Convection in Fluids (eBook)

Contents 7
Preface and Acknowledgments 11
1 Short Preliminary Comments and Summary of Chapters 2 to 10 16
1.1 Introduction 16
1.2 Summary of Chapters 2 to 10 18
References 41
2 The Navier –Stokes–Fourier System of Equations and Conditions 44
2.1 Introduction 44
2.2 Thermodynamics 45
2.3 NS–F System for a Thermally Perfect Gas 50
2.4 NS–F System for an Expansible Liquid 53
2.5 Upper Free Surface Conditions 55
2.6 Influence of Initial Conditions and Transient Behavior 64
2.7 The Hills and Roberts’ (1990) Approach 67
References 69
3 The Simple Rayleigh (1916) Thermal Convection Problem 70
3.1 Introduction 70
3.2 Formulation of the Starting Problem for Thermal Convection 75
3.3 Dimensionless Dominant Rayleigh Problem and the Boussinesq Limiting Process 77
3.4 The Rayleigh–Bénard Rigid-Rigid Problem as a Leading-Order Approximate Model 81
3.5 Second-Order Model Equations Associated with the RB Shallow Convection Equations (3.25a–c) 86
3.6 Second-Order Model Equations Following from the Hills and Roberts Equations (2.70a–c) 88
3.7 Some Comments 91
References 97
4 The Bénard (1900, 1901) Convection Problem, Heated from Below 99
4.1 Introduction 99
4.2 Bénard Problem Formulation, Heated from Below 106
4.3 Rational Analysis and Asymptotic Modelling 118
4.4 Some Complements and Concluding Remarks 124
References 144
5 The Rayleigh–Bénard Shallow Thermal Convection Problem 147
5.1 Introduction 147
5.2 The Rayleigh–Bénard System of Model Equations 152
5.3 The Second-Order Model Equations, Associated to RB Equations 157
5.4 An Amplitude Equation for the RB Free-Free Thermal Convection Problem 158
5.5 Instability and Route to Chaos in RB Thermal Convection 166
5.6 Some Complements 176
References 183
6 The Deep Thermal Convection Problem 186
6.1 Introduction 186
6.2 The Deep Bénard Thermal Convection Problem 187
6.3 Linear – Deep – Thermal Convection Theory 189
6.4 Routes to Chaos 195
6.5 Rigorous Mathematical Results 202
References 206
7 The Thermocapillary, Marangoni, Convection Problem 207
7.1 Introduction 207
7.2 The Formulation of the Full Bénard–Marangoni Thermocapillary Problem 212
7.3 Some ‘BM Long-Wave’ Reduced Convection Model 217
7.4 Lubrication Evolution Equations for the Dimensionless Thickness of the Film 226
7.5 Benney, KS, KS–KdV, IBL Model Equations Revisited 230
7.6 Linear andWeakly Nonlinear Stability Analysis 252
7.7 Some Complementary Remarks 264
References 272
8 Summing Up the Three Significant Models Related with the Bénard Convection Problem 274
8.1 Introduction 274
8.2 A Rational Approach to the Rayleigh–Bénard Thermal 276
Shallow Convection Problem 276
8.3 The Deep Thermal Convection with Viscous Dissipation 281
8.4 The Thermocapillary Convection with 283
Temperature-Dependent Surface Tension 283
References 286
9 Some Atmospheric Thermal Convection Problems 287
9.1 Introduction 287
9.2 The Formulation of the Breeze Problem via the Boussinesq 289
Hydrostatic Approximation 289
9.3 Model Problem for the Local Thermal Prediction – A Triple 302
Deck Viewpoint 302
9.4 Free Convection over a Curved Surface – A Singular 308
Problem 308
9.5 Complements and Remarks 314
References 332
10 Miscellaneous: Various Convection Model Problems 335
10.1 Introduction 335
10.2 Convection Problem in the Earth’s Outer Core 337
10.3 Magneto-Hydrodynamic, Electro, Ferro, Chemical, Solar, 341
Oceanic, Rotating, Penetrative Convections 341
10.4 Averaged Integral Boundary Layer Approach: 355
Non-Isothermal Case 355
10.5 Interaction between Short-Scale Marangoni Convection 359
and Long-Scale Deformational Instability 359
10.6 Some Aspects of Thermosolutal Convection 364
10.7 Anelastic (Deep) Non-Adiabatic and Viscous Equations for 369
the Atmospheric Thermal Convection ( 369
Zeytounian) 369
10.8 Flow of a Thin Liquid Film over a Rotating Disk 373
10.9 SolitaryWaves Phenomena in Bénard–Marangoni 379
Convection 379
10.10 Some Comments and Complementary References 387
References 394
Subject Index 401