Molecular Gas Dynamics

by Sone, Yoshio

• ISBN13: 9780817643454
• ISBN10: 0817643451
• Format: Hardcover
• Copyright: 2006-11-30
• Publisher: Birkhauser

Summary

This self-contained work is an up-to-date treatment of the basic theory of molecular gas dynamics and its various applications. Recent progress in the field has greatly enhanced the original theory and stimulated interesting and critical gas dynamic phenomena and problems. This book, unique in the literature, presents working knowledge, theory, techniques, and typical phenomena in rarefied gases for theoretical development and applications. Basic theory is developed in a systematic way and presented in a form easily applied to practical use. After presenting basic theory and various simple flows, such as unidirectional or quasi-unidirectional flows and flows around a sphere, the author discusses additional topics, including flows induced by temperature fields, which are typical in rarefied gases; flows with evaporation and condensation; and bifurcation of flows in rarefied gases. The appendix contains many useful fundamental formulae, as well as an explanation of the theoretical background for the direct simulation Monte Carlo (DSMC) method, easily accessible to nonmathematicians and not found elsewhere in the literature. Existence of the ghost effect has made molecular gas dynamics indispensable to the study of a gas in the continuum limit, traditionally treated by classical fluid dynamics. In this book, the ghost and non-Navier-Stokes effects are demonstrated for typical examples'such as Bènard and Taylor-Couette problems'in the context of a new framework. An infinitesimal curvature effect is also discussed, with a long-standing problem of the bifurcation of the plane Couette flow worked out as an example. Molecular Gas Dynamics is useful for those working in different communities where kinetic theory or fluid dynamics is important: graduate students, researchers, and practitioners in theoretical physics, applied mathematics, and various branches of engineering. The work may be used as a self-study reference or as a textbook in graduate-level courses on fluid dynamics, gas dynamics, kinetic theory, molecular or rarefied gas dynamics, microflows, and applied mathematics.

Table of Contents

Preface	p. xi
Boltzmann Equation	p. 1
Velocity distribution function and macroscopic variables	p. 1
Boltzmann equation	p. 3
Conservation equations	p. 6
Maxwell distribution (Equilibrium distribution)	p. 6
Mean free path	p. 7
Kinetic boundary condition	p. 8
Simple boundary	p. 8
Interface	p. 9
H theorem	p. 11
Model equation	p. 12
Nondimensional expressions I	p. 13
Nondimensional expressions II	p. 19
Linearized Boltzmann equation	p. 23
Highly Rarefied Gas: Free Molecular Gas and Its Correction	p. 29
General solution of a free molecular flow	p. 29
Initial-value problem	p. 30
Boundary-value problem	p. 30
Preparation	p. 30
Free molecular gas around a convex body	p. 31
Arbitrary body shape and arrangement	p. 34
Initial and boundary-value problem	p. 42
Statics of a free molecular gas: Effect of the temperature of the boundary	p. 45
Construction of the velocity distribution function	p. 45
Condition of applicability	p. 49
Macroscopic variables	p. 49
Flow velocity	p. 50
Principle of superposition	p. 50
Simple applications	p. 51
Forces acting on heated bodies in a free molecular gas	p. 54
Effect of intermolecular collisions	p. 63
Slightly Rarefied Gas: Asymptotic Theory of the Boltzmann System for Small Knudsen Numbers	p. 73
Linear problem	p. 74
Problem	p. 74
Grad-Hilbert expansion and fluid-dynamic-type equations	p. 74
Stress tensor and heat-flow vector of the Grad-Hilbert solution	p. 78
Analysis of Knudsen layer	p. 79
Slip boundary condition and Knudsen-layer correction	p. 83
Discontinuity of the velocity distribution function and layer	p. 91
Force and mass and energy transfers on a closed body	p. 93
Summary	p. 94
Supplement: viscosity and thermal conductivity	p. 95
Weakly nonlinear problem	p. 96
Problem	p. 96
S expansion and fluid-dynamic-type equations	p. 97
Knudsen layer and slip boundary condition	p. 102
Rarefaction effect of a gas	p. 107
Force and mass and energy transfers on a closed body	p. 108
Summary	p. 110
Nonlinear problem I: Finite temperature variations and ghost effect	p. 112
Problem	p. 112
Outline of the analysis	p. 113
Fluid-dynamic-type equations and their boundary conditions	p. 117
Ghost effect and incompleteness of the classical gas dynamics	p. 119
Illustrative example	p. 124
Nonlinear problem II: Flow with a finite Mach number around a simple boundary	p. 126
Problem and the outline of analysis	p. 126
Fluid-dynamic-type equations and their boundary conditions and the recipe for solution	p. 132
Nonlinear problem III: Flow with a finite speed of evaporation or condensation	p. 137
Problem and the outline of analysis	p. 137
System of fluid-dynamic-type equations and boundary conditions in the continuum limit	p. 140
Review of the fluid-dynamic-type systems	p. 144
Classification	p. 144
Supplementary discussion	p. 148
Time-dependent problem	p. 149
Fluid-dynamic-type equations I: Sh = 0(1)	p. 150
Fluid-dynamic-type equations II: Sh = 0(k)	p. 155
Slip boundary condition and Knudsen-layer correction	p. 163
Initial layer and others	p. 165
Simple Flows	p. 169
Couette-flow and heat-transfer problems between two parallel plates	p. 169
Flows through a channel or pipe I: Straight pipe	p. 178
Analysis by a similarity solution	p. 178
Example	p. 181
Slowly varying approximation	p. 186
Flow through a channel or pipe II: Quasi-unidirectional flow	p. 189
Gas over a plane wall	p. 196
Uniform flow past a sphere with a uniform temperature	p. 200
Uniform flow past a sphere with an arbitrary thermal conductivity	p. 207
Formulation	p. 207
A gas around a sphere with a nonuniform temperature	p. 210
Solution for a sphere with an arbitrary thermal conductivity	p. 217
Shock wave	p. 219
Formation and propagation of a shock wave	p. 222
Flows Induced by Temperature Fields	p. 233
Flows in a slightly rarefied gas	p. 233
Thermal creep flow	p. 233
Thermal-stress slip flow	p. 239
Nonlinear-thermal-stress flow	p. 242
Thermal edge flow	p. 244
Flow between elliptic cylinders with different temperatures	p. 246
Thermophoresis	p. 248
A spherical particle with a uniform temperature	p. 249
A spherical particle with an arbitrary thermal conductivity	p. 253
One-way flows induced through a pipe without average pressure and temperature gradients	p. 261
Background	p. 261
Pipe with ditches	p. 261
Pipe with shelves	p. 267
Compressors without a moving part	p. 272
Knudsen compressor	p. 272
Performance	p. 274
Discussion	p. 275
Thermal-edge compressor	p. 277
Summary	p. 280
Flows with Evaporation and Condensation	p. 281
Evaporation from or condensation onto a plane condensed phase	p. 281
Problem and basic equations	p. 281
Behavior of evaporating flows	p. 283
Behavior of condensing flows	p. 294
Evaporation from a cylindrical condensed phase into a vacuum	p. 302
Problem and basic equation	p. 302
Outline of numerical computation	p. 304
The behavior of the gas	p. 306
Evaporation from a cylindrical condensed phase into a gas	p. 315
Problem and basic equation	p. 315
The behavior of the gas	p. 315
Evaporation from a spherical condensed phase into a vacuum	p. 321
Problem and basic equation	p. 321
The behavior of the gas	p. 325
Negative temperature gradient phenomenon	p. 338
Generalized kinetic boundary condition	p. 344
Bifurcation in the Half-Space Problem of Evaporation and Condensation	p. 355
Problem	p. 355
Transition from evaporation to condensation	p. 356
Basic equation and boundary condition	p. 356
Slowly varying solution	p. 357
Knudsen-layer correction	p. 359
Solution	p. 360
Transonic condensation	p. 362
Preparation	p. 362
Slowly varying solution	p. 365
Construction of the solution of the half-space problem	p. 371
Existence range of a solution	p. 377
Supplementary discussion	p. 378
Ghost Effect and Bifurcation I: Benard and Taylor-Couette Problems	p. 379
Benard problem I: Finite Knudsen number	p. 379
Introduction	p. 379
Existence range of nonstationary solutions and their flow patterns	p. 381
Array of rolls and its stability	p. 382
Benard problem II: Continuum limit	p. 389
Introduction	p. 389
One-dimensional solution	p. 390
Bifurcation from the one-dimensional solution	p. 391
Two-dimensional temperature field under infinitesimal flow velocity	p. 396
Discussions	p. 399
Taylor-Couette problem	p. 403
Problem and basic equation	p. 403
Analysis of bifurcation	p. 406
Bifurcated temperature field under infinitesimal speeds of rotation of the cylinders	p. 410
Discussion	p. 413
Flows between rotating circular cylinders with evaporation and condensation	p. 417
Introduction	p. 417
Axially symmetric and uniform case	p. 418
Axially symmetric and nonuniform case I: Finite Knudsen number	p. 430
Axially symmetric and nonuniform case II: Limiting solution as Kn [rightarrow] 0	p. 438
Ghost Effect and Bifurcation II: Ghost Effect of Infinitesimal Curvature and Bifurcation of the Plane Couette Flow	p. 449
Problem and basic equations	p. 450
Asymptotic analysis	p. 452
[Characters not reproducible] solution	p. 453
Knudsen-layer analysis	p. 455
Asymptotic fluid-dynamic-type equations and their boundary conditions	p. 457
Supplementary notes	p. 460
System for small Mach numbers and small temperature variations	p. 462
Bifurcation of the plane Couette flow	p. 466
Bifurcation analysis	p. 466
Bifurcated flow field under infinitesimal curvature	p. 471
Summary and supplementary discussion	p. 472
Supplement to the Boltzmann Equation	p. 481
Derivation of the Boltzmann equation	p. 481
Collision integral	p. 494
Binary collision	p. 494
Symmetry relation and its applications	p. 495
Summational invariant	p. 499
Function B[Characters not reproducible]	p. 501
Spherically symmetric field of a symmetric tensor	p. 509
Isotropic property of collision operator	p. 511
Parity of the linearized collision integral [Lambda](0)	p. 514
Linearized collision integral [Lambda](0) and integral equation [Lambda](0) = Ih	p. 517
Functions defined by [Lambda](0) = Ih and transport coefficients	p. 520
Kernel representation of the linearized collision integral [Lambda](0)	p. 523
Boltzmann equation in the cylindrical and spherical coordinate systems	p. 528
Integral form of the Boltzmann equation	p. 531
General case	p. 531
Linearized BKW equation with the diffuse-reflection or complete-condensation condition	p. 532
Similarity solution	p. 539
Reduced BKW equation	p. 544
Maxwell distribution	p. 546
Equilibrium distribution	p. 546
Local Maxwell distribution	p. 548
Mean free path for a Maxwellian	p. 551
Kinetic boundary condition in the linearized problem	p. 553
Darrozes-Guiraud inequality	p. 558
Equation for Knudsen layer	p. 562
Uniqueness of solution of the boundary-value problem of the linearized Boltzmann equation	p. 566
Methods of Solution	p. 571
Direct simulation Monte Carlo method	p. 571
Introduction	p. 571
Preparation	p. 571
Process of DSMC method	p. 574
Theoretical background of DSMC method	p. 579
Economy of computation	p. 591
Example	p. 595
Moment method	p. 601
Basic idea	p. 601
Examples	p. 603
Modified Knudsen number expansion	p. 606
Chapman-Enskog expansion	p. 607
Hypersonic approximation	p. 612
Some Data	p. 617
Some integrals	p. 617
Some numerical data	p. 618
Bibliography	p. 621
List of Symbols	p. 643
Index	p. 647
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