnetic field is

135 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 111 Table 3.3: The critical values of Gr, α and disturbance wave speed c = σ I / α for the upward propagating wave of mixed convection in an oblique magnetic fields for Gr m = 12, γ =, Pr = 55, H e = 1 (odd-numbered lines), H e = 1 (even-numbered lines) and various values of χ and χ. δ = 5 δ = 1 δ = 15 χ χ α c Gr c c c α c Gr c c c α c Gr c c c Table 3.4: Same as Table 3.3 but for the downward propagating wave. δ = 5 δ = 1 δ = 15 χ χ α c Gr c c c α c Gr c c c α c Gr c c c

136 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 112 proportional to parameter N, which is inversely proportional to the pyromagnetic coefficient that characterizes the sensitivity of the fluid magnetization to the temperature variation. The large value of non-dimensional magnetic field corresponds to the weaker temperature dependence of the fluids magnetization and vice versa. Thus, the magnitudes of non-dimensional external magnetic field H e = 1 and H e = 1 correspond to thermomagnetically less and more sensitive fluids, respectively. As follows from Tables 3.3 and 3.4 regardless of whether the fluid magnetization is linear or not the upward propagating wave in a thermo-magnetically more sensitive fluid is characterized by a larger wavenumber and the basic flow becomes less stable there than in a thermo-magnetically less sensitive fluid when an oblique magnetic field is applied. Whatever the fluid magnetization law is, the upward disturbance wave in a less thermo-magnetically sensitive fluid propagates quicker than that in a more thermo-magnetically sensitive fluid (compare the data for H e = 1 in odd-numbered and H e = 1 in even-numbered lines in Table 3.3). On the other hand, regardless of whether the fluid magnetization is linear or not the downward propagating wave in a more thermo-magnetically sensitive fluid is characterized by a smaller wavenumber compared to that in a less thermo-magnetically sensitive fluid in an arbitrarily inclined magnetic field. The basic flow in a more thermo-magnetically sensitive fluid becomes more stable with respect to a downward propagating wave and its disturbance wave speed becomes faster compared to that in a less thermo-magnetically sensitive fluid (see the data in Table 3.4). To this point the dependence of flow stability characteristics on the values of Pr, χ, χ and the field inclination angle δ has been investigated for the zero azimuthal field orientation angle γ. To investigate the influence of the field orientation angle γ stability results are computed for a representative value of Gr m = 12. In the case of linear magnetization law the critical parameter values as functions of the magnetic field inclination angles are shown in figure The flow is stable in the regions below the respective curves in figure 3.11(a). Therefore according to the discussion given in Section 3.6 this type of instability occurs in the form of two-dimensional patterns that are α = α-periodic in the vertical y direction and in this case γ = γ. Regardless of the field orientation

137 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 113 Gr c (a) δ =5 δ =1 δ =15 αc (b) cc (c) γ, γ, γ, Figure 3.11: Comparison among the critical parameter values: (a) Grashof number Gr (the flow is stable under the respective curves), (b) wavenumber α and (c) wave speeds c as functions of the field inclination angles δ and γ for Gr m = 12, H e = 1, Pr = 55 and χ = χ = 3. characterised by γ the basic flow becomes more stable at larger field inclination angles δ. This is primarily due to the geometric reduction of the active normal component of the applied magnetic field, which is proportional to cos δ (see discussion in Chapter 2). With the increase of the field inclination angle δ the wavenumber decreases (see figure 3.11(b)) and the distance between the instability rolls increases. It follows from figures 3.11(c) that as the field inclination angle increases the wave speed also increases. The numerical results for a stronger magnetizable fluid with χ = χ = 5 are shown in figure While the qualitative behaviour of the parametric curves remains similar a quantitative change occurs in the values of critical parameters. Namely, the basic convection flow of a stronger magnetizable fluid is found to be generally more stable for all field orientation angles γ, and its instabilities patterns are characterised by a smaller wavenumber and a faster wavespeed (compare figures 3.11 and 3.12). The critical values of parameters for the case of non-linear magnetization law closer to magnetic saturation i.e. for χ = χ are presented as functions of the magnetic field inclination angles in figure In this case the comparison with the linear magnetization case presented in figure 3.11 does not reveal any clear general trends but rather indicates that the stability characteristics depend on a particular combination of the values of δ,

138 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 114 Gr c (a) δ =5 δ =1 δ = γ, αc γ, (b) cc Figure 3.12: Same as figure 3.11 but for χ = χ = (c) γ, Grc (a) δ =5 δ =1 δ = γ, αc Figure 3.13: Same as figure 3.11 but for χ = 1.5 and χ = 2.5. γ, (b) cc (c) γ, γ, χ and χ. For example, the proximity of fluid s magnetic saturation makes the basic flow less stable compared with the regimes away from it when (δ, γ) = (5, ), but the influence of the magnetic saturation at (δ, γ) = (5, 18 ), and even more so for (δ, γ) = (15, 18 ), is opposite: the basic flows becomes more stable. Yet the comparison of all three figures 3.11, 3.12 and 3.13 indicates that generally stability of the basic flow is influenced by the value of the differential magnetic susceptibility χ to a larger degree than by the value of the integral susceptibility χ, and generally the larger χ is the more stable the basic flow is, the smaller the critical wavenumber is and the larger the critical wave speed is. As seen from figures 3.11(a), 3.12(a) and 3.13(a) the instabilities detected for γ = 18

139 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 115 occur at noticeably higher values of Gr c than those for γ =. This indicates that the updown symmetry of the field influence is broken. This could be traced back to the curvature of magnetic field lines within the layer of ferrofluid discussed in Chapter 2. Specifically, as follows from figures in absence of the gravitational field changing the field orientation angle γ from to 18 degrees reverses the sign of the curvature of magnetic field lines and this in turn leads to the reversal of the sign of the wave speed of thermomagnetically driven disturbances. At the same time when the gravity is accounted for, the computational data reported so far indicates that the upward propagating wave near the hot wall remains most dangerous. Thus changing the field orientation angle γ by 18 degrees leads to the change from the situation when gravitationally and thermo-magnetically induced wave speeds are in the same direction to that when they oppose each other and the overall instability is suppressed in the latter case compared to the former situation Optimal Orientation of Wave-like Instabilities The most prominent feature of figures 3.11(a), 3.12(a) and 3.13(a) is the existence of the minima of the Gr c (γ) curves. Such minima are more-pronounced in stronger magnetizable fluids characterised by the larger value of χ. Their existence demonstrates that for each field inclination angle δ there exists a preferred field orientation angle γ that promotes the onset of magneto-gravitational instability the most. As shown in Chapter 2 in zero gravity environment the most dangerous instability patterns are always stationary and are aligned with the in-layer component of the applied magnetic field. It is also known that in the absence of magnetic field (i.e. when Gr m = ) the arising thermogravitational waves consist of the propagating structures with horizontal axes. Therefore it is intuitive to expect that the least stable situation when both Gr and Gr m are nonzero would occur when the direction of the in-layer component of the applied oblique magnetic field is horizontal that is when γ = 9. Yet the computational results presented in figures 3.14(a), 3.15(a) and 3.16(a) show that regardless of the degree of fluid magnetisation or its proximity to saturation the field orientation angle γ min for which the

140 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION (a) (b) γcmin, 4 Gr cmin δ, δ, αcmin (c) ccmin (d) δ, δ, Figure 3.14: (a) The value of the field orientation angle γ min at which the instability first occurs and (b d) the corresponding critical parameters as functions of the field inclination angle δ for Gr m = 12, H e = 1, Pr = 55 and χ = χ = 3. instability first occurs tends to 9 only for sufficiently large field inclination angles δ, that is when the in-layer component of the magnetic field becomes sufficiently large. When such a component is small (δ 2 4 ) the instability is virtually insensitive to the field orientation, but for larger field inclination angles the behaviour of γ min undergoes a bifurcation-type change and it becomes a sensitive function of δ. The likely reason for such a peculiar behaviour can be traced back to the fact reported in Chapter 2 for pure magnetic convection: the most amplified instability patterns there are aligned with the in-layer component of the applied magnetic field. Therefore intuitively it is expected that horizontally uniform thermal waves appearing in the differentially heated vertical layer of a large-prandtl-number fluid would be most supported when the applied magnetic field has a fully horizontal in-layer component, that is if Hy e =, or γ = 9. However as discussed in Chapter 2 the most amplified magnetically driven component of the instability for the so-applied field would be stationary and thus would, to some degree,

141 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 117 γcmin, (a) Gr cmin (b) δ, δ, 1.2 (c) 4.1 (d) 4 αcmin 1.18 ccmin δ, δ, Figure 3.15: Same as figure 3.14 but for χ = χ = (a) (b) γcmin, 4 2 Gr cmin δ, δ, (c) 2.5 (d) αcmin ccmin δ, δ, Figure 3.16: Same as figure 3.14 but for χ = 1.5 and χ = 2.5.

142 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 118 hinder the development of the propagating thermogravitational waves. It was also shown in Chapter 2 that the instability patterns that are not aligned with the in-layer component of the magnetic field (i.e. observed for γ = 9 in the current context) while characterised by a smaller growth rate have a non-zero wave speed. Such non-stationary wave-like patterns therefore are favoured when magnetic instability overlaps with the vertically propagating horizontally uniform thermogravitational waves. Therefore the choice of γ min = 9 appears to be due to the competition between the two optimality criteria: maximizing the amplification rate of combined thermogravitational and thermo-magnetic instabilities and matching their propagation speeds. It follows from figures 3.14(b), 3.15(b) and 3.16(b) that there always exists the overall optimal orientation of magnetic field (δ min, γ min ), which minimizes the value of the critical Grashof number Gr c,min. In particular, for Gr m = 12 and Pr = 55, δ min 2 and γ min =. Such a global minimum corresponds to a disturbance with a wavenumber that is close to α c,min 1.21, see figures 3.14(c), 3.15(c) and 3.16(c). Figures 3.14(d), 3.15(d) and 3.16(d) indicate another noteworthy feature of the wave-like instabilities detected in an oblique field: the upward propagating wave remains the most dangerous for all optimal field orientations. While the computational results show that the up-down symmetry remains broken in all flow regimes and the basic flow becomes unstable with respect to the upward propagating wave first when the oblique magnetic field is applied the difference between the critical characteristics of the upward and downward propagating waves remains relatively small. Therefore from an experimental point of view considering both waves is important. The comparison of their critical parameters is presented in figures 3.17, 3.18 and 3.19 for a representative field inclination angle δ = 5. When the fluid s magnetization law is linear the critical parameter curves for both waves have qualitatively similar forms regardless of the degree of fluid magnetization, compare the results for χ = 3 and χ = 5 shown in figures 3.17 and Quantitatively though

143 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION (a) (b) (c) Gr c 6 59 γ =39, Gr = γ, γ =39, Gr = αc γ, cc Figure 3.17: Comparison of the critical parameter values for the upward (solid line) and downward (dashed line) propagating waves: (a) Grashof number Gr (the flow is stable 63 under the respective curves), (b) wavenumber α and (c) wave speeds c as functions of the azimuthal angle γ for Gr m = 12, H e = 1, Pr = 55, δ = 5 and χ = χ = 3. γ, Gr c (a) γ c, (b) (c) Gr c 62 αc 1.16 cc γ =6, Gr = γ, γ, γ, Figure 3.18: Same as figure 3.17 but for χ = χ = Gr c γ =6, Gr = 6.45 γ =6, Gr = 6.45

144 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 12 Gr c (a) 58 γ =39, Gr = γ, αc γ, Figure 3.19: Same as figure 3.17 but for χ = 1.5 and χ = 2.5. (b) cc γ, (c) Gr c 64 the differences between the critical parameters for the two waves are more evident for 63 γ < 9 that is when magnetic field lines cross the fluid layer from hot to cold wall slightly 6 shorter than that of the downward waves. For γ > 9 the basic flow becomes 59 unstable with respect to both waves almost at the same values of parameters. The main 58 γ =39, Gr = upward. For such a field orientation the wavelength of the upward propagating waves is quantitative difference between the results obtained for χ = 3 and χ = 5 is in the values of the optimal field γ c, orientation angle γ min. For a weaker magnetizable fluid with χ = 3 it is about 39 while for χ = 5 it is close to 6. Consistent with the previous discussion the optimal field orientation angle approaches 9 when the strength of magnetic effects increases. The main qualitative observation for the case of a nonlinear fluid s magnetization law is that the critical values for the two waves are clearly distinguished regardless of the field orientation angle, see the solid and dashed lines in figure Therefore the degree of the up-down symmetry breaking increases in regimes near fluid s magnetic saturation. In conclusion of this section the stability characteristics of the basic flow with respect to wave-like disturbances will be compared for thermo-magnetically less (H e = 1) and more (H e = 1) sensitive fluids, see figures 3.2, 3.21 and The upward propagating wave remains the most dangerous in both types of fluids in all regimes. Thus only critical parameters for the upward propagating waves are shown in these figures. There are a

145 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 121 number of general trends that are evident from figures Firstly, the flows of thermo-magnetically more sensitive fluids are less stable than those of their less sensitive counterparts. This distinction in the critical values of magnetic Grashof number is most evident when the applied magnetic field has a relatively large vertical component (γ close to or 18 degrees). When the applied oblique magnetic field is mostly horizontal γ 9 that is when the curvature of the magnetic field lines within the fluid layer is in the plane perpendicular to the direction of gravity the magnetic sensitivity of a fluid does not appear to play a significant role in defining the flow stability parameters. Secondly, the wave-like instability patterns arising in a more thermo-magnetically sensitive fluid are characterised by a larger wavenumber and thus by convection structures that are closer packed in the direction of gravity. Thirdly, instability waves arising in a thermo-magnetically more sensitive fluid have a somewhat smaller wave speed. Therefore increasing fluid s thermo-magnetic sensitivity effectively quenches the propagation of disturbance waves which is consistent with the findings reported in Chapter 2 that the most amplified thermo-magnetically driven instability patterns remain stationary in the absence of gravity. These trends are not affected when a fluid approaches its magnetic saturation and the magnetization law becomes nonlinear and is characterized by non-equal values of χ and χ, see figure Complete Stability Diagrams for an Equivalent Two-dimensional Problem The flow instability properties associated with the wave-like disturbances have been discussed in detail in Section The goal of this section is to identify parametric regions where different physical mechanisms lead to the onset of instability in the considered geometry. To do that the complete stability diagrams for an equivalent two-dimensional problem are computed. We start with the discussion of typical stability diagrams for a normal magnetic field

146 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION (a) H e = 1 H e = (b) 4 (c) Gr c γ, αc γ, cc Figure 3.2: Comparison of the critical parameter values for thermo-magnetically less (H e = 1, solid line) and more (H e = 1, dashed line) sensitive fluids: (a) Grashof 61.5 number Gr (the flow H e = is 1 stable under the respective curves), (b) wavenumber α and (c) wave speeds c as functions of the azimuthal angle γ for Gr 61 m = 12, Pr = 55, δ = 5 and χ = χ = 3. Type I (thermogravitational convection) instability. γ, Gr c Gr c (a) H e = 1 H e =1 γ, αc (b) cc (c) γ, γ, γ, Figure 3.21: Same as figure 3.2 but for χ = χ = (a) H e = 1 H e = 1 H e = (b) (c) Gr Gr c c αc cc γ, γ, γ, γ, Figure 3.22: Same as figure 3.2 but for χ = 1.5 and χ = H e = 1 61

147 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION (a) (b) 5 (c) Grc 4 αc 2.5 cc Gr mc Gr mc Gr mc Figure 3.23: (a) Stability diagram for an equivalent two-dimensional problem; (b) the critical wavenumber α c and (c) the corresponding wave speeds along the stability boundaries shown in plot (a) for H e = 1, Pr = 55 and χ = χ = 5 in a normal magnetic field (δ = ). Parametric curves corresponding to the Type I, II and III instabilities are shown by the solid, dashed and dash-dotted lines, respectively. Grc (a) αc (b) cc (c) Gr mc Gr mc Gr mc Figure 3.24: Same as Figure 3.23 but for χ = χ = 3. shown in figures 3.23, 3.24 and While computed for different values of Prandtl number and magnetic susceptibilities, qualitatively, all diagrams are similar to that given in Fig. 2 for Pr = 13 in Suslov (28) demonstrating that the dependence of the flow stability characteristics on thermo-viscous properties of the fluid placed in the normal magnetic field and on its magnetic susceptibilities is just quantitative. Therefore only the detailed discussion of figure 3.23 will be given below and this discussion will follow closely that of Suslov (28). The stability diagram consists of three lines each representing a different type of instability characterized by its own wavenumber as follows from figure 3.23(b). The solid line in figure 3.23(a) starts from Gr m =, which corresponds to the threshold of a classical thermogravitational convection instability. As discussed earlier this is the Type I instability characterised by two counterpropagating waves investigated in (Gershuni et al. 1989,

148 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 124 Grc Gr mc (a) αc (b) Gr mc cc Gr mc (c) Figure 3.25: Same as Figure 3.23 but for χ = 1.5 and χ = 2.5. Chait & Korpela 1989, Wakitani 1996, Suslov 28). The basic flow is unstable with respect to such an instability above the solid line in figure 3.23(a). Therefore as discussed in Section 3.6 the Type I instability corresponds to vertically propagating patterns with α = α and β = that are y-periodic and uniform in the horizontal z-direction. Recollect that in this case γ = γ. The dashed line in figure 3.23(a) starts from Gr = and therefore corresponds to the threshold of magneto-convection. In this case, the disturbance amplification rate σ R is real (see figure 3.3). As follows from an earlier discussion this is the Type II instability that is characterized by a stationary pattern (see also Finlayson (197), Smorodin et al. (27)). The basic flow is unstable below the dashed line in figure 3.23(a) and therefore to determine the spatial orientation of such patterns an additional analysis of the inverse Squire s transformation is required. To perform it refer to figure 3.26(a), where the linear amplification rate σ = σ is plotted as the function of the Squire-transformed ( two-dimensional ) Grashof number Gr (see relationships (3.6.1)). The maximum amplification rate σ R is detected for the Type II instability (the dashed line) when Gr. Since according to Squire s transformation σ = σ the same amplification rate σ R = σ R will be observed in the original non-transformed three-dimensional system provided that a non-transformed ( three-dimensional ) Grashof number Gr is linked to Gr as αgr = α 2 + β 2 Gr, that is if αgr or α. This means that the maximum amplification rate for the

149 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION (a) 2.5 (b) 5 (c).2 2 σ R.1 α c Gr Gr Gr Figure 3.26: (a) Maximum amplification rate for an equivalent two-dimensional problem; (b) the corresponding wave numbers α and (c) wave speeds for Gr m = 12, H e = 1, Pr = 55 and χ = χ = 5 in a normal magnetic field (δ = ). The solid and dashed lines correspond to the Type I and II instabilities, respectively. Type II instability at any value of Grashof number will be observed for vertically oriented rolls corresponding to α and β α, where α is the left-most value along the dashed line in figure 3.26(b). In other words, the Type II instability in the form of vertical magneto-convection rolls aligned with the y-axis (so that γ = γ + 9 for this instability mode) will arise for arbitrary values of Gr once Gr m = Gr m excedes the critical value corresponding to the left-most point along the dashed curve in figure 3.23(a). For any values of Gr exceeding the values corresponding to the solid line in figure 3.23(a) the stationary vertical rolls of the Type II instability will overlap with the vertically propagating Type I instability waves. Given that the inverse Squire s transformation states that for sufficiently large values of Gr m the flow is always unstable with respect to vertical thermo-magnetic rolls regardless of the value of Gr, the physical meaning of the dashed line in figure 3.23(a) needs to be clarified. Simple arguments can be developed that while vertical rolls with α = and β = are always most dangerous, rolls of all other orientations are also unstable below the dashed line, but not above it. Indeed, when the gravity is absent and Gr = the arising magneto-convection rolls can be arbitrarily oriented as all directions along the fluid layer are fully equivalent. When the gravity is switched on and Gr becomes non-zero the basic gravitational convection flow arises, which removes the spatial degeneracy so that the vertically oriented rolls are preferred. Yet it is clear from the physical

150 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 126 Table 3.5: The representative critical values of the Squire-transformed Grashof number Gr, wavenumber α and disturbance wave speed c = σ I / α for the Type II istability in a normal magnetic field (δ = ) for Gr m = 12, H e = 1, Pr = 55 and various values of χ and χ. χ χ α c Gr c c c continuity arguments that at small values of Gr rolls of all other orientations still can exist as in the Gr limit even though vertical rolls now have a larger growth rate. As the value of Grashof number increases the growth rate of the vertical rolls remains the same as long as the value of Gr m remains fixed, but the growth rate of non-vertical rolls becomes smaller. As Grashof number increases to the value corresponding to the dashed line in figure 3.23(a) horizontal rolls cannot grow anymore and disappear. Above the dashed line there exists a maximum value of the Type II roll orientation angle beyond which they cannot be observed. To illustrate that refer to the data in Table 3.5. It shows that in the normally applied field the flow stabilisation occurs when the value of the Squire-transformed Grashof number Gr exceeds 3.24 for Gr m = Gr m = 12 and χ = χ = 5 (this corresponds to the point in figure 3.26(a) where the dashed line crosses zero and to the respective critical point ( Gr mc, Gr c ) = (12, 3.24) on the dashed line in figure 3.26 (a)). Say, the experimental value of interest is Gr = 1. Then the presented analysis enables one to conclude that in such experimental conditions it is expected that the instability patterns will be in the form of vertical stationary rolls that however could be modulated by weaker rolls with the axes forming the angle of up to sin 1 α α c = sin 1 Gr c Gr = sin with the vertical y-direction. The above example also shows that the larger the experi-

151 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 127 mental value of Gr is the smaller the modulation angle for vertical rolls becomes. This has a straightforward physical explanation: the increase in the value of Grashof number leads to the increase of the vertical basic flow velocity component (see equation (3.4.2)), which in turn leads to a stronger vertical alinement of the vertical instability patterns. The data in Table 3.5 also shows that such an oblique modulation of the preferred vertical rolls becomes non-stationary when the magnetisation law of a fluid becomes non-linear: due to the symmetry breaking effects in this case the stationary vertical rolls may be modulated with oblique patterns predominantly propagating in the positive y direction (against the direction of gravity). The Type III instability boundary is shown in figure 3.23(a) by the dash-dotted line. The basic flow is stable below it. As seen from figure 3.23(b), the corresponding instability patterns have larger wavenumbers (dash-dotted line) than those of the Type I and Type II instabilities (solid and dashed lines, respectively). As follows from figure 3.23(c)), similar to the Type I instability, the Type III instability arises in the form of two counterpropagating waves. It becomes larger than that of the Type I waves for sufficiently large values of Gr m. The peculiar feature of the Type III instability that is evident from figure 3.23(a) is that the corresponding instability boundary ends abruptly at certain values of gravitational and magnetic Grashof numbers. To explain this unusual behaviour refer to figures 3.27 and The Type III instability arises when the disturbance amplification rate becomes positive at relatively large wavenumbers, see the maximum labelled as 2 in the left plot in figure However such a maximum disappears abruptly at the left end of the dashdotted line in figure 3.23(a) when the eigenvalue branch denoted by 1 in figure 3.27 and corresponding to the Type II instability extends toward the large wavenumbers as Gr decreases and absorbs the maximum labelled by 2. In contrast, for large values of Gr the amplification rates maxima for the Type I and Type III instabilities coalesce, which also leads to the abrupt disappearance of the Type III instability. In other words, the Type III instability appears as a result of a sudden qualitative change in the problem s

152 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION σ R.2.4 σ I α α Figure 3.27: Leading disturbance temporal amplification rates σ R (left) and frequencies σ I (right) as functions of the combined wavenumber α for ( Gr m, Gr) = (31.37, 5.) at δ = γ =, χ = χ = 5 and Pr = σ R.6 σ I α α Figure 3.28: Same as figure 3.27 but for ( Gr m, Gr) = (51.3, 21.).

153 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION (a) 4 3 (b) 5 (c) σ R.5 α 2 c Gr Gr Gr Figure 3.29: Same as figure 3.26 but for Gr m = 35. The solid, dashed and dash-dotted lines correspond to the Type I, II and III instabilities, respectively. dispersion relation when its branches corresponding to either Type I or Type II instabilities bifurcate resulting in the appearance of the Type III instability. Experimentally, the above discussion suggests that the appearance of the Type III instability can be detected either by a sudden transition from stationary (Type II) to unstationary (Type III) patterns at relatively small Gr (see figure 3.27) or from one unsteady pattern (Type I) to another (Type III) with a shorter wave length (see figure 3.28). Another feature distinguishing the Type III instability from its Type I and Type II counterparts, whose patterns are characterised by fixed a spatial orientation, is that the main periodicity direction for the Type III instability depends on the value of Grashof number. For example, as follows from figure 3.29(a) for Gr m = Gr m = 35 the Type III instability first occurs at Gr c = 4.84 (the left end of the dash-dotted line) in the form of vertically propagating waves with α = α and β =. However according to the inverse Squire s transformation, for any larger value of Grashof number it will be seen as a pair of oblique waves counterpropagating along the direction forming the angle cos 1 Gr c Gr with the vertical y direction. In other words, as Grashof number (and thus the vertical basic flow velocity) increases the Type III instability roll axes become closer to vertical and propagate almost horizontally while the Type I rolls remain horizontal and propagate vertically. For example, at Gr = 1 the Type III instability waves are expected to propagate along the direction forming the angle of cos 1 (4.84/1) 61 with the vertical direction rather than vertically and their axes are expected to form the angle of sin 1 (4.84/1) 29 with the vertical direction.

154 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION (a) (b) 5 4 (c) Gr c 4 αc cc Gr mc Gr mc Gr mc Figure 3.3: (a) Stability diagram for an equivalent two-dimensional problem; (b) the critical wavenumber α c and (c) the corresponding wave speeds along the stability boundaries shown in plot (a) for H e = 1, Pr = 55 and χ = χ = 5 in an oblique magnetic field for δ = 5 and γ =. The solid and dash-dotted lines correspond to the Type I/II and III instabilities, respectively. Grc (a) αc (b) cc (c) Gr mc Gr mc Gr mc Figure 3.31: Same as figure 3.3 but for χ = χ = 3. Gr c Gr mc (a) αc Gr mc (b) cc Gr mc (c) Figure 3.32: Same as figure 3.3 but for χ = 1.5 and χ = 2.5.

155 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 131 The magnetic field inclination adds further complexity to the already quite complicated instability picture in the presence of both gravitational and magnetic effects. The stability diagrams for an oblique magnetic field (δ = 5, γ = ) shown in figures 3.3, 3.31 and 3.32 are computed using the same parameter values as in figures 3.23, 3.24 and 3.25, respectively. The dependence of the stability characteristics on the type of fluid s magnetization law is found to be only quantitative so that all three figures are qualitatively similar. Thus only the first one of them will be discussed in detail. The diagrams show that the flow stability region becomes larger in an oblique magnetic field (compare, for example, the stability regions in figures 3.23(a) and 3.3(a)) and this is consistent with the numerical results given in Tables 3.2, 3.3 and 3.4. As follows from figure 3.3(b), similar to the normal field case the Type I instability is characterized by a smaller wavenumber (the solid line) compared to that of the Type III instability (the dash-dotted line). As discussed earlier, regardless of whether the magnetization law is linear or not the symmetry of the disturbance thermal waves propagation is broken in oblique field and the upward wave becomes more dangerous. Therefore, in figure 3.3(c) only the critical wave speed for this wave is shown. It increases monotonically with Gr. It is remarkable that the qualitative change in stability diagram occurs even for such small field inclination angles. The solid and dashed stability boundary lines distinguished in figure 3.23(a) merge in figure 3.3(a) indicating that the distinction between the Type I and Type II instabilities becomes blurred when the applied magnetic field is inclined in a vertical plane. The dash-dotted line in the lower right corner in figure 3.23(a) almost completely disappears meaning that the Type III instability could be hard to detect in inclined-field experiments. As follows from figure 3.3(a) even though the solid line originates from Gr c = it corresponds to non-stationary magneto-convection as the corresponding eigenvalues shown in figure 3.33 have non-zero imaginary parts. This is consistent with the results presented in Chapter 2 where it has been shown that the arising in a ferrofluid thermo-magnetic instability patterns not aligned with the in-layer component of the applied magnetic field are always non-stationary.

156 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION x σ R σ I α α Figure 3.33: Same as Figure 3.27 but for ( Gr m, Gr) = (6.4, ) and δ = 5, γ =. The solid and dashed lines represent the first and second leading eigenvalues σ of the linearised stability problem, respectively. The merging of the Type I and Type II instabilities is illustrated in figure As the values of magnetic and gravitational Grashof numbers increase along the solid line in figure 3.3 from there initial values ( Gr m, Gr) = (6.4, ) the two leading branches of the problem dispersion relation depicted in the left panel of figure 3.33 and typical to the Type II instability (compare with figure 3.3) deform and eventually meet at ( Gr m, Gr) = (13.88, 1.7) as seen from the left and right panels of figure The middle panel of figure 3.34 confirms that starting from this parametric point the two counterpropagating waves are formed, which is typical of the Type I instability. Further qualitative changes brought about by the field inclination occur in the orientation of the basic instability patterns. The analysis of figure 3.35 similar to that conducted earlier in this section shows that when the oblique applied magnetic field has only the vertical in-layer component (i.e. if γ = ), for relatively small values of magnetic Grashof numbers, the instability patterns consist of predominantly vertical rolls that overlap with vertically propagating waves appearing at sufficiently large values of Gr. When Gr m increases the vertical thermomagnetic rolls (Type II instability) are always present (they have positive growth rates as evidenced by the dashed lines in figures 3.36(a) and 3.37(a)), but the

157 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION σ R.15 σ I Δ σ R α α α Figure 3.34: Leading disturbance temporal amplification rates σ R (left), frequencies σ I (middle) and the amplification rate difference σ R = σ 1 R σr 2 as functions of the combined wavenumber α for ( Gr m, Gr) = (13.88, 1.7), δ = 5 and γ =. The solid and dashed lines represent the first and second leading eigenvalues σ of the linearised stability problem, respectively. orientation of the Type I instability patterns changes for smaller values of Gr. Namely, since the corresponding σ R ( Gr) is a decreasing function (see the segment of a solid line labeled as 2 in figure 3.37(a)) it follows from the inverse Squire s transformation that the Type I instability patterns tend to re-orient to approach vertical rolls. This is consistent with the conclusion made in Chapter 2 that the magnetic field, which has a vertical component in the discussed case, tends to align the instability patterns with it. To investigate the influence of the fluid s magnetic sensitivity the stability diagram shown in figure 3.38 has been computed for the same parameter values as in figure 3.3 but for a H e = 1. The two stability diagrams are very similar with the only qualitative difference being that the Type III instability completely disappeares from figure Due to this similarity in what follows we focus mostly on the influences of the field inclination and orientation angles so that only the results for H e = 1 will be discussed in detail unless stated otherwise. To facilitate the further discussion the stability diagrams computed for identical physical parameters but for various field inclination angles are collected in figure For conve-

158 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 134 σ R (a) α (b) c (c) Gr Gr Gr Figure 3.35: (a) Maximum amplification rate for an equivalent two-dimensional problem; (b) the corresponding wave numbers α and (c) wave speeds for Gr m = 12, H e = 1, Pr = 55 and χ = χ = 5 in an oblique magnetic field for δ = 5 and γ =. The solid and dashed lines correspond to the Type I and II instabilities, respectively..8.6 (a) (b) 5 4 (c) σ R.4.2 α 2 c Gr Gr Gr Figure 3.36: Same as figure 3.35 but for Gr m = 35. σ R (a) Gr α (b) Gr c (c) Gr Figure 3.37: Same as figure 3.35 but for Gr m = 5.

159 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 135 Gr c (a) αc (b) cc (c) Gr mc Gr mc Figure 3.38: Same as Figure 3.3 but for H e = 1. Gr mc nience of comparison figures 3.23 and 3.3 that have been discussed in detail earlier in this section are repeated in the top two rows of figure The magnetic field inclination angle continues to play a very important role in shaping the parametric stability boundaries of the considered flow. The stability diagram for an oblique magnetic field (δ = 8.5, γ = ) shown in the third row of figure 3.39 demonstrates that a significant stabilisation of the flow is observed when δ increases (the area bounded by the dashed and solid lines increases). The Type III instability disappears completely at the larger field inclination angles by merging with the Type I instability, see the solid line curve in the wavenumber plot (h) in the third row in figure 3.39 that now joins large and small wavenumber values. However the Type I and Type II instabilities are again easily distinguished even though the Type II instability boundary now corresponds to a non-stationary pattern, see the dashed line in plot (i) in the third row in figure The upward propagating waves still remain the most dangerous for the non-zero values of magnetic Grashof number. The lower segment of the solid line in plot (g) in the the third row in figure 3.39 still corresponds to the obliquely propagating Type I/III waves, see the discussion of figure 3.3 given above. The Type I instability arising at larger values of Gr (above the upper segment of the solid line in plot (g) in the third row) is characterised by vertically propagating waves and the most prominent Type II instability patterns are still vertically oriented. Figures 3.4, 3.41 and 3.42 clarify how the transition between the Type I/III and the Type II instabilities occurs in terms of the deformation of the branches of the problem dispersion relation. Near the transition pont the two types of instability correspond to the two distinct maxima of the disturbance amplification rate σ R. As the value of the gravitational Grashof number

160 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION (a) (b) 5 (c) Grc 4 αc 2.5 cc Gr mc Gr mc Gr mc 8 6 (d) (e) 5 4 (f) Grc 4 αc cc Gr mc Gr mc Gr mc 8 6 (g) (h) 5 4 (i) Grc 4 αc cc Grc 14 (j) Gr mc 5 1 αc 3.5 (k) Gr mc cc (l) Gr mc 5 1 Grc 2 (m) Gr mc 5 1 αc 2.5 (n) Gr mc cc 14 (o) Gr mc 5 1 Gr mc Gr mc Gr mc Figure 3.39: Variation of stability diagrams for H e = 1, Pr = 55 and χ = χ = 5 in an oblique magnetic field for γ = and (top to bottom rows) δ =, 5, 8.5, 1 and 15. The solid, dashed and dash-dotted lines correspond to the Type I, II and III instabilities, respectively.

161 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION σ R.15 σ I α α Figure 3.4: Leading disturbance temporal amplification rates σ R (left) and frequencies σ I (right) as functions of the combined wavenumber α for χ = χ = 5, Pr = 55, δ = 8.5, γ = and ( Gr m, Gr) = (197, 1.35). Type I/III instability. The solid and dashed lines represent the first and second leading eigenvalues σ of the linearised stability problem, respectively. decreases the maximum corresponding to the larger wavenumber (figure 3.4, the Type I/III instability) is suppressed and the maximum corresponding to the smaller wavenumber (figure 3.41, the Type II instability) becomes dominant. The wavenumbers of the two maxima remain distinct and therefore the switch between the two modes leads to the discontinuity of the wavenumber curves in the third row in figure With the further decrease of the gravitational Grashof number only one maximum survives (figure 3.42, the Type II instability). At even larger field inclination angles another qualitative change occurs. As seen from the stability diagrams in plots (j) and (m) in figure 3.39 for δ = 1 and δ = 15, respectively, the critical Gr values for the Type I instability now increase monotonically with Gr m. This is likely to be traced back to the aligning influence of the applied magnetic field. With the increasing field inclination angle δ and γ = the vertical in-layer field component increases as well and so does its pattern aligning effect. Thus the vertically propagating Type I waves aligned horizontally are suppressed by the applied oblique field and require a much stronger graviational buoyancy characterised by Gr to arise. The

162 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION σ R.15 σ I α α Figure 3.41: Same as Figure 3.4 but for ( Gr m, Gr) = (187.3, 8.9). The The Type II instability x σ R.25 σ I α α Figure 3.42: Same as Figure 3.4 but for ( Gr m, Gr) = (2.3, ). The Type II instability.

163 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 139 Type II instability characteristics presented for the field inclined at δ = 1 and shown in the fourth row in figure 3.39 change in a peculiar manner. For small values of magnetic Grashof number the Type II instability boundary (the basic flow is unstable below the dashed line) rises almost linearly and the critical wavenumber remains almost constant at α 2.5. This instability remains nearly stationary up to Gr m 3. However for larger values of Gr m the slope of the stability boundary changes rapidly (even though in a continuous manner) to a larger value and so does the value of α. The critical wave speed becomes nonzero and starts growing. Unfortunately, eigenvalue diagrams similar to those shown in figures do not provide a clear explanation to such a behaviour. Thus we postpone the further discussion of stability diagrams shown in figure 3.39 until Section 3.9, where the disturbance energy balance will be considered. The stability analysis presented so far shows that the upward propagating disturbance waves remain more dangerous compared to their counterparts moving downward. In the stability diagrams shown in figure 3.43 where the stability regions and characteristics for the two waves are compared for the complete range of Gr and Gr m for δ = 5 and γ =. Indeed the stability regions corresponding to upward propagating waves (enlooped by the solid lines in plots (a) ) are somewhat smaller than those for downward waves (enclosed by the dashed lines). This conclusion holds true for both stronger (χ = 5) and weaker (χ = 3) magnetizable fluids. The other qualitative feature distinguishing the complete stability diagrams is that at small values of the gravitational Grashof numbers the Type I wave instability characterised by the upward propagating waves smoothly transitions to the Type II instability, which, as discussed earlier in this section, for non-zero field inclination δ also leads to the wave propagating upward, although with a much smaller wave speed (see plots (c) in figure In contrast the downward propagating wave of the Type I instability ceases to exist at small values of Gr and is replaced by the Type III instability with a larger wavenumber, see plots (b) in figure Overall, however, from an experimental point of view it is very likely that both waves will always be visible after the onset of instability because the parametric difference between the transition points for the two waves is relatively small and might undetectable within the accuracy of a realistic

164 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 14 Gr c (a) αc (b) cc 5 (c) Gr c Gr mc Gr mc (a) αc Gr mc (b) Gr mc cc Gr mc Gr mc (c) Figure 3.43: Comparison of the critical values for the upward (solid line) and downward (dashed line) wave modes in an oblique field for H e = 1, Pr = 55, γ = and δ = 5 and χ = χ = 5 (top row), χ = χ = 3 (bottom row). experiment. All stability diagrams discussed so far have been computed for γ =. It is of interest now to compare the flow stability characteristics in the applied fields of different orientations. This is done in figures 3.44 and 3.45 for δ = 5 and δ = 1, respectively. Both figures demonstrate a sensitive dependence on the choice of γ and that the variation of the critical parameters with γ becomes stronger as δ increases, that is as the in-layer component of the magnetic field grows. However this variation is not monotonic. As γ increases from the parametric stability region in figures 3.44(a) and 3.45(a) initially shrinks and then starts growing. Therefore the general conclusion is that there exists an optimal field orientation angle γ for which the basic flow becomes most unstable. The intermediate angles γ = 4 and γ = 65 chosen in figures 3.44 and 3.45, respectively, correspond to the most unstable field orientations for the Type I instability at Gr m = 12, but in general they are expected to be different for each value of Gr m and for each of the instability types and thus need to be determined on a case-by-case basis. The existence of optimal field orientation angles γ min has been already discussed in detail before and here

165 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 141 Gr c γ = γ =4 γ =9 (a) αc (b) cc (c) Gr mc Gr mc Gr mc Figure 3.44: Comparison of the critical values for H e = 1, Pr = 55, χ = χ = 3, δ = 5 and various field orientation angles γ: (a) stability diagram for an equivalent two-dimensional problem; (b) critical wavenumber α c and (c) the corresponding wave speeds along the stability boundaries shown in plot (a). Gr c γ = γ =65 γ =9 (a) αc (b) cc (c) Gr mc Gr mc Figure 3.45: Same as figure 3.44 but for δ = 1. Gr mc (a) (b) σ R.18 α δ = γ, γ, Figure 3.46: (a) The maximum amplification rate of thetype II instability and (b) the corresponding wavenumber α as functions of the field orientation angle γ for δ = 5, Gr =, Gr m = 12, H e = 1, Pr = 55 and χ = χ = 5.

166 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION (a) 2.16 (b).5 (c) Gr c αc 2.1 cc γ = 115, Gr = γ, γ, γ, Gr c Figure 3.47: The critical parameter values: (a) Grashof number Gr (the flow is stable above 2.8 the curve), (b) wavenumber α and (c) wave speeds c as functions of the field orientation angle γ for δ = 5, Gr 2.6 m = 12, H e = 1, Pr = 55 and χ = χ = 5. The Type II instability we focus on the Type II instability As discussed previously for the Type II instability the most unstable pattern corresponds 1.6 to vertical rolls with α =. Figure 3.46 confirms that the maximum growth rate of this 1.4 instability 1.2 is observed for γ = 9 ± γ = or 18. This is a result that is expected γ, from the analysis given in Chapter 2 because in this case the in-layer component of the applied magnetic field is aligned with the axes of the arising instability rolls. Figure 3.47 illustrates the variation with γ of the critical point corresponding to Gr m = 12 along the lower segment of the Type II instability boundary shown in figure 3.3(a). While it has been already established that the most prominent Type II instability pattern for these values of physical parameters always consists of vertical rolls, rolls of all other orientations can also exist up to Gr = Gr corresponding to the values shown in figure 3.47(a). The vertical alignement of instability patterns occurs for larger values of Gr. As follows from figure 3.47(a) such an alignment for Gr m = Gr m = 12 and δ = 5 is most delayed when the field is oriented at the angle γ = 9 ± γ = 25 or (or, equivalently, 25 ) to the vertical y axis. For larger values of Gr m the dependence of the critical values on the field orientation angle

167 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION (a) 2.2 (b).3 (c) Gr c 6.5 αc 2 cc γ =52.5, Gr = γ = 125, Gr = γ, γ, γ, Figure 3.48: Same as figure 3.47 but for Gr m = 35. The Type I (solid lines) and II (dashed lines) instabilities. 8 Gr c 7.5 γ becomes more complicated. As seen from figures 3.48 and 3.49 unlike for smaller 7 values of Gr m both the Type II (dashed lines) and Type I (solid lines) instabilities can 6.5 be detected when varying γ and both instabilities have its own optimal field orientation 6 (shown by symbolic markers in the figures). Namely, as follows from the inverse Squire s 5.5 transformation discussed earlier, for both Gr m = Gr m = 35 and Gr m = Gr m = 4 the narrowing of the orientation of the Type II instability patterns toward vertical is delayed γ c, the most (up to the Gr = for Gr m = Gr m = 35 and up to Gr = for Gr m = Gr m = 4) if the field is oriented at γ = 9 ± γ = 215 or 35. At the same time such a delay for the Type I instability is always most pronounced for the field orientation angles in the range < γ < 18 (the maxima of the solid curves in figures 3.48(a) and 3.49(a) are found when γ < 9 ). Note also that as Gr m increases the narrowing of the Type I instability at the optimal field orientation is found to occur at the larger values of Gr than that of the Type II instability (compare the heights of the maxima of the solid and dashed curves in figures 3.48(a) and 3.49(a)). This is expected as Type II instability is of the pre-dominantly thermomagnetic nature and the strengthening of the magnetic effects for larger Gr m leads to a stronger alignment of its patterns. Lastly, note from figure 3.5 that the instability types become harder to distinguish as the field orientation changes when the field inclination angle increases to δ = 1.

168 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION (a) (b).7.6 (c) Grc 1 αc 1.9 cc γ =57.5, Gr = γ = 125, Gr = γ, γ, γ, Figure 3.49: Same as figure 3.47 but for Gr m = 4. The Type I (solid lines) and II (dashed lines) instabilities Grc γ c, (a) (b).5 (c) Gr c 3 αc 2.3 cc γ, γ, γ = 17.5, Gr = γ, Figure 3.5: Same as figure 3.47 but for Gr m = 4 and δ = 1. The Type I (solid lines) and II (dashed lines) instabilities Gr c 3 2

169 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION Perturbation Energy Balance and Perturbation Fields In Section the stability characteristics and the three-dimensional orientation of instability patterns have been discussed in detail. However a number of aspects still require clarifications. In particular, while the physical nature of instabilities is clearly defined in the limits of Gr m (the thermogravitational Type I instability in the form of counterpropagating thermal waves) and Gr m (the thermomagnetic Type II instability in the form of stationary rolls) it is not clear what the dominant physical mechanism is when both Gr m = and Gr =, whether the dominant roles of various mechanisms swap and if so at what exactly values of the governing parameters. It is also not clear what the nature of the Type III instability is as its parametric boundary is not continuously connected to either the Gr m = or Gr = regimes. In order to answer these and other remaining questions the perturbation kinetic energy balance equations will be derived and the corresponding energy balance for linearized disturbances will be considered next. Applying the same technique as the one presented in Section 2.7.4, namely, multiplying equations (3.6.3) and (3.6.4) by the complex conjugate velocity components ũ and ṽ, respectively, adding the two equations together and integrating their sum over the width of the fluid layer leads to σσ k = Σ uv + Σ Gr + Σ vis + Σ m1 + Σ m2, (3.9.1) where Σ k = Σ vis = Σ uv = ( ũ 2 + ṽ 2) d x >, (3.9.2) }{{} E k ( α 2 ( ũ 2 + ṽ 2 ) + Dũ 2 + Dṽ 2) }{{} E vis d x = 1, (3.9.3) ( ) Dṽ ũ ṽ + i αṽ ( ũ 2 + ṽ 2 ) d x, (3.9.4) }{{} E uv

170 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 146 Σ Gr = Σ m1 = Σ m2 = } Gr θ {{} ṽ d x, (3.9.5) E Gr Gr m D H x ẽ }{{ 1 θ } ũ d x, (3.9.6) E m1 E m2 d x (3.9.7) and θ ( ) E m2 = Gr m D H x ũ (1 ẽ1 2 H )D φ i αẽ 1 ẽ 2 φ ) 2 Gr m θ (ẽ 1 ũd φ + i α(ẽ 1 ṽ + ẽ2 ũ)d φ α 2 ẽ 2 ṽ φ. The terms defined by (3.9.2) (3.9.7) unambiguously represent the perturbation kinetic energy, the viscous dissipation, the energy exchange with the basic flow velocity field, the buoyancy, the variation of the fluid magnetization with the temperature and the variation of a magnetic field induced by the motion of ferrofluid, respectively. Equation (3.9.1) is just an integrated form of the linearized perturbation momentum equations (3.6.3) and (3.6.4) and thus it does not contain any new information on instabilities. However it presents it in a straightforward and easy to interpret form. Indeed at the critical points σ R. Thus the real part of the right-hand side of equation (3.9.1) evaluated at ( Gr c, Gr mc ) must be zero as well, and the positive terms in the right-hand side of equation (3.9.1) unambiguously indicate that the corresponding physical mechanism plays a destabilizing role and vice versa. The magnitude of each of the terms is an easy-to-read measure of the relative importance of the corresponding physical mechanism. Note that the viscous dissipation term (3.9.3) is negatively defined and since the eigenfunctions of the linearized problem are defined up to an arbitrary multiplicative constant, for convenience, the perturbation energy balance terms are normalized so that Σ vis 1. Table 3.6 contains the numerical data for the perturbation energy integrals for various values of χ and χ at the critical points for the Type I instability at Gr m = 12 in a normal field. The data in the table enables one to make a number of general conclusions regard-

171 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 147 ing the driving mechanisms of the Type I instability that are independent of whether the fluid is strongly or weakly magnetizable and of whether its magnetization law is linear or not. It is evident from Table 3.6 that the dominant physical mechanism leading to this type of instability is the gravitational buoyancy caused by the thermal disturbances because Re(Σ Gr ) > and the magnitude of this term is much larger than that of any other energy terms (apart from the viscous dissipation term whose magnitude is set to 1). Thus even when the Gr m = the Type I instability is similar to that caused by thermal waves observed in large-prandtl-number nonmagnetic fluid. The dependence of the ferrofluid s magnetization on the temperature also plays a destabilizing role (Re(Σ m1 ) > ). Therefore in general the Type I instability is caused by thermal perturbations that affect the fluid s density and magnetization so that both Archemedean and Kelvin forces act together to destabilize the parallel basic flow. However both Re(Σ m2 ) and Re(Σ uv ) remain negative. Therefore the perturbation kinetic energy is lost not only due to the viscous dissipation, but also because it is used to modify the applied magnetic field (field induction by moving ferrofluid) and to feed back to the parallel base velocity field, even though these two negative energy fluxes remain relatively weak. It also follows from Table 3.6 that the Type I instability waves propagating upward and downward in a normal field are identical from the energy flux point of view. However closer to magnetic saturation when χ < χ the upward propagating wave is characterised by a somewhat larger values of Re(Σ m1 ) and smaller values of Re(Σ Gr ). This is consistent with the general observation made in the previous section that waves propagating upward become more dangerous when the relative role of magnetic effects increases. The imaginary parts of the energy integrals define the sign of σ I and thus of the disturbance wave speed c = σ I / α. The corresponding data presented in Table 3.6 shows that Im(Σ Gr ) has the largest magnitude and therefore its sign defines the sign of c. Physically, this means that the motion of the Type I instability patterns is mostly due to the gravitational buoyancy effects with a relatively weak contribution from the magnetic field (Im(Σ m2 )) and the basic velocity field (Im(Σ uv )) effects. At the same time the variation of the fluid magnetisation (Im(Σ m1 )) hinders this motion.

172 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 148 The disturbance energy data presented in Table 3.7 for the Type I instability waves in an oblique field reveals similar general trends, however, with one remarkable qualitative change: the Re(Σ m2 ) values become positive. This indicates that in an oblique field the coupling between magnetic and velocity fields can play a destabilising role. The possible physical interpretation of this fact is that the inclined magnetic field forces the arising convection flow patterns to re-orient aligning with the in-layer component of the applied field. This conclusion also holds for a more magnetically sensitive fluid, see the data in Table 3.8. While the data in Tables shows that for relatively small values of Gr m the Type I instability is mostly due to the gravitational buoyancy it is intuitively clear that magnetic effects must become progressively more importnat as the value of Gr m increases. This is indeed confirmed by figure In a normal field a sharp transition from the predominantly buoyancy to predominatly magnetically driven Type I instability occurs when the ratio Gr m / Gr approaches a certain value (e.g. Gr m / Gr.6 for the fluid with properties chosen in figure 3.51(a), see also Suslov (28) for the similar result for a ferrofluid with different properties).

173 Table 3.6: Values of the perturbation energy integrals Σ k, Σ m1, Σ m2, Σ Gr and Σ uv computed for the Type I instability in a normal magnetic field (δ = ) at Gr m = 12, Pr = 55, H e = 1 and various values of χ and χ and at the corresponding critical values of α, Gr given in Table 3.2 for the upward (odd-numbered lines) and downward (even-numbered lines) propagating waves. χ χ Σ k Re(Σ m1 ) Im(Σ m1 ) Re(Σ m2 ) Im(Σ m2 ) Re(Σ Gr ) Im(Σ Gr ) Re(Σ uv ) Im(Σ uv ) CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 149

174 Table 3.7: Same as Table 3.6 but for an oblique magnetic field (δ = 5, γ = ) and for the critical values of α, Gr given in Tables 3.3 and 3.4. χ χ Σ k Re(Σ m1 ) Im(Σ m1 ) Re(Σ m2 ) Im(Σ m2 ) Re(Σ Gr ) Im(Σ Gr ) Re(Σ uv ) Im(Σ uv ) Table 3.8: Same as Table 3.7 but for H e = 1. χ χ Σ k Re(Σ m1 ) Im(Σ m1 ) Re(Σ m2 ) Im(Σ m2 ) Re(Σ Gr ) Im(Σ Gr ) Re(Σ uv ) Im(Σ uv ) CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 15

175 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 151 Σm1 + Σm2, ΣGr, Σuv (a) Gr m /Gr Σm1 + Σm2, ΣGr, Σuv (b) Gr m /Gr Figure 3.51: The perturbation energy integrals entering equation (3.9.1) characterizing the thermomagnetic (Σ m1 + Σ m2, the solid line) and thermogravitational (Σ Gr, the dashdotted line) mechanisms of convection and the exchange with the basic flow (Σ uv, the dashed line) as functions of the ratio Gr m / Gr along the stability boundaries shown by the solid lines (a) in figure 3.23(a) for the normal field and (b) in figure 3.3(a) for the oblique field. In fact, for large values of Gr m the gravitational buoyancy not only becomes less important, but it even starts playing a slightly stabilising role (the dash-dotted line crosses the zero level in figure 3.51(a) demonstrating that Re(Σ Gr ) becomes negative). At the same time the overall magnetic contribution to the perturbation energy is always non-negative. Thus magnetic effects always play a destabilising role. The interaction of perturbations with the basic flow velocity field also starts playing a slightly destabilizing role for large Gr m but this instability mechanism remains very weak. Therefore we conclude that the basic flow has little to do with the Type I instability in a complete parametric range. The similar energy diagram presented in figure 3.51(b) constructed for an oblique magnetic field looks different because of the different shape of the stability boundary shown in figure 3.3(a) for an oblique field. Note that if the solid curve in figure 3.3(a) is followed in the clockwise direction then so is dash-dotted line in figure 3.51(b), while the solid line in figure 3.51(b) needs to be traced anti-clockwise. The two curves intersect at Gr m / Gr.75. Therefore the magnetic effects causing the Type I instability overcome the gravitational buoyancy at much larger values of Gr m when the magnetic field is applied obliquely. This explains the qualitative change in the shape of the instability boundary that is observed when the field inclination angle is increased and is demonstrated in figure 3.39, compare the third and fourth rows there: starting from some

176 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 152 field inclination angle 8.5 < δ < 1 the Type I instability remains dominated by the gravitational buoyancy regardless of the value of Gr m. This can be traced back to the geometric reduction of the cross-layer x component of the applied magnetic field that is mostly responsible for thermomagnetic mechanism of instability. The interaction of the Type I instability fields with that of the basic flow velocity remains negligible. The consideration of disturbance energy integrals serves as a useful tool for clarifying and distinguishing between the physical mechanisms behind various instabilities. The data for several parametric points of interest is collected in Table 3.9. Rows 1 3 correspond to the three σ R maxima in figure 3.4 that distinguish the Types I, II and III instabilities. The comparison of real parts of the integrals confirms that in normal field the destabilisation of the basic flow is achieved primarily due to the thermal variation of fluid s magnetisation Re(Σ m1 ). The main feature that distinguishes the Type I instability from its Type III counterpart is that even though as Gr m increases the magnetic effects become dominant for both of them the gravitational buoyancy still remains a destabilising factor for the Type I instability while for the Type III instabilities the driving mechanism is purely magnetic. As expected from the previous discussion this is also so for the Type II instability. The energy exchange with the basic flow velocity field remains negligible for all three instability types. Thus the presence of a specific cubic velocity profile is rather inconsequential in the considered problem. Note also that while the roles of the buoyancy (Σ Gr ) and the magnetic field variation (Σ m2 ) in supporting the instabilities at the large values of Gr m are relatively weak the inspection of imaginary parts of the energy integrals shows that the combination of these two influences defines the propagation direction of the disturbance wave for both Type I and Type III instabilities. Rows 4 12 in Table 3.9 characterise the Type I instability observed for various orientations of the applied magnetic field as shown in figure In this case the ratio of Gr m / Gr does not exceed.2 and consistent with figure 3.51 the instability is mostly driven by the gravitational buoyancy with a relatively small contribution from magnetic effects. However the data in Table 3.9 suggets a remarkable conjecture. The optimal

177 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 153 orientation γ c min of the magnetic field when the Type I instability is promoted the most corresponds to the regime for which Re(Σ m2 ) vanishes that is to the situation when the secondary flow arising due to the developing disturbances does not lead to the integral variation of the applied magnetic field. Rows 13 and 14 in Table 3.9 illustrate a switch between the two instability patterns with significantly different wavenumbers that occur when the field inclination angle becomes sufficiently large as seen in the third row in figure The data presented in the table indicate that neither buoyancy nor the basic flow velocity field are responsible for such a transition. The spatial distribution of the energy integrands presented in figure 3.52 also confirms that (see the solid and dashed lines in the middle panels). Qualitatively, the disturbance energy distributions for both patterns remain similar, which allows one to conclude that physical mechanism of instability is the same for both type of patterns. As seen from middle panels in figure 3.52 the destabilising magnetic field variation effect (Σ m2 ) is most pronounced in the near-wall regions, especially near the right cold wall (dotted line in the middle panels). However the viscous dissipation there is also stronger than that near the hot wall (dashed line in the left panels). As a result, the overall instability pattern shifts toward the hot wall as seen in the bottom row in figure The main observable features distinguishing the two switching patterns are their wavenumbers (compare perturbation patterns shown in figure 3.53) and propagation speeds: both increase discontinuously at the transition point. Such an increase is fully due to magnetic effects, compare the corresponding entries for Im(Σ m1 ) and Im(Σ m2 ) in rows 13 and 14 of the table. The data in rows in Table 3.9 that refer to the fourth row of plots in figure 3.39 indicate that similar magnetically influenced transitions occur at larger values of the field inclination angle as the magnetic Grashof number increases. Even though the pattern wavenumber changes in a stepwise manner the transitions now are continuous.

178 Table 3.9: Selected critical values of parameters and perturbation energy integrals for Pr = 55, H e = 1 and χ = χ = 5. # δ, γ, α c Gr c Gr mc c c Σ k Re(Σ m1 ) Im(Σ m1 ) Re(Σ m2 ) Im(Σ m2 ) Re(Σ Gr ) Im(Σ Gr ) Re(Σ uv ) Im(Σ uv ) CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 154

179 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 155 The data in rows also demonstrates that as the field inclination angle increases the thermomagnetic destabilisation due to the variation of the fluid magnetisation with temperature (Re(Σ m1 )) gives way to the destabilisation due to the interaction of the perturbed velocity field with the induced variations of the magnetic field (Re(Σ m2 )). This is also confirmed by the energy integrand plots in the middle panels in figure As the value of Gr m increases the maximum of the disturbance energy production Re(E m2 ) (dotted line) shifts toward the cold wall where the basic flow magnetic field is weaker and its relative height increases. At the same time the maximum of the disturbance energy 2 2 Ek,Evis Ek,Evis Re(Euv,EGr,Em1,Em2) 1 1 E 2 m1 3 E m1 2.5 E m2 E m2 E k E uv E uv E vis E Gr 4 E Gr x x x.5 1 Re(Euv,EGr,Em1,Em2) E m1 3 E m1 E m2 E m2 E k E uv E 2 E vis 3 E 4 uv Gr E Gr x x x Figure 3.52: Disturbance energy integrands at the critical points corresponding to parameters listed in rows 13 (top) and 14 (bottom) in Table 3.9. Im(Euv,EGr,Em1,Em2) Im(Euv,EGr,Em1,Em2)

180 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 156 Figure 3.53: Disturbance fields at the critical points corresponding to parameters listed in rows 13 (top) and 14 (bottom) in Table 3.9. Cross-section along the main periodicity direction.

181 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 157 production Re(E m1 ) (dash-dotted line) shifts to the hot wall where the basic flow fluid magnetisation is weaker. Thus the thermo-magnetic perturbations arising in the layer of ferrofluid tend to make the magnetic and magnetisation fields there more uniform. Following a hydrodynamic analogy when perturbations lead to mixing the fluid making its velocity and temperature fields more uniform here one can introduce the concept of thermomagnetic mixing when magnetically driven perturbations tend to make magnetic and magnetisation fields more uniform. This is also illustrated in disturbance field plots in figure 3.55 where the shift of the perturbation fields is seen to occur as Gr m increases, which is followed by the decrease of the disturbance wavelength. A further increase of Gr m leads to yet another qualitative change. As is evidenced by figures 3.56 and 3.57 the cross-layer symmetry of the perturbation fields is lost completely with all perturbations shifted closer to the hot wall. Such a cross-layer localisation (that is consistent with the previously made conclusion that the thermo-magnetic instability waves are most dangerous near the hot wall) is followed by the corresponding decrease of the disturbance wavelength or, equivalently, by the increase in the disturbance wavenumber, which is seen in the α plot in the fourth row in figure Finally, rows 18 and 19 in Table 3.9 corresponding to points shown by symbols in figure 3.48 confirm that in an oblique field at relatively large values of the magnetic Grashof number either the Type I or Type II instability can arise and dominate the flow depending on the field orientation angle γ: row 18 corresponds to the Type I instability because both magnetic and gravitational buoyancy effects are destabilising, while row 19 illustrates the Type II instability that is fully magnetically driven with buoyancy playing a stabilising role.

182 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 158 Ek,Evis Ek,Evis Re(Euv,EGr,Em1,Em2) E m1 E m1 2 E m2 4 E m2 E k E 3 uv E uv E vis E Gr E 3 5 Gr x x x Re(Euv,EGr,Em1,Em2) E m1 3 E m1 2.5 E m2 E m2 E k E uv E E vis E 4 uv Gr E Gr x x x Figure 3.54: Disturbance energy integrands at the critical points corresponding to parameters listed in rows 15 (top) and 16 (bottom) in Table 3.9. Im(Euv,EGr,Em1,Em2) Im(Euv,EGr,Em1,Em2)

183 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION 159 Figure 3.55: Disturbance fields at the critical points corresponding to parameters listed in rows 15 (top) and 16 (bottom) in Table 3.9. Cross-section along the main periodicity direction.

184 CHAPTER 3. MAGNETO-GRAVITATIONAL THERMAL CONVECTION Ek,Evis Re(Euv,EGr,Em1,Em2) E m1 6 E m1 1 E E 4 m2 E m2 k E uv E uv E vis E Gr E Gr x x x Figure 3.56: Disturbance energy integrands at the critical points corresponding to parameters listed in row 17 in Table 3.9. Im(Euv,EGr,Em1,Em2) 2 4 Figure 3.57: Disturbance fields at the critical points corresponding to parameters listed in row 17 in Table 3.9. Cross-section along the main periodicity direction.