Table of Contents
1 Basic Principles of the Kinetic Theory 1
1.1 Introduction 1
1.2 Molecular Structure of a Gas 7
1.2.1 Basic Principles 7
1.2.2 The Maxwellian Distribution Function 8
1.2.3 Determination of the Characteristic Velocities 12
1.2.4 Molecular Flux 14
1.2.5 Elementary Theory of Transport Processes 16
1.2.6 Characteristic Dimensions 20
1.2.7 Potentials of Molecular Interactions 21
1.3 Brownian Motion 23
1.4 Dynamics of a Binary Collision 26
1.4.1 Conservation Laws 26
1.4.2 Asymptotic Post-Collisional Velocities 28
1.4.3 Asymptotic Velocities for Gas Mixtures 29
1.4.4 Scattering Angle X 30
1.4.5 Differential Cross Section 32
1.5 Appendix 35
2 The Boltzmann Equation 37
2.1 The Boltzmann Equation 37
2.2 The BBGKY Hierarchy 42
2.2.1 The Liouville Theorem 42
2.2.2 Equations of the BBGKY Hierarchy 44
2.2.3 The Boltzmann Equation 47
2.3 The Transfer Equation 51
2.4 Summational Invariants 53
2.5 Macroscopic Description 56
2.5.1 Moments of the Distribution Function 56
2.5.2 Balance Equations for the Moments 58
2.6 The Definition of Equilibrium 59
2.6.1 The Maxwellian Distribution Function 59
2.6.2 Equilibrium States 60
2.7 Entropy and Entropy Flux 66
2.8 The H-Theorem 69
2.8.1 Interactions of Gas Molecules with Solid Surfaces 69
2.8.2 Scattering Kernels 71
2.8.3 The H-Theorem 76
2.8.4 The Paradoxes of Loschmidt and Zermelo 77
2.9 The Many Faces of Entropy 78
2.10 Appendix 79
3 The Chapman-Enskog Method 81
3.1 Thermodynamics of a Single Fluid 81
3.2 Simplified Version of the Chapman-Enskog Method 83
3.2.1 The Integral Equation 83
3.2.2 Solution of the Integral Equation 86
3.2.3 Constitutive Equations and Transport Coefficients 89
3.3 Formal Version of the Chapman-Enskog Method 93
3.3.1 The dimensionless Boltzmann Equation 93
3.3.2 The Integral Equations 94
3.3.3 The Second Approximation 96
3.3.4 Expansion of the Scalar Coefficients A and B 98
3.3.5 Transport Coefficients 101
3.4 The BGK Model 104
3.5 Appendix 107
4 Moment Methods 109
4.1 Grad's Moment Method 109
4.1.1 Balance Equations 109
4.1.2 Grad's Distribution Function 110
4.1.3 Grad's Distribution from Entropy Maximization 113
4.1.4 Determination of the Non-convective Fluxes, Production Terms, Entropy Density and Entropy Flux 114
4.1.5 Field Equations 117
4.2 The Method of Maxwell and Ikenberry-Truesdell 119
4.2.1 Calculation of the Production Terms 119
4.2.2 The Maxwellian Iteration 120
4.3 The Chapman-Enskog-Grad Combined Method 122
4.4 Non-inertial Reference Frames 125
4.4.1 Objective Tensors 125
4.4.2 The Boltzmann Equation in Non-inertial Reference Frames 128
4.4.3 Frame Dependence of the Heat Flux Vector 130
4.5 Appendix 132
5 Polyatomic Gases 133
5.1 Some Properties of Polyatomic Gases 133
5.2 Semi-classical Model 135
5.2.1 Boltzmann and Transfer Equations 135
5.2.2 Macroscopic Description 138
5.2.3 The Equilibrium Distribution Function 140
5.2.4 Equilibrium States 142
5.2.5 The Non-equilibrium Distribution Function 143
5.2.6 The Laws of Navier-Stokes and Fourier 145
5.2.7 A Limiting Case 149
5.3 Classical Model 151
5.3.1 Basic Fields 151
5.3.2 Boltzmann and Transfer Equations 152
5.3.3 Transport Coefficients 155
5.4 Rough Spherical Molecules 156
5.4.1 Dynamics of a Binary Collision 157
5.4.2 Transport Coefficients 159
5.5 Appendix 163
6 Dense Gases 165
6.1 The Thermal Equation of State 165
6.1.1 The Van der Waals Equation 165
6.1.2 The Virial Equation of State 169
6.2 Enskog's Dense Gas 171
6.2.1 The Enskog's Equation 171
6.2.2 The Transfer Equation 172
6.2.3 Macroscopic Description 173
6.2.4 Determination of the Potential Contributions 174
6.2.5 Equilibrium Constitutive Equations 176
6.2.6 Determination of the Kinetic Contributions 177
6.2.7 The Laws of Navier-Stokes and Fourier 179
6.3 The Modified Enskog Equation 180
7 Granular Gases 185
7.1 Dynamics of a Binary Collision 185
7.2 The Boltzmann Equation 186
7.3 Macroscopic Description of a Granular Gas 187
7.4 The Chapman-Enskog Method 188
7.4.1 Integral Equations 188
7.4.2 First Approximation f(0) 190
7.4.3 Second Approximation f(1) 193
7.4.4 Constitutive Equations for the Pressure Tensor and the Heat Flux Vector 197
7.5 Granular Gases of Rough Spherical Molecules 199
8 Mixtures of Monatomic Gases 203
8.1 Boltzmann and Transfer Equations 203
8.2 Macroscopic Description 204
8.3 Thermodynamics of Fluid Mixtures 208
8.4 The Equilibrium Distribution Function 210
8.5 Equilibrium States 213
8.6 Grad's Distribution Function 215
8.7 The Combined Chapman-Enskog-Grad Method 217
8.8 The Navier-Stokes Law 218
8.9 The Laws of Fick and Fourier 219
8.10 Matrices as Functions of the Collision Integrals 223
8.11 Binary Mixtures 226
8.11.1 Coefficients of Shear Viscosity and Thermal Conductivity 226
8.11.2 Coefficients of Diffusion and Thermal-Diffusion Ratio 228
8.11.3 Coefficients for Some Intermolecular Potentials 229
8.12 Appendix 231
9 Chemically Reacting Gas Mixtures 235
9.1 Thermodynamics of Chemically Reacting Systems 235
9.1.1 Extent of Reaction and Affinity 235
9.1.2 Chemical Potentials 237
9.1.3 The Law of Mass Action 238
9.1.4 The Arrhenius Equation 240
9.2 Boltzmann Equations 242
9.3 Transfer and Balance Equations 244
9.4 Models for Differential Cross Sections 249
9.5 Equilibrium Distribution Function 250
9.6 Transport Coefficients for H2 + Cl + HCl + H 252
9.6.1 Chapman-Enskog Method 252
9.6.2 Transport Coefficients 257
9.6.3 Quaternary Mixture H2 Cl, HCl, H 260
9.6.4 Remarks on the Reactive Contributions to the Transport Coefficients 267
9.7 Trend to Equilibrium of H2 + Cl + HCl + H 268
9.7.1 Determination of the Production Terms 268
9.7.2 Constituents at Same Temperature 270
9.8 The H-Theorem and the Tendency to Equilibrium 275
9.9 Symmetric Reactions 280
9.9.1 The Influence of the Heat of Reaction on Slow Reactions 281
9.9.2 Chemical Reactions without Activation Energy 287
9.9.3 Remarks on the Geometry of the Collisions 291
9.9.4 Remarks on Inelastic Reactive Collisions 293
References 297
Index 299
