Preface and Acknowledgments .................................... xi
1 Short Preliminary Comments and Summary of Chapters 2
to 10 ........................................................ 1
1.1 Introduction ............................................ 1
1.2 Summary of Chapters 2 to 10 ............................. 3
References .................................................. 26
2 The Navier-Stokes-Fourier System of Equations and
Conditions .................................................. 29
2.1 Introduction ........................................... 29
2.2 Thermodynamics ......................................... 30
2.3 NS-F System for a Thermally Perfect Gas ................ 35
2.4 NS-F System for an Expansible Liquid ................... 38
2.5 Upper Free Surface Conditions .......................... 40
2.6 Influence of Initial Conditions and Transient
Behavior ............................................... 49
2.7 The Hills and Roberts' (1990) Approach ................. 52
References .................................................. 54
3 The Simple Rayleigh (1916) Thermal Convection Problem ....... 55
3.1 Introduction ........................................... 55
3.2 Formulation of the Starting д la Rayleigh Problem for
Thermal Convection ..................................... 60
3.3 Dimensionless Dominant Rayleigh Problem and the
Boussinesq Limiting Process ............................ 62
3.4 The Rayleigh-Benard Rigid-Rigid Problem as a
Leading-Order Approximate Model ........................ 66
3.5 Second-Order Model Equations Associated with the RB
Shallow Convection Equations (3.25a-c) ................. 71
3.6 Second-Order Model Equations Following from the Hills
and Roberts Equations (2.70a-c) ........................ 73
3.7 Some Comments .......................................... 76
References .................................................. 82
4 The Benard (1900,1901) Convection Problem, Heated from
Below ....................................................... 85
4.1 Introduction ........................................... 85
4.2 Benard Problem Formulation, Heated from Below .......... 92
4.3 Rational Analysis and Asymptotic Modelling ............ 104
4.4 Some Complements and Concluding Remarks ............... 110
References ................................................. 130
5 The Rayleigh-Benard Shallow Thermal Convection Problem ..... 133
5.1 Introduction .......................................... 133
5.2 The Rayleigh-Benard System of Model Equations ......... 138
5.3 The Second-Order Model Equations, Associated to RB
Equations ............................................. 143
5.4 An Amplitude Equation for the RB Free-Free Thermal
Convection Problem .................................... 144
5.5 Instability and Route to Chaos in RB Thermal
Convection ............................................ 152
5.6 Some Complements ...................................... 162
References ................................................. 169
6 The Deep Thermal Convection Problem ........................ 173
6.1 Introduction .......................................... 173
6.2 The Deep Benard Thermal Convection Problem ............ 174
6.3 Linear - Deep - Thermal Convection Theory ............. 176
6.4 Routes to Chaos ....................................... 182
6.5 Rigorous Mathematical Results ......................... 189
References ................................................. 193
7 The Thermocapillary, Marangoni, Convection Problem ......... 195
7.1 Introduction .......................................... 195
7.2 The Formulation of the Full Benard-Marangoni
Thermocapillary Problem ............................... 200
7.3 Some 'BM Long-Wave' Reduced Convection Model
Problems .............................................. 205
7.4 Lubrication Evolution Equations for the
Dimensionless Thickness of the Film ................... 214
7.5 Benney, KS, KS-KdV, IBL Model Equations Revisited ..... 218
7.6 Linear and Weakly Nonlinear Stability Analysis ........ 240
7.7 Some Complementary Remarks ............................ 252
References ................................................. 260
8 Summing Up the Three Significant Models Related with the
Benard Convection Problem .................................. 263
8.1 Introduction .......................................... 263
8.2 A Rational Approach to the Rayleigh-Bénard Thermal
Shallow Convection Problem ............................ 265
8.3 The Deep Thermal Convection with Viscous
Dissipation ........................................... 270
8.4 The Thermocapillary Convection with Temperature-
Dependent Surface Tension ............................. 272
References ................................................. 275
9 Some Atmospheric Thermal Convection Problems ............... 277
9.1 Introduction .......................................... 277
9.2 The Formulation of the Breeze Problem via the
Boussinesq Hydrostatic Approximation .................. 279
9.3 Model Problem for the Local Thermal Prediction -
A Triple Deck Viewpoint ............................... 292
9.4 Free Convection over a Curved Surface - A Singular
Problem ............................................... 298
9.5 Complements and Remarks ............................... 304
References ................................................. 322
10 Miscellaneous: Various Convection Model Problems ........... 325
10.1 Introduction .......................................... 325
10.2 Convection Problem in the Earth's Outer Core .......... 327
10.3 Magneto-Hydrodynamic, Electro, Ferro, Chemical,
Solar, Oceanic, Rotating, Penetrative Convections ..... 331
10.4 Averaged Integral Boundary Layer Approach: Non-
Isothermal Case ....................................... 345
10.5 Interaction between Short-Scale Marangoni Convection
and Long-Scale Deformational Instability .............. 349
10.6 Some Aspects of Thermosolutal Convection .............. 354
10.7 Anelastic (Deep) Non-Adiabatic and Viscous Equations
for the Atmospheric Thermal Convection
(à la Zeytounian) ..................................... 359
10.8 Flow of a Thin Liquid Film over a Rotating Disk ....... 363
10.9 Solitary Waves Phenomena in Benard-Marangoni
Convection ............................................ 369
10.10 Some Comments and Complementary References ........... 377
References ................................................. 384
Subject Index ................................................. 391
|
