The mathematical theory of non-uniform gases. An account of the kinetic theory of viscosity, thermal conduction and diffusion in gases.
3rd ed. prepared in co-operation with D. Burnett.

*(English)*Zbl 0726.76084
Cambridge Mathematical Library. Cambridge etc.: Cambridge University Press. 423 p. £19.50; $ 32.50 (1991).

A detailed mathematical description is given, complemented with an appropriate historical account about the gas theory. The resultant of this combination is a classic, written in admirable form.

This book represents a very good option to mathematicians interested to follow the asymptotic methods of solution for nonlinear partial differential equations in gas theory, and also to those that want to find the original ideas and mathematical formulation for basic problems in this field. The main mathematical aspect in this book is the Chapmann- Enskog way of solution via asymptotic expansions of Boltzmann equation in a variety of mathematical models for gases.

In order to derive and solve the equations is needed a background in operations with vectors and second-order tensors, like products involving both, appearing in the first chapter in junction with some basic integrals related with vectors and dyadic tensors and useful notations of dyadics involving differential operators, particularly the rate-of-strain and the rate-of-shear tensors associated to the velocity-gradient tensor. The variables that represent physical properties of gases: density, velocity-distribution function, hydrostatic and boundary pressures, pressure tensor, temperature, thermal flux vector, are presented in the second part. The equations of Boltzmann and Maxwell are derived in Ch. III. The relation between Boltzmann’s H-function and entropy and a description of the important Maxwellian velocity-distribution function appear in the next chapter. Now one follows with deductions to the expressions for the mean free path: frequency of collisions and mean persistence-ratio of velocities. The ideas are continuated with deductions to approximate expressions for the coefficients of viscosity, thermal conduction and diffusion, giving a discussion of the treatments based on the free-path phenomena and on the basis of collision interval theory.

The Enskog’s method of successive approximations in order to solve Boltzmann’s equation of the nonuniform state for a simple gas is given in Ch. VII. This way of solution keeps the interest and beauty to these days. Here are obtained expressions for the coefficients of thermal conduction and viscosity of a gas on the basis of the solution of Boltzmann equations and using expansions in series in Sonine polynomials. This method of solution is applied in the next two parts in order to solve the equations of Boltzmann for a binary mixture giving expressions for the viscosity, thermal conduction and thermal diffusion coefficients, leading to the explicit expressions for their respective successive approximations. After this are obtained theoretical formulae for some of the above coefficients for special molecular models, and one can find some applications of eigenvalue theory to the collision operator. The authors consider molecules with internal energy in Ch. XI, giving mathematical expressions for the “volume” viscosity, opposing motions of contraction or expansion of the gas. The generalized Boltzmann equation in generalized coordinates appears using the Hamiltonian equations. To a comparison with the experiment for theoretical formulae of viscosity, thermal conductivity and diffusion are devoted, respectively, the next three chapters.

The exposition follows with detailed considerations of the third approximation in the asymptotic expansion of the velocity-distribution function, the thermal flux and stress tensor. A discussion about the method of Grad to solve the Boltzmann equation and, particularly, the 13- moment approximation is given. In the next chapter the formulae for viscosity and thermal conductivity of a dense gas are given, and a comparison with experiments is presented. A brief sketch of a theory based on multiple velocity-distribution function leading to the BBGKY equations, and the generalization of the method of successive approximations due to Bogoliubov appear also in this part. Chapter VII is devoted to the quantum methods applied to molecular encounters and the SchrĂ¶dinger equation for the distribution of molecular deflections is solved by separating variables and using Legendre polynomials and Bessel functions. A comparison with experiment for helium is shown. Degeneracy for Fermi-Dirac and for Bose-Einstein particles is treated. The Boltzmann equation taking into account the quantum effects is presented here.

A generalization of methods in Ch. VIII for a mixture of three or more constituent gases is given in Ch.XVIII, using the Enskog’s method of solution of the Boltzmann equation.

In the last chapter one can find the Boltzmann equation for an ionized gas in the presence of a magnetic field, and an approximate theory of diffusion, heat conduction and viscosity. Assuming small deflections in all encounters, and using Taylor expansions of the velocity-distribution function in the Boltzmann integral, the Landau and Fokker-Planck equations are derived. The end of this chapter is devoted to the collisionless Boltzmann equation for the velocity-distribution function of rarefied plasmas, using expansions in powers of the inverse of the spiral-frequencies.

This book represents a very good option to mathematicians interested to follow the asymptotic methods of solution for nonlinear partial differential equations in gas theory, and also to those that want to find the original ideas and mathematical formulation for basic problems in this field. The main mathematical aspect in this book is the Chapmann- Enskog way of solution via asymptotic expansions of Boltzmann equation in a variety of mathematical models for gases.

In order to derive and solve the equations is needed a background in operations with vectors and second-order tensors, like products involving both, appearing in the first chapter in junction with some basic integrals related with vectors and dyadic tensors and useful notations of dyadics involving differential operators, particularly the rate-of-strain and the rate-of-shear tensors associated to the velocity-gradient tensor. The variables that represent physical properties of gases: density, velocity-distribution function, hydrostatic and boundary pressures, pressure tensor, temperature, thermal flux vector, are presented in the second part. The equations of Boltzmann and Maxwell are derived in Ch. III. The relation between Boltzmann’s H-function and entropy and a description of the important Maxwellian velocity-distribution function appear in the next chapter. Now one follows with deductions to the expressions for the mean free path: frequency of collisions and mean persistence-ratio of velocities. The ideas are continuated with deductions to approximate expressions for the coefficients of viscosity, thermal conduction and diffusion, giving a discussion of the treatments based on the free-path phenomena and on the basis of collision interval theory.

The Enskog’s method of successive approximations in order to solve Boltzmann’s equation of the nonuniform state for a simple gas is given in Ch. VII. This way of solution keeps the interest and beauty to these days. Here are obtained expressions for the coefficients of thermal conduction and viscosity of a gas on the basis of the solution of Boltzmann equations and using expansions in series in Sonine polynomials. This method of solution is applied in the next two parts in order to solve the equations of Boltzmann for a binary mixture giving expressions for the viscosity, thermal conduction and thermal diffusion coefficients, leading to the explicit expressions for their respective successive approximations. After this are obtained theoretical formulae for some of the above coefficients for special molecular models, and one can find some applications of eigenvalue theory to the collision operator. The authors consider molecules with internal energy in Ch. XI, giving mathematical expressions for the “volume” viscosity, opposing motions of contraction or expansion of the gas. The generalized Boltzmann equation in generalized coordinates appears using the Hamiltonian equations. To a comparison with the experiment for theoretical formulae of viscosity, thermal conductivity and diffusion are devoted, respectively, the next three chapters.

The exposition follows with detailed considerations of the third approximation in the asymptotic expansion of the velocity-distribution function, the thermal flux and stress tensor. A discussion about the method of Grad to solve the Boltzmann equation and, particularly, the 13- moment approximation is given. In the next chapter the formulae for viscosity and thermal conductivity of a dense gas are given, and a comparison with experiments is presented. A brief sketch of a theory based on multiple velocity-distribution function leading to the BBGKY equations, and the generalization of the method of successive approximations due to Bogoliubov appear also in this part. Chapter VII is devoted to the quantum methods applied to molecular encounters and the SchrĂ¶dinger equation for the distribution of molecular deflections is solved by separating variables and using Legendre polynomials and Bessel functions. A comparison with experiment for helium is shown. Degeneracy for Fermi-Dirac and for Bose-Einstein particles is treated. The Boltzmann equation taking into account the quantum effects is presented here.

A generalization of methods in Ch. VIII for a mixture of three or more constituent gases is given in Ch.XVIII, using the Enskog’s method of solution of the Boltzmann equation.

In the last chapter one can find the Boltzmann equation for an ionized gas in the presence of a magnetic field, and an approximate theory of diffusion, heat conduction and viscosity. Assuming small deflections in all encounters, and using Taylor expansions of the velocity-distribution function in the Boltzmann integral, the Landau and Fokker-Planck equations are derived. The end of this chapter is devoted to the collisionless Boltzmann equation for the velocity-distribution function of rarefied plasmas, using expansions in powers of the inverse of the spiral-frequencies.

Reviewer: M.Rodriguez Ricard (Ciudao de La Habana)

##### MSC:

76P05 | Rarefied gas flows, Boltzmann equation in fluid mechanics |

76-02 | Research exposition (monographs, survey articles) pertaining to fluid mechanics |

76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |

76N15 | Gas dynamics (general theory) |

76N20 | Boundary-layer theory for compressible fluids and gas dynamics |

76Rxx | Diffusion and convection |

82B40 | Kinetic theory of gases in equilibrium statistical mechanics |