Kinetic Theory and Fluid Dynamics

by Sone, Yoshio

• ISBN13: 9780817642846
• ISBN10: 0817642846
• Format: Hardcover
• Copyright: 2002-09-01
• Publisher: Birkhauser

Summary

This monograph gives a comprehensive description of the relationship and connections between kinetic theory and fluid dynamics, mainly for a time-independent problem in a general domain. Ambiguities in this relationship are clarified, and the incompleteness of classical fluid dynamics in describing the behavior of a gas in the continuum limit'”recently reported as the ghost effect'”is also discussed. The approach used in this work engages an audience of theoretical physicists, applied mathematicians, and engineers.By a systematic asymptotic analysis, fluid-dynamic-type equations and their associated boundary conditions that take into account the weak effect of gas rarefaction are derived from the Boltzmann system. Comprehensive information on the Knudsen-layer correction is also obtained. Equations and their boundary conditions are carefully classified depending on the physical context of problems. Applications are presented to various physically interesting phenomena, including flows induced by temperature fields, evaporation and condensation problems, examples of the ghost effect, and bifurcation of flows.Key features:* many applications and physical models of practical interest* experimental works such as the Knudsen compressor are examined to supplement theory* engineers will not be overwhelmed by sophisticated mathematical techniques* mathematicians will benefit from clarity of definitions and precise physical descriptions given in mathematical terms* appendices collect key derivations and formulas, important to the practitioner, but not easily found in the literatureKinetic Theory and Fluid Dynamics serves as a bridge 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 can be used in graduate-level courses in fluid dynamics, gas dynamics, and kinetic theory; some parts of the text can be used in advanced undergraduate courses.

Table of Contents

Preface	ix
Introduction	1	(4)
Boltzmann Equation	5	(22)
Velocity distribution function and macroscopic variables	5	(2)
Boltzmann equation	7	(1)
Conservation equations	8	(1)
Maxwell distribution (Equilibrium distribution)	9	(1)
Mean free path	9	(1)
Boundary condition	10	(3)
Simple boundary	10	(1)
Interface	11	(2)
H theorem	13	(1)
Model equation	14	(1)
Nondimensional expressions I	15	(4)
Nondimensional expressions II	19	(5)
Linearized Boltzmann equation	24	(2)
Boltzmann equation in the cylindrical and spherical coordinate systems	26	(1)
Linear Theory - Small Reynolds Numbers	27	(58)
Problem	27	(1)
Grad-Hilbert solution and fluid-dynamic-type equations	28	(4)
Stress tensor and heat-flow vector of the Grad-Hilbert solution	32	(3)
Analysis of the Knudsen layer	35	(9)
Slip condition and Knudsen-layer correction	44	(8)
On a simple solid boundary	44	(5)
On an interface of a gas and its condensed phase with evaporation or condensation	49	(3)
Determination of macroscopic variables	52	(1)
Discontinuity of the velocity distribution function and S layer	53	(4)
Force and mass and energy transfers on a closed body	57	(2)
Viscosity and thermal conductivity	59	(1)
Summary of the asymptotic theory	60	(1)
Applications	60	(25)
Thermal creep flow and thermal transpiration	60	(3)
Thermal-stress slip flow	63	(1)
Nonlinear thermal-stress flow	64	(1)
Thermal edge flow	65	(5)
Thermophoresis	70	(2)
Knudsen compressor	72	(9)
Negative-temperature-gradient phenomenon	81	(4)
Weakly Nonlinear Theory - Finite Reynolds Numbers	85	(38)
Problem	86	(1)
S solution	86	(6)
Fluid-dynamic-type equations	92	(5)
Knudsen-layer analysis	97	(4)
Slip condition and Knudsen layer	101	(3)
On a simple solid boundary	102	(1)
On an interface of a gas and its condensed phase	103	(1)
Determination of macroscopic variables	104	(2)
Rarefaction effect	106	(1)
Force and mass and energy transfers on a closed body	106	(3)
Summary of the asymptotic theory and a comment on a time-dependent problem	109	(4)
Applications	113	(10)
Half space problem of evaporation and condensation	113	(3)
Evaporation and condensation around a cylindrical or spherical condensed phase	116	(3)
The difference of the temperature field for the S expansion and for the incompressible Navier-Stokes set in a time-dependent problem	119	(4)
Nonlinear Theory I - Finite Temperature Variations and Ghost Effect	123	(44)
Problem	123	(2)
SB solution	125	(10)
Fluid-dynamic-type equations	135	(2)
Knudsen layer and slip condition	137	(9)
Determination of macroscopic variables	146	(2)
Ghost effect: Incompleteness of the system of the classical gas dynamics	148	(12)
Nonlinear thermal-stress flow and inappropriateness of the heat-conduction equation	148	(3)
Ambiguity in the continuum world	151	(1)
Reflection on the Navier-Stokes set of equations	152	(1)
Illustrative examples	153	(4)
Supplementary discussion	157	(3)
Half-space problem of evaporation and condensation	160	(7)
Nonlinear Theory II - Flow with a Finite Mach Number around a Simple Boundary	167	(36)
Problem	167	(1)
Hilbert solution	168	(5)
Viscous boundary-layer solution	173	(12)
Knudsen-layer solution and slip condition	185	(7)
Connection of Hilbert and viscous boundary-layer solutions	192	(2)
Recipe for construction of solution	194	(2)
Discussions	196	(7)
Nonlinear Theory III - Finite Speed of Evaporation and Condensation	203	(32)
Problem	203	(1)
Hilbert solution	204	(2)
Knudsen layer	206	(3)
Half-space problem of evaporation and condensation	209	(4)
System of equations and boundary conditions in the continuum limit	213	(3)
Generalized kinetic boundary condition	216	(4)
Boundary-condition functions h1(Mn), h2(Mn), Fs(Mn, Mn, Mt, T/Tw), and Fb(Mn, Mt, T/Tw)	220	(5)
Applications	225	(10)
Two-surface problem of evaporation and condensation	225	(1)
Evaporating flow from a spherical condensed phase into a vacuum	226	(5)
Evaporating flow from a cylindrical condensed phase into a vacuum	231	(4)
Bifurcation of Cylindrical Couette Flow with Evaporation and Condensation	235	(22)
Problem	235	(2)
Solution type I	237	(7)
Analysis	237	(5)
Solution	242	(2)
Solution type II	244	(6)
Bifurcation diagram and transition solution	250	(3)
Discussions for the other parameter range	253	(1)
Concluding remark and supplementary comment	253	(4)
A Supplementary Explanations and Formulas	257	(44)
Formal derivation of the Boltzmann equation from the Liouville equation	257	(12)
Solution of integral equation L(φ) = Ih	269	(2)
Derivation of the Stokes set of equations	271	(9)
Golse's theorem on a one-way flow	280	(5)
Functions A(ζ, TSB0), B(ζ, Tv0), etc	285	(4)
Viscosity and thermal conductivity	289	(3)
Linear integral equations J(fh0, fhm) = Ihhm, J(fSB0, fSBm) = IhSBm, etc	292	(3)
Ja(ζiE, ζjAE), Ja(ζ2 E, (ζiζj - ζ2δij/3)BE), etc	295	(2)
Equation for the Knudsen layer and Bardos's theorem	297	(1)
The boundary condition for the linearized Euler set of equations	298	(3)
Spherically Symmetric Field of Symmetric Tensor	301	(14)
Problem	301	(1)
TI(ζi) on ζ1 axis	302	(2)
TI(ζi) at an arbitrary point	304	(8)
Preparation	304	(2)
Derivation of the expression of TI(ζi)	306	(5)
Summary	311	(1)
Applications	312	(3)
Definite integral ∞∫∫∫ - ∞ ζi1 ... ζi2s exp(-ζ2)dζ1dζ2dζ3	312	(1)
Axially symmetric field	313	(2)
Kinetic-Equation Approach to Fluid-Dynamic Equations	315	(12)
Introduction	315	(1)
Exact kinetic-equation approach	316	(8)
Discussion on numerical systems	324	(3)
Bibliography	327	(18)
Index	345