Slow Rarefied Flows

by Cercignani, Carlo

• ISBN13: 9783764375348
• ISBN10: 3764375345
• Format: Hardcover
• Copyright: 2006-05-01
• Publisher: Birkhauser

Summary

The book presents the mathematical tools used to deal with problems related to slow rarefied flows, with particular attention to basic concepts and problems which arise in the study of micro- and nanomachines. The mathematical theory of slow flows is presented in a practically complete fashion and provides a rigorous justification for the use of the linearized Boltzmann equation, which avoids costly simulations based on Monte Carlo methods. The book surveys the theorems on validity and existence, with particular concern for flows close to equilibria, and discusses recent applications of rarefied lubrication theory to micro-electro-mechanical systems (MEMS). It gives a general acquaintance of modern developments of rarefied gas dynamics in various regimes with particular attention to low speed microscale gas dynamics. Senior students and graduates in applied mathematics, aerospace engineering, and mechanical mathematical physics will be provided with a basis for the study of molecular gas dynamics. The book will also be useful for scientific and technical researchers engaged in the research on gas flow in MEMS.

Table of Contents

Preface	vii
Introduction	ix
1 The Boltzmann Equation	1	(28)
1.1 Historical Introduction	1	(3)
1.2 The Boltzmann Equation	4	(7)
1.3 Molecules Different from Hard Spheres	11	(1)
1.4 Collision Invariants	12	(3)
1.5 The Boltzmann Inequality and the Maxwell Distributions	15	(1)
1.6 The Macroscopic Balance Equations	16	(4)
1.7 The H-theorem	20	(2)
1.8 Equilibrium States and Maxwellian Distributions	22	(2)
1.9 The Boltzmann Equation in General Coordinates	24	(1)
1.10 Mean Free Path	25	(1)
References	26	(3)
2 Validity and Existence	29	(12)
2.1 Introductory Remarks	29	(1)
2.2 Lanford's Theorem	30	(6)
2.3 Existence and Uniqueness Results	36	(3)
2.4 Remarks on the Mathematical Theory of the Boltzmann Equation	39	(1)
References	39	(2)
3 Perturbations of Equilibria	41	(28)
3.1 The Linearized Collision Operator	41	(2)
3.2 The Basic Properties of the Linearized Collision Operator	43	(7)
3.3 Some Spectral Properties	50	(10)
3.4 Asymptotic Behavior	60	(3)
3.5 The Global Existence Theorem for the Nonlinear Equation	63	(2)
3.6 The Periodic Case and Problems in One and Two Dimensions	65	(1)
References	66	(3)
4 Boundary Value Problems	69	(34)
4.1 Boundary Conditions	69	(5)
4.2 Initial-Boundary and Boundary Value Problems	74	(7)
4.3 Properties of the Free-streaming Operator	81	(3)
4.4 Existence in a Vessel with an Isothermal Boundary	84	(1)
4.5 The Results of Arkeryd and Maslova	85	(3)
4.6 Rigorous Proof of the Approach to Equilibrium	88	(2)
4.7 Perturbations of Equilibria	90	(1)
4.8 A Steady Flow Problem	91	(6)
4.9 Stability of the Steady Flow Past an Obstacle	97	(2)
4.10 Concluding Remarks	99	(1)
References	100	(3)
5 Slow Flows in a Slab	103	(28)
5.1 Solving the Linearized Boltzmann Equation in a Slab	103	(6)
5.2 Model Equations	109	(2)
5.3 Linearized Collision Models	111	(2)
5.4 Transformation of Models into Pure Integral Equations	113	(2)
5.5 Variational Methods	115	(8)
5.6 Poiseuille Flow	123	(5)
References	128	(3)
6 Flows in More Than One Dimension	131	(14)
6.1 Introduction	131	(1)
6.2 Linearized Steady Problems	131	(5)
6.3 Linearized Solutions of Internal Problems	136	(3)
6.4 External Problems	139	(1)
6.5 The Stokes Paradox in Kinetic Theory	140	(3)
References	143	(2)
7 Rarefied Lubrication in Mems	145	(20)
7.1 Introductory Remarks	145	(1)
7.2 The Modified Reynolds Equation	146	(3)
7.3 The Reynolds Equation and the Flow in a Microchannel	149	(2)
7.4 The Poiseuille-Couette Problem	151	(5)
7.5 The Generalized Reynolds Equation for Unequal Walls	156	(4)
7.6 Concluding remarks	160	(1)
References	161	(4)
Index	165