| Abstract: | Some features of the derivation of the gasdynamic equations of higher (Burnett and super-Burnett) approximations of the Chapman-Enskog
method are discussed for non-dense monatomic gases in the absence of external forces. The equations are given, and their properties
and the results of applying them to various problems of gas dynamics are described.
The problems of deriving, justifying, and testing macroscopic models are among the fundamental problems of not only continuum
mechanics but physical kinetics in general. In the latter case it is a question of reducing more general molecular-kinetic
models to simpler macroscopic ones. Historically, the classical Chapman-Enskog method of solution (as the Knudsen number Kn
→ 0) of the kinetic Boltzmann-Maxwell equation is of primary importance. Its application yields a chain of Euler and Navier-Stokes
equations and equations of higher (Burnett, super-Burnett, etc.) approximations of the method.
The publication of this review is a result of the increasing interest in the equations of higher approximations of the Chapman-Enskog
method, in particular, in the Burnett equations. Primarily, this is a consequence of advances in the application of the latter
equations to the problem of shock wave structure and the possible (by virtue thereof) expansion of the region of applicability
of the macroscopic rarefied gas flow models. In addition, using the Burnett approximation, a series of new interesting effects
has been established. In particular, it has been shown that in the classical problem of heat transfer between bodies heated
to different temperatures it is necessary to take into account not only the temperature creep but also the Burnett temperature
stresses. At the same time, a number of fundamental problems concerning the status of these equations has not yet been solved.
An extensive literature is devoted to this problem. In the present paper, we shall mainly consider studies carried out since
the end of the sixties and concerned with the best developed case of neutral monatomic (structureless) non-dense gas flow
in the absence of external forces. Purely mathematical problems are not considered. Preference is given to final or review
studies appearing in the most accessible publications.
In Section 1 we consider the problem of taking out-of-order terms in the Chapman-Enskog method into account and make a brief
comparison with other perturbation methods. Section 2 is devoted to the expressions for the transport properties in the Burnett
and super-Burnett approximations. The relations between the Burnett approximation and axiomatic continuum mechanics theory,
the thermodynamics of irreversible processes, and nonequilibrium statistical mechanics are discussed in Section 3. New effects
due to specific terms of the formulas for the Burnett transport properties as Kn → 0 are considered in Section 4. The basic
features of formulations of the boundary value problems for the systems of gasdynamic equations considered are outlined in
Section 5. Finally, we describe the results of applying these equations to particular problems: some exact results of the
kinetic theory (Section 6), the propagation of sound waves and cylindrical Couette flow (Section 7), shock wave structure
and hypersonic hydrodynamics (Section 8).
Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 3–28, July–August, 1998.
The work was carried out with financial support from the Russian Foundation for Basic Research (project No. 96-01-01244). |
