Slow Rarefied Flows (eBook)

Contents 5
Preface 7
Introduction 8
1 The Boltzmann Equation 11
1.1 Historical Introduction 11
1.2 The Boltzmann Equation 14
1.3 Molecules Di.erent from Hard Spheres 21
1.4 Collision Invariants 22
1.5 The Boltzmann Inequality and the Maxwell 25
Distributions 25
1.6 The Macroscopic Balance Equations 26
1.7 The 30
theorem 30
1.8 Equilibrium States and Maxwellian Distributions 32
1.9 The Boltzmann Equation in General Coordinates 34
1.10 Mean Free Path 35
References 36
2 Validity and Existence 39
2.1 Introductory Remarks 39
2.2 Lanford’s Theorem 40
2.3 Existence and Uniqueness Results 46
2.4 Remarks on the Mathematical Theory of the 49
Boltzmann Equation 49
References 49
3 Perturbations of Equilibria 50
3.1 The Linearized Collision Operator 50
3.2 The Basic Properties of the Linearized Collision 52
Operator 52
3.3 Some Spectral Properties 59
3.4 The Asymptotic Behavior of the Solution of the 69
Cauchy Problem for the Linearized Boltzmann 69
Equation 69
3.5 The Global Existence Theorem for the Nonlinear 72
Equation 72
3.6 Extensions: The Periodic Case and Problems in One 74
and Two Dimensions 74
References 75
4 Boundary Value Problems 77
4.1 Boundary Conditions 77
4.2 Initial-Boundary and Boundary Value Problems 82
4.3 Properties of the Free-streaming Operator 89
4.4 Existence in a Vessel with an Isothermal Boundary 92
4.5 The Results of Arkeryd and Maslova 93
4.6 Rigorous Proof of the Approach to Equilibrium 96
4.7 Perturbations of Equilibria 98
4.8 A Steady Flow Problem 99
4.9 Stability of the Steady Flow Past an Obstacle 105
4.10 Concluding Remarks 107
References 108
5 Slow Flows in a Slab 111
5.1 Solving the Linearized Boltzmann Equation in a 111
Slab 111
5.2 Model Equations 117
5.3 Linearized Collision Models 119
5.4 Transformation of Models into Pure Integral 121
Equations 121
5.5 Variational Methods 123
5.6 Poiseuille Flow 131
References 136
6 Flows in More Than One Dimension 138
6.1 Introduction 138
6.2 Linearized Steady Problems 138
6.3 Linearized Solutions of Internal Problems 143
6.4 External Problems 146
6.5 The Stokes Paradox in Kinetic Theory 147
References 150
7 Rarefied Lubrication in Mems 152
7.1 Introductory Remarks 152
7.2 The Modi.ed Reynolds Equation 153
7.3 The Reynolds Equation and the Flow in a 156
Microchannel 156
7.4 The Poiseuille-Couette Problem 158
7.5 The Generalized Reynolds Equation for Unequal 163
Walls 163
7.6 Concluding remarks 167
References 168
Index 171