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Summary
Summary
The kinetic theory of gases as we know it dates to the paper of Boltzmann in 1872. The justification and context of this equation has been clarified over the past half century to the extent that it comprises one of the most complete examples of many-body analyses exhibiting the contraction from a microscopic to a mesoscopic description. The primary result is that the Boltzmann equation applies to dilute gases with short ranged interatomic forces, on space and time scales large compared to the corresponding atomic scales. Otherwise, there is no a priori limitation on the state of the system. This means it should be applicable even to systems driven very far from its eqUilibrium state. However, in spite of the physical simplicity of the Boltzmann equation, its mathematical complexity has masked its content except for states near eqUilibrium. While the latter are very important and the Boltzmann equation has been a resounding success in this case, the full potential of the Boltzmann equation to describe more general nonequilibrium states remains unfulfilled. An important exception was a study by Ikenberry and Truesdell in 1956 for a gas of Maxwell molecules undergoing shear flow. They provided a formally exact solution to the moment hierarchy that is valid for arbitrarily large shear rates. It was the first example of a fundamental description of rheology far from eqUilibrium, albeit for an unrealistic system. With rare exceptions, significant progress on nonequilibrium states was made only 20-30 years later.
Author Notes
Vicente Garzo: Departamento de Fisica, Universidad de Extremadura, Badajoz, Spain
Andres Santos: Departamento de Fisica, Universidad de Extremadura, Badajoz, Spain
Table of Contents
List of Figures | p. ix |
List of Tables | p. xxiii |
Foreword | p. xxv |
Acknowledgments | p. xxix |
Introduction | p. xxxi |
1. Kinetic Theory of Dilute Gases | p. 1 |
1 Introduction | p. 1 |
2 Derivation of the Boltzmann equation | p. 3 |
3 General properties of the Boltzmann equation. H-theorem | p. 12 |
4 Chapman-Enskog expansion | p. 18 |
5 Boltzmann equation for gas mixtures | p. 29 |
6 Kinetic models for a single gas | p. 33 |
7 Kinetic models for gas mixtures | p. 40 |
2. Solution of the Boltzmann Equation for Uniform Shear Flow | p. 55 |
1 Introduction | p. 55 |
2 The Boltzmann equation for uniform shear flow | p. 57 |
3 Moment equations for a gas of Maxwell molecules. Rheological properties | p. 61 |
4 Third- and fourth-degree velocity moments | p. 71 |
5 Singular behavior of the velocity moments | p. 79 |
6 Perturbation expansion of the distribution function | p. 86 |
7 Nonequilibrium entropy | p. 89 |
3. Kinetic Model for Uniform Shear Flow | p. 95 |
1 Introduction | p. 95 |
2 The BGK equation for uniform shear flow. Maxwell molecules | p. 96 |
3 Power-law repulsive potentials. Hard spheres | p. 107 |
3.1 Velocity distribution function | p. 107 |
3.2 Rheological properties | p. 110 |
4 The thermostatted state | p. 117 |
5 Nonequilibrium entropy of the thermostatted state | p. 126 |
6 Small perturbations from the thermostatted state | p. 133 |
7 Heat transport under uniform shear flow | p. 139 |
7.1 Boltzmann description for Maxwell molecules | p. 140 |
7.2 BGK description for general interactions | p. 144 |
7.3 Heat flux induced by an external force | p. 148 |
8 Stability of the uniform shear flow | p. 149 |
8.1 Theoretical analysis | p. 149 |
8.2 Monte Carlo simulations | p. 153 |
4. Uniform Shear Flow in a Mixture | p. 165 |
1 Introduction | p. 165 |
2 Maxwell molecules | p. 166 |
2.1 Transient regime | p. 168 |
2.2 Rheological properties | p. 174 |
3 General repulsive interactions | p. 178 |
3.1 Rheological properties | p. 180 |
3.2 Velocity distribution functions | p. 188 |
4 Nonequilibrium phase transition in the tracer limit | p. 193 |
5 Generalized diffusion and Dufour coefficients | p. 202 |
5.1 Diffusion tensor | p. 206 |
5.2 Dufour tensor | p. 209 |
5. Planar Couette Flow in a Single Gas | p. 213 |
1 Introduction | p. 213 |
2 Hydrodynamic description | p. 215 |
3 The Boltzmann equation for the planar Couette flow | p. 222 |
4 BGK kinetic model description | p. 231 |
4.1 Generalized transport coefficients | p. 232 |
4.2 Velocity distribution function | p. 242 |
5 Nonequilibrium entropy of the Couette flow | p. 246 |
6 Other kinetic theories | p. 248 |
7 Comparison with computer simulations | p. 254 |
6. Planar Couette Flow in a Mixture | p. 271 |
1 Introduction | p. 271 |
2 Kinetic model description for a mixture | p. 272 |
3 Application to the case of a binary mixture | p. 280 |
4 Diffusion and mobility in the tracer limit | p. 289 |
Appendices | p. 298 |
List of symbols | p. 299 |
References | p. 305 |
Index | p. 315 |