An Introduction to the Boltzmann Equation and Transport Processes in Gases (eBook)

Gilberto M. Kremer (Autor)

eBook Download: PDF

2010 | 2010
XVIII, 310 Seiten
Springer Berlin (Verlag)
978-3-642-11696-4 (ISBN)

Infos

This book covers classical kinetic theory of gases, presenting basic principles in a self-contained framework and from a more rigorous approach based on the Boltzmann equation. Uses methods in kinetic theory for determining the transport coefficients of gases.

Inhalt

Title Page 2
Preface 7
Contents 10
Chapter 1 Basic Principles of the Kinetic Theory 15
Introduction 15
Molecular Structure of a Gas 21
Basic Principles 21
The Maxwellian Distribution Function 22
Determination of the Characteristic Velocities 26
Molecular Flux 28
Elementary Theory of Transport Processes 30
Characteristic Dimensions 34
Potentials of Molecular Interactions 35
Brownian Motion 37
Dynamics of a Binary Collision 40
Conservation Laws 40
Asymptotic Post-Collisional Velocities 42
Asymptotic Velocities for Gas Mixtures 43
Scattering Angle ? 44
Differential Cross Section 46
Appendix 49
Chapter 2 The Boltzmann Equation 50
The Boltzmann Equation 50
The BBGKY Hierarchy 55
The Liouville Theorem 55
Equations of the BBGKY Hierarchy 57
The Boltzmann Equation 60
The Transfer Equation 64
Summational Invariants 66
Macroscopic Description 69
Moments of the Distribution Function 69
Balance Equations for the Moments 71
The Definition of Equilibrium 72
The Maxwellian Distribution Function 72
Equilibrium States 73
Entropy and Entropy Flux 79
The $H$-Theorem 82
Interactions of Gas Molecules with Solid Surfaces 82
Scattering Kernels 84
The $H$-Theorem 89
The Paradoxes of Loschmidt and Zermelo 90
The Many Faces of Entropy 91
Appendix 92
Chapter 3 The Chapman–Enskog Method 94
Thermodynamics of a Single Fluid 94
Simplified Version of the Chapman–Enskog Method 96
The Integral Equation 96
Solution of the Integral Equation 99
Constitutive Equations and Transport Coefficients 102
Formal Version of the Chapman–Enskog Method 106
The dimensionless Boltzmann Equation 106
The Integral Equations 107
The Second Approximation 109
Expansion of the Scalar Coefficients A and B 111
Transport Coefficients 114
The BGK Model 117
Appendix 120
Chapter 4 Moment Methods 121
Balance Equations 121
Grad’s Distribution Function 122
Grad’s Distribution from Entropy Maximization 125
Determination of the Non-convective Fluxes, Production Terms, Entropy Density and Entropy Flux 126
Field Equations 129
The Method of Maxwell and Ikenberry–Truesdell 131
Calculation of the Production Terms 131
The Maxwellian Iteration 132
The Chapman–Enskog–Grad Combined Method 134
Non-inertial Reference Frames 137
Objective Tensors 137
The Boltzmann Equation in Non-inertial Reference Frames 140
Frame Dependence of the Heat Flux Vector 142
Appendix 144
Chapter 5 Polyatomic Gases 145
Some Properties of Polyatomic Gases 145
Semi-classical Model 147
Boltzmann and Transfer Equations 147
Macroscopic Description 150
The Equilibrium Distribution Function 152
Equilibrium States 154
The Non-equilibrium Distribution Function 155
The Laws of Navier–Stokes and Fourier 157
A Limiting Case 161
Classical Model 163
Basic Fields 163
Boltzmann and Transfer Equations 164
Transport Coefficients 167
Rough Spherical Molecules 168
Dynamics of a Binary Collision 169
Transport Coefficients 171
Appendix 175
Chapter 6 Dense Gases 177
The Thermal Equation of State 177
The Van der Waals Equation 177
The Virial Equation of State 181
Enskog’s Dense Gas 183
The Enskog’s Equation 183
The Transfer Equation 184
Macroscopic Description 185
Determination of the Potential Contributions 186
Equilibrium Constitutive Equations 188
Determination of the Kinetic Contributions 189
The Laws of Navier–Stokes and Fourier 191
The Modified Enskog Equation 192
Chapter 7 Granular Gases 197
Dynamics of a Binary Collision 197
The Boltzmann Equation 198
Macroscopic Description of a Granular Gas 199
The Chapman–Enskog Method 200
Integral Equations 200
First Approximation f$^(0)$ 202
Second Approximation f$^(1)$ 205
Constitutive Equations for the Pressure Tensor and the Heat Flux Vector 209
Granular Gases of Rough Spherical Molecules 211
Chapter 8 Mixtures of Monatomic Gases 215
Boltzmann and Transfer Equations 215
Macroscopic Description 216
Thermodynamics of Fluid Mixtures 220
The Equilibrium Distribution Function 222
Equilibrium States 225
Grad’s Distribution Function 227
The Combined Chapman–Enskog–Grad Method 229
The Navier–Stokes Law 230
The Laws of Fick and Fourier 231
Matrices as Functions of the Collision Integrals 235
Binary Mixtures 238
Coefficients of Shear Viscosity and Thermal Conductivity 238
Coefficients of Diffusion and Thermal–Diffusion Ratio 240
Coefficients for Some Intermolecular Potentials 241
Appendix 243
Chapter 9 Chemically Reacting Gas Mixtures 246
Thermodynamics of Chemically Reacting Systems 246
Extent of Reaction and Affinity 246
Chemical Potentials 248
The Law of Mass Action 249
The Arrhenius Equation 251
Boltzmann Equations 253
Transfer and Balance Equations 255
Models for Differential Cross Sections 260
Equilibrium Distribution Function 261
Transport Coefficients for $H2 + Cl HCl + H$ 263
Chapman–Enskog Method 263
Transport Coefficients 268
Quaternary Mixture H2, Cl, HCl, H 271
Remarks on the Reactive Contributions to the Transport Coefficients 278
Trend to Equilibrium of $H2 + Cl HCl + H$ 279
Determination of the Production Terms 279
Constituents at Same Temperature 281
The $H$-Theorem and the Tendency to Equilibrium 286
Symmetric Reactions 291
The Influence of the Heat of Reaction on Slow Reactions 292
Chemical Reactions without Activation Energy 298
Remarks on the Geometry of the Collisions 302
Remarks on Inelastic Reactive Collisions 304
References 308
Index 309

Produkt-Details

• Erscheint lt. Verlag: 18.8.2010
• Reihe/Serie: Interaction of Mechanics and Mathematics
• Reihe/Serie: Interaction of Mechanics and Mathematics
• Zusatzinfo: XVIII, 310 p. 43 illus.
• Verlagsort: Berlin
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Statistik
• Themenwelt: Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik ► Maschinenbau
• Schlagworte: Boltzmann equation • Non-Equilibrium Effects • reactive system • Reactive Systems • transport theory
• ISBN-10: 3-642-11696-5 / 3642116965
• ISBN-13: 978-3-642-11696-4 / 9783642116964