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An Introduction to the Boltzmann Equation and Transport Processes in Gases : Interaction of Mechanics and Mathematics - Gilberto M. Kremer

An Introduction to the Boltzmann Equation and Transport Processes in Gases

Interaction of Mechanics and Mathematics


Published: 29th June 2010
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Thisbookdealswith theclassicalkinetictheoryofgases.Itsaimisto present the basic principles of this theory within an elementary framework and from a more rigorous approach based on the Boltzmann equation. The subjects are presented in a self-contained manner such that the readers can und- stand and learn some methods used in the kinetic theory of gases in order to investigate the Boltzmann equation. In Chapter 1, a sketch on the evolution of the ideas of the kinetic theory is presented. Afterwards, the basic principles of an elementary kinetic theory areintroduced, which arebasedonthe concepts ofmean freepath, molecular mean velocity and mean free time. The Maxwellian distribution function is determinedfromstatisticalarguments, andthetransportcoe?cients ofshear viscosity, thermal conductivity and self-di?usion are obtained from the e- mentary theory. The most common interaction potentials used in the kinetic theory of gases are also introduced in this chapter, and the dynamics of a binary collision is analyzed. Chapter 2 is dedicated to the study of the Boltzmann equation. First, the BoltzmannequationisderivedandtheequationsoftheBBGKYhierarchyare determined.Fromtheknowledgeofthetransferequation-whichfollowsfrom theBoltzmannequation-themacroscopicbalanceequationsforthemoments ofthedistributionfunctionarederived.Theequilibriumdistributionfunction is determined from the Boltzmann equation and the equilibrium states of a rare?ed gas are also analyzed. In this chapter, theH-theorem and the paradoxes of Loschmidt and Zermelo are discussed. The chapter ends with an analysis of the di?erent forms of the entropy which are used in statistical mechanics to describe the canonical and microcanonical ensembles.

From the reviews:

"Kremer has written a book with a focus on engineering perspectives and applications of the nonlinear Boltzmann equation, well suited for an advanced course. ... this book is very well suited as a graduate introduction to the Boltzmann equation and its variants. It is also a good reference for those who wish to apply kinetic theory to, say, chemically reacting gases, or who are looking for a good way to comprehend standard closure procedures for the macroscopic balance equations, like Chapman-Enskog or moment methods."--- (Reinhard Illner, SIAM Review, Vol. 53 (4), 2011)

Basic Principles of the Kinetic Theoryp. 1
Introductionp. 1
Molecular Structure of a Gasp. 7
Basic Principlesp. 7
The Maxwellian Distribution Functionp. 8
Determination of the Characteristic Velocitiesp. 12
Molecular Fluxp. 14
Elementary Theory of Transport Processesp. 16
Characteristic Dimensionsp. 20
Potentials of Molecular Interactionsp. 21
Brownian Motionp. 23
Dynamics of a Binary Collisionp. 26
Conservation Lawsp. 26
Asymptotic Post-Collisional Velocitiesp. 28
Asymptotic Velocities for Gas Mixturesp. 29
Scattering Angle Xp. 30
Differential Cross Sectionp. 32
Appendixp. 35
The Boltzmann Equationp. 37
The Boltzmann Equationp. 37
The BBGKY Hierarchyp. 42
The Liouville Theoremp. 42
Equations of the BBGKY Hierarchyp. 44
The Boltzmann Equationp. 47
The Transfer Equationp. 51
Summational Invariantsp. 53
Macroscopic Descriptionp. 56
Moments of the Distribution Functionp. 56
Balance Equations for the Momentsp. 58
The Definition of Equilibriump. 59
The Maxwellian Distribution Functionp. 59
Equilibrium Statesp. 60
Entropy and Entropy Fluxp. 66
The H-Theoremp. 69
Interactions of Gas Molecules with Solid Surfacesp. 69
Scattering Kernelsp. 71
The H-Theoremp. 76
The Paradoxes of Loschmidt and Zermelop. 77
The Many Faces of Entropyp. 78
Appendixp. 79
The Chapman-Enskog Methodp. 81
Thermodynamics of a Single Fluidp. 81
Simplified Version of the Chapman-Enskog Methodp. 83
The Integral Equationp. 83
Solution of the Integral Equationp. 86
Constitutive Equations and Transport Coefficientsp. 89
Formal Version of the Chapman-Enskog Methodp. 93
The dimensionless Boltzmann Equationp. 93
The Integral Equationsp. 94
The Second Approximationp. 96
Expansion of the Scalar Coefficients A and Bp. 98
Transport Coefficientsp. 101
The BGK Modelp. 104
Appendixp. 107
Moment Methodsp. 109
Grad's Moment Methodp. 109
Balance Equationsp. 109
Grad's Distribution Functionp. 110
Grad's Distribution from Entropy Maximizationp. 113
Determination of the Non-convective Fluxes, Production Terms, Entropy Density and Entropy Fluxp. 114
Field Equationsp. 117
The Method of Maxwell and Ikenberry-Truesdellp. 119
Calculation of the Production Termsp. 119
The Maxwellian Iterationp. 120
The Chapman-Enskog-Grad Combined Methodp. 122
Non-inertial Reference Framesp. 125
Objective Tensorsp. 125
The Boltzmann Equation in Non-inertial Reference Framesp. 128
Frame Dependence of the Heat Flux Vectorp. 130
Appendixp. 132
Polyatomic Gasesp. 133
Some Properties of Polyatomic Gasesp. 133
Semi-classical Modelp. 135
Boltzmann and Transfer Equationsp. 135
Macroscopic Descriptionp. 138
The Equilibrium Distribution Functionp. 140
Equilibrium Statesp. 142
The Non-equilibrium Distribution Functionp. 143
The Laws of Navier-Stokes and Fourierp. 145
A Limiting Casep. 149
Classical Modelp. 151
Basic Fieldsp. 151
Boltzmann and Transfer Equationsp. 152
Transport Coefficientsp. 155
Rough Spherical Moleculesp. 156
Dynamics of a Binary Collisionp. 157
Transport Coefficientsp. 159
Appendixp. 163
Dense Gasesp. 165
The Thermal Equation of Statep. 165
The Van der Waals Equationp. 165
The Virial Equation of Statep. 169
Enskog's Dense Gasp. 171
The Enskog's Equationp. 171
The Transfer Equationp. 172
Macroscopic Descriptionp. 173
Determination of the Potential Contributionsp. 174
Equilibrium Constitutive Equationsp. 176
Determination of the Kinetic Contributionsp. 177
The Laws of Navier-Stokes and Fourierp. 179
The Modified Enskog Equationp. 180
Granular Gasesp. 185
Dynamics of a Binary Collisionp. 185
The Boltzmann Equationp. 186
Macroscopic Description of a Granular Gasp. 187
The Chapman-Enskog Methodp. 188
Integral Equationsp. 188
First Approximation f(0)p. 190
Second Approximation f(1)p. 193
Constitutive Equations for the Pressure Tensor and the Heat Flux Vectorp. 197
Granular Gases of Rough Spherical Moleculesp. 199
Mixtures of Monatomic Gasesp. 203
Boltzmann and Transfer Equationsp. 203
Macroscopic Descriptionp. 204
Thermodynamics of Fluid Mixturesp. 208
The Equilibrium Distribution Functionp. 210
Equilibrium Statesp. 213
Grad's Distribution Functionp. 215
The Combined Chapman-Enskog-Grad Methodp. 217
The Navier-Stokes Lawp. 218
The Laws of Fick and Fourierp. 219
Matrices as Functions of the Collision Integralsp. 223
Binary Mixturesp. 226
Coefficients of Shear Viscosity and Thermal Conductivityp. 226
Coefficients of Diffusion and Thermal-Diffusion Ratiop. 228
Coefficients for Some Intermolecular Potentialsp. 229
Appendixp. 231
Chemically Reacting Gas Mixturesp. 235
Thermodynamics of Chemically Reacting Systemsp. 235
Extent of Reaction and Affinityp. 235
Chemical Potentialsp. 237
The Law of Mass Actionp. 238
The Arrhenius Equationp. 240
Boltzmann Equationsp. 242
Transfer and Balance Equationsp. 244
Models for Differential Cross Sectionsp. 249
Equilibrium Distribution Functionp. 250
Transport Coefficients for H2 + Cl + HCl + Hp. 252
Chapman-Enskog Methodp. 252
Transport Coefficientsp. 257
Quaternary Mixture H2 Cl, HCl, Hp. 260
Remarks on the Reactive Contributions to the Transport Coefficientsp. 267
Trend to Equilibrium of H2 + Cl + HCl + Hp. 268
Determination of the Production Termsp. 268
Constituents at Same Temperaturep. 270
The H-Theorem and the Tendency to Equilibriump. 275
Symmetric Reactionsp. 280
The Influence of the Heat of Reaction on Slow Reactionsp. 281
Chemical Reactions without Activation Energyp. 287
Remarks on the Geometry of the Collisionsp. 291
Remarks on Inelastic Reactive Collisionsp. 293
Referencesp. 297
Indexp. 299
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9783642116957
ISBN-10: 3642116957
Series: Interaction of Mechanics and Mathematics
Audience: General
Format: Paperback
Language: English
Number Of Pages: 303
Published: 29th June 2010
Publisher: Springer-Verlag Berlin and Heidelberg Gmbh & Co. Kg
Country of Publication: DE
Dimensions (cm): 23.39 x 15.6  x 1.75
Weight (kg): 0.46