Preface ........................................................ xv
Acknowledgements .............................................. xix
1 Equations, General Concepts and Methods of Analysis .......... 1
1.1 Pattern Formation and Nonlinear Dynamics ................ 1
1.1.1 Some Fundamental Concepts: Pattern,
Interrelation and Scale .......................... 2
1.1.2 PDEs, Symmetry and Nonequilibrium Phenomena ...... 4
1.2 The Navier-Stokes Equations ............................. 6
1.2.1 A Satisfying Microscopic Derivation of the
Balance Equations ................................ 6
1.2.2 A Statistical Mechanical Theory of Transport
Processes ........................................ 7
1.2.3 The Continuity Equation .......................... 9
1.2.4 The Momentum Equation ........................... 10
1.2.5 The Total Energy Equation ....................... 11
1.2.6 The Budget of Internal Energy ................... 13
1.2.7 Newtonian Fluids ................................ 13
1.2.8 Some Considerations About the Dynamics of
Vorticity ....................................... 15
1.2.9 Incompressible Formulation of the Balance
Equations ....................................... 18
1.2.10 Nondimensional Form of the Equations for
Thermal Problems ................................ 19
1.3 Energy Equality and Dissipative Structures ............. 21
1.4 Flow Stability, Bifurcations and Transition to Chaos ... 25
1.5 Linear Stability Analysis: Principles and Methods ...... 27
1.5.1 Conditional Stability and Infinitesimal
Disturbances .................................... 27
1.5.2 The Exponential Matrix and the Eigenvalue
Problem ......................................... 28
1.5.3 Linearization of the Navier-Stokes Equations .... 30
1.5.4 A Simple Example: The Stability of a Parallel
Flow with an Inflectional Velocity Profile ...... 32
1.5.5 Weaknesses and Limits of the Linear Stability
Approach ........................................ 35
1.6 Energy Stability Theory ................................ 36
1.6.1 A Global Budget for the Generalized
Disturbance Energy .............................. 36
1.6.2 The Extremum Problem ............................ 39
1.7 Numerical Integration of the Navier-Stokes Equations ... 40
1.7.1 Vorticity Methods ............................... 41
1.7.2 Primitive Variables Methods ..................... 42
1.8 Some Universal Properties of Chaotic States ............ 46
1.8.1 Feigenbaum, Ruelle-Takens and Manneville-
Pomeau Scenarios ................................ 46
1.8.2 Phase Trajectories, Attractors and Strange
Attractors ...................................... 47
1.8.3 The Lorenz Model and the Butterfly Effect ....... 48
1.8.4 A Possible Quantification of SIC: The Lyapunov
Spectrum ........................................ 51
1.8.5 The Mandelbrot Set: The Ubiquitous Connection
Between Chaos and Fractals ...................... 53
1.9 The Maxwell Equations .................................. 58
2 Classical Models, Characteristic Numbers and Scaling
Arguments ................................................... 63
2.1 Buoyancy Convection and the Boussinesq Model ........... 64
2.2 Convection in Space .................................... 66
2.2.1 A Definition of Microgravity .................... 66
2.2.2 Experiments in Space ............................ 67
2.2.3 Surface Tension-driven Flows .................... 68
2.2.4 Acceleration Disturbances on Orbiting
Platforms and Vibrational Flows ................. 68
2.3 Marangoni Flow ......................................... 70
2.3.1 The Genesis and Relevant Nondimensional
Numbers ......................................... 71
2.3.2 Microzone Facilities and Microscale
Experimentation ................................. 75
2.3.3 A Paradigm Model: The Liquid Bridge ............. 75
2.4 Exact Solutions of the Navier-Stokes Equations for
Thermal Problems ....................................... 78
2.4.1 Thermogravitational Convection: The Hadley
Flow ............................................ 80
2.4.2 Marangoni Flow .................................. 80
2.4.3 Hybrid States ................................... 83
2.4.4 General Properties .............................. 83
2.4.5 The Infinitely Long Liquid Bridge ............... 85
2.4.6 Inclined Systems ................................ 86
2.5 Conductive, Transition and Boundary-layer Regimes ...... 89
3 Examples of Thermal Fluid Convection and Pattern Formation
in Nature and Technology .................................... 95
3.1 Technological Processes: Small-scale Laboratory and
Industrial Setups ...................................... 95
3.1.1 Crystal Growth from the Melt: Typical
Techniques ...................................... 96
3.1.2 Detrimental Effects Induced by Convective
Phenomena ...................................... 101
3.2 Examples of Thermal Fluid Convection and Pattern
Formation at the Mesoscale ............................ 103
3.3 Planetary Structure and Dynamics: Convective
Phenomena ............................................. 103
3.3.1 Earth's 'Layered' Structure .................... 103
3.3.2 Earth's Mantle Convection ...................... 104
3.3.3 Plate Tectonics Theory ......................... 104
3.3.4 Earth's Core Convection ........................ 106
3.3.5 The Icy Galilean Satellites .................... 107
3.4 Atmospheric and Oceanic Phenomena ..................... 108
3.4.1 A Fundamental Model: The Hadley Circulation .... 108
3.4.2 Mesoscale Shallow Cellular Convection:
Collection of Clouds and Related Patterns ...... 110
3.4.3 The Planetary Boundary Layer ................... 112
3.4.4 Atmospheric Convection in Other Solar System
Bodies ......................................... 116
4 Thermogravitational Convection: The Rayleigh-Bénard
Problem .................................................... 119
4.1 Nonconfined Fluid Layers and Ideal Straight Rolls ...... 119
4.1.1 The Linearized Problem: Primary Convective
Modes .......................................... 119
4.1.2 Systems Heated from Above: Internal Gravity
Waves .......................................... 122
4.2 The Busse Balloon ..................................... 124
4.2.1 Toroidal-Poloidal Decomposition ................ 125
4.2.2 The Zoo of Secondary Modes ..................... 127
4.3 Some Considerations About the Role of Dislocation
Dynamics .............................................. 133
4.4 Tertiary and Quaternary Modes of Convection ........... 135
4.5 Spoke Pattern Convection .............................. 138
4.6 Spiral Defect Chaos, Hexagons and Squares ............. 142
4.7 Convection with Lateral Walls ......................... 149
4.8 Two-dimensional Models ................................ 151
4.8.1 Distinct Modes of Convection and Possible
Symmetries ..................................... 151
4.8.2 Higher Modes of Convection and Oscillatory
Regimes ........................................ 155
4.9 Three-dimensional Parallelepipedic Enclosures:
Classification of Solutions and Possible Symmetries ... 157
4.9.1 The Cubical Box ................................ 160
4.9.2 The Onset of Time Dependence ................... 161
4.10 The Circular Cylindrical Problem ...................... 165
4.10.1 Moderate Aspect Ratios: Azimuthal Structure
and Effect of Lateral Boundary Conditions ...... 165
4.10.2 Small Aspect Ratios: Targets and PanAm
Textures ....................................... 170
4.11 Spirals: Genesis, Properties and Dynamics ............. 173
4.11.1 The Archimedean Spiral ......................... 175
4.11.2 Spiral Wavenumber .............................. 175
4.11.3 Multi-armed Spirals and Spiral Core
Instability .................................... 176
4.12 From Spirals to SDC: The Extensive Chaos Problem ...... 179
4.13 Three-dimensional Convection in a Spherical Shell ..... 182
4.13.1 Possible Patterns of Convection and Related
Symmetries ..................................... 183
4.13.2 The Heteroclinic Cycles ........................ 183
4.13.3 The Highly Viscous Case ........................ 185
4.13.4 The Geodynamo Problem .......................... 188
5 The Dynamics of Thermal Plumes and Related Regimes of
Motion ..................................................... 195
5.1 Introduction .......................................... 195
5.2 Free Plume Regimes .................................... 196
5.2.1 The Diffusive-Viscous Regime ................... 197
5.2.2 The Viscous-Nondiffusive Regime ................ 198
5.2.3 The Inviscid-Diffusive Regime .................. 198
5.2.4 The Inviscid-Nondiffusive Regime ............... 200
5.2.5 Sinuous Instabilities Created by Horizontal
Shear .......................................... 200
5.2.6 Geometric Constraints .......................... 201
5.3 The Flywheel Mechanism: The 'Wind' of Turbulence ...... 202
5.3.1 Upwelling and Downward Jets and Alternating
Eruption of Thermal Plumes ..................... 203
5.3.2 Geometric Effects .............................. 204
5.3.3 The Origin of the Large-scale Circulation:
The Childress and Villermaux Theories .......... 205
5.3.4 The Role of Thermal Diffusion in Turbulent
Rayleigh-Benard Convection ..................... 208
5.4 Multiplume Configurations Originated from Discrete
Sources of Buoyancy ................................... 208
6 Systems Heated from the Side: The Hadley Flow .............. 215
6.1 The Infinite Horizontal Layer ......................... 215
6.1.1 The Hadley Flow and its General Perturbing
Mechanisms ..................................... 216
6.1.2 Hydrodynamic Modes and Oscillatory
Longitudinal Rolls ............................. 219
6.1.3 The Rayleigh Mode .............................. 223
6.1.4 Competition of Disturbances and Tertiary
Modes of Convection ............................ 225
6.2 Two-dimensional Horizontal Enclosures ................. 228
6.2.1 Geometric Constraints and Multiplicity of
Solutions ...................................... 228
6.2.2 Instabilities Originating from Boundary
Layers and Patterns with Internal Waves ........ 235
6.3 The Infinite Vertical Layer: Cats-eye Patterns and
Temperature Waves ..................................... 247
6.4 Three-dimensional Parallelepipedic Enclosures ......... 253
6.5 Cylindrical Geometries under Various Heating
Conditions ............................................ 262
7 Thermogravitational Convection in Inclined Systems ......... 271
7.1 Inclined Layer Convection ............................. 272
7.1.1 The Codimension-two Point ...................... 273
7.1.2 Tertiary and High-order Modes of Convection .... 275
7.2 Inclined Side-heated Slots ............................ 279
7.2.1 Stationary Longitudinal Long-wavelength
Instability .................................... 281
7.2.2 Stationary Transversal Instability ............. 282
7.2.3 Oscillatory Transversal Long-wavelength
Instability .................................... 284
7.2.4 Stationary Longitudinal Short-wavelength
Instability .................................... 284
7.2.5 Oscillatory Longitudinal Instability ........... 284
7.2.6 Interacting Longitudinal and Transversal
Multicellular Modes ............................ 286
8 Thermovibrational Convection ............................... 289
8.1 Equations and Relevant Parameters ..................... 289
8.2 Fields Decomposition .................................. 290
8.3 The TFD Distortions ................................... 291
8.4 High Frequencies and the Thermovibrational Theory ..... 293
8.5 States of Quasi-equilibrium and Related Stability ..... 294
8.5.1 The Vibrational Hydrostatic Conditions ......... 294
8.5.2 The Linear Stability Problem ................... 295
8.5.3 Solutions for the Infinite Layer ............... 297
8.6 Primary and Secondary Patterns of Symmetry ............ 299
8.7 Medium and Low Frequencies: Possible Regimes and
Flow Patterns ......................................... 303
8.7.1 Synchronous, Subharmonic and Nonperiodic
Response ....................................... 303
8.7.2 Reduced Equations and Related Ranges of
Validity ....................................... 305
9 Marangoni-Benard Convection ................................ 317
9.1 Introduction .......................................... 317
9.2 High Prandtl Number Liquids: Patterns with Hexagons,
Squares and Triangles ................................. 320
9.3 Liquid Metals: Inverted Hexagons and High-order
Solutions ............................................. 325
9.4 Effects of Lateral Confinement ........................ 326
9.4.1 Circular Containers ............................ 328
9.4.2 Rectangular Containers ......................... 331
9.5 Temperature Gradient Inclination ...................... 334
10 Thermocapillary Convection ................................. 341
10.1 Basic Features of Steady Marangoni Convection ......... 342
10.2 Stationary Multicellular Flow and Hydrothermal
Waves ................................................. 345
10.2.1 Basic Velocity Profiles: The Linear and
Return Flows ................................... 346
10.2.2 Linear Stability Analysis ...................... 346
10.2.3 Weakly Nonlinear Analysis ...................... 354
10.2.4 Boundary Effects: 2D and 3D Numerical
Studies ........................................ 359
10.3 Annular Configurations ................................ 368
10.4 The Liquid Bridge ..................................... 375
10.4.1 Historical Perspective ......................... 375
10.4.2 Liquid Metals and Semiconductor Melts .......... 378
10.4.3 The First Bifurcation: Structure of the
Secondary 3D Steady Flow ....................... 379
10.4.4 Effect of Geometric Parameters ................. 381
10.4.5 A Generalized Theory for the Azimuthal
Wavenumber ..................................... 389
10.4.6 The Second Bifurcation: Tertiary Modes of
Convection ..................................... 390
10.4.7 High Prandtl Number Liquids .................... 393
10.4.8 Standing Waves and Travelling Waves ............ 399
10.4.9 Symmetric and Asymmetric Oscillatory Modes of
Convection ..................................... 407
10.4.10 System Dynamic Evolution ...................... 412
10.4.11 The Hydrothermal Mechanism in Liquid
Bridges ........................................ 417
10.4.12 Noncylindrical Liquid Bridges ................. 421
10.4.13 The Intermediate Range of Prandtl Numbers ..... 423
11 Mixed Buoyancy-Marangoni Convection ........................ 427
11.1 The Canonical Problem: The Infinite Horizontal
Layer ................................................. 429
11.2 Finite-sized Systems Filled with Liquid Metals ........ 436
11.3 Typical Terrestrial Laboratory Experiments with
Transparent Liquids ................................... 449
11.4 The Rectangular Liquid Layer .......................... 450
11.4.1 Waves and Multicellular Patterns ............... 450
11.4.2 Tertiary Modes of Convection: OMC and HTW
with Spatiotemporal Dislocations ............... 456
11.5 Effects Originating from the Walls .................... 458
11.5.1 Lateral Boundaries as a Permanent Stationary
Disturbance .................................... 459
11.5.2 Collision Phenomena of HTW and Wall-generated
Steady Patterns ................................ 460
11.5.3 Streaks Generated by a Lift-up Process and
Instabilities of a Mechanical Nature ........... 464
11.6 The Open Vertical Cavity .............................. 468
11.6.1 Volume Driving Actions and Rising Thermal
Plumes ......................................... 470
11.6.2 Aiding Marangoni and Buoyant Flows ............. 470
11.6.3 Counteracting Driving Forces and Separation
Phenomena ...................................... 472
11.6.4 Surface Driving Actions and Vertical
Temperature Gradients .......................... 474
11.7 The Annular Pool ...................................... 475
11.7.1 Target-like Wave Patterns (HW2) ................ 476
11.7.2 Waves with Spiral Pattern (HW1) ................ 478
11.7.3 Stationary Radial Rolls ........................ 480
11.7.4 Progression Towards Chaos and Fractal
Behaviour ...................................... 483
11.7.5 The Reverse Annular Configuration: Incoherent
Spatial Dynamics ............................... 487
11.7.6 Some Considerations About the Role of
Curvature, Heating Direction and Gravity ....... 488
11.8 The Liquid Bridge on the Ground ....................... 491
11.8.1 Microscale Experiments ......................... 492
11.8.2 Heating from Above or from Below ............... 499
11.8.3 The Route to Aperiodicity ...................... 510
12 Hybrid Regimes with Vibrations ............................. 517
12.1 RB Convection with Vertical Shaking ................... 519
12.2 Complex Order, Quasi-periodic Crystals and
Superlattices ......................................... 525
12.2.1 Purely Harmonic Patterns ....................... 527
12.2.2 Purely Subharmonic Patterns .................... 529
12.2.3 Coexistence and Complex Order .................. 529
12.3 RB Convection with Horizontal or Oblique Shaking ...... 533
12.4 Laterally Heated Systems and Parametric Resonances .... 538
12.4.1 The Infinite Horizontal Layer .................. 538
12.4.2 Domains with Vertical Walls .................... 544
12.4.3 The Infinite Vertical Layer .................... 548
12.4.4 Inclined Systems ............................... 550
12.5 Control of Thermogravitational Convection ............. 550
12.5.1 Cell Orientation as a Means to Mitigate
Convective Disturbances on Orbiting
Platforms ...................................... 551
12.5.2 Control of Convection Patterning and
Intensity in Shallow Enclosures ................ 553
12.5.3 Modulation of Thermal Boundary Conditions ...... 559
12.6 Mixed Marangoni-Thermovibrational Convection .......... 561
12.6.1 Basic Solutions ................................ 561
12.6.2 Control of Convection Patterning and
Intensity in Shallow Enclosures ................ 566
12.6.3 Control of Hydrothermal Waves .................. 567
12.7 Modulation of Marangoni-Benard Convection ............. 575
13 Flow Control by Magnetic Fields ............................ 581
13.1 Static and Uniform Magnetic Fields .................... 582
13.1.1 Physical Principles and Governing Equations .... 582
13.1.2 Hartmann Boundary Layers ....................... 584
13.2 Historical Developments and Current Status ............ 584
13.2.1 Stabilization of Thermogravitational Flows ..... 584
13.2.2 Stabilization of Surface Tension-driven
Flows .......................................... 597
13.3 Rotating Magnetic Fields .............................. 604
13.4 Gradients of Magnetic Fields and Virtual
Microgravity .......................................... 607
References .................................................... 609
Index ......................................................... 659
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