Approximating kinetic equation for weakly nonequilibrium states of polyatomic gases

Abstract

The aim of the present paper is to construct an approximate kinetic equation that, first, takes into account correctly the possibility of excitation of both rotational and vibrational degrees of freedom of the molecules and, second, is valid for any law of intermolecular interaction.The successes of the kinetic theory of polyatomic gases are associated primarily with the calculation of transport coefficients, the derivation of equations of hydrodynamic type, and the description of relaxation processes in an unbounded space. In recent years, there has been a considerable growth of interest in boundary problems of the dynamics of rarefied gases, including polyatomic gases (Knudsen layer, exterior and interior flows at intermediate values of the Knudsen number, etc.).The theoretical solution of such problems is appreciably simplified if approximating kinetic equations are used instead of the Boltzmann equation.In [i], the method of entropy maximization was developed to construct model kinetic equations.The main shortcoming of the model equation obtained in [i] is that, possessing too few free parameters, it does not give a correct description of the process of relaxation of some important macroscopic parameters, for example, the heat flux component due to the presence of internal degrees of freedom of the molecules.This shortcoming was corrected in [2], in which an approximating kinetic equation was obtained for a diatomic gas (only rotational degrees of freedom considered) with Maxwellian law of intermolecular interaction.In [2], the kinetic equation is averaged over all possible directions of the vector of the intrinsic angular momentum of the molecules and over the energy of the rotational motion.In [3], approximating kinetic equations are constructed for polyatomic gases by the Gross--Jackson method.The equation of the third approximation is derived explicitly by an expansion of the collision integral with respect to eigenfunctions and eigenvalues. A similar equation is obtained in [4] by the method of equal moments.The difference from the results of [3] is due to the fact that in [4] different expressions are used for the moments of the collision integral and, further, there is an error in the expression for the equilibrium value of the mean internal energy of the molecules.A general shortcoming of the existing approximating equations is that they make it possible to take into account correctly only one form of the internal energy of the molecules, for example, the rotational energy.In addition, some of these equations are obtained for Maxwellian molecules and, after averaging, do not take into account the details of the intermolecular interaction. This last circumstance can lead in the solution of concrete problems to appreciable errors, at least quantitatively.The aim of the present paper is to construct an approximating kinetic equation that, first, takes into account correctly the possibility of excitation of both rotational and vibrational degrees of freedom and, second, is valid for any law of intermolec...

Heat and mass transfer of a multiatomic gas in a cylindrical capillary with arbitrary Knudsen numbers

“…In the present work, the heat and mass transfer of a multiatomic gas in a cylindrical capillary is described on the basis of the model equation of [3], which, in contrast to the Hanson--Morse equation, takes account of the possibility of excitation of two forms of internal energy: rotational and vibrational. The problem is solved by a variational method, which allows sufficiently accurate results to be obtained over the whole range of Knudsen numbers (Kn) and, in addition, leads to a simple analytical expressions for the fluxes in the two limiting cases: Kn << i and Kn ~ 1.…”

“…Then, taking account of Eq. (i), the model kinetic equation [3] for the perturbation function hij is written in the following dimensionless form…”