The use of the moment method to derive the gas and plasma transport equations with transport coefficients in higher-order approximations

Abstract

Grad's moment method is used to derive the linear equations of mass momentum and energy transfer of the components and to obtain all the transport coefficients (kinetic coefficients) for a multicomponent mixture of monatomic gases. A system of equations for the expansion coefficients of the non-equilibrium correction to the distribution function using a system of irreducible tensorial Hermite polynomials (the equations of moments) is obtained on the basis of the linearized Boltzmann equations for the components of the mixture. The assumptions under which these equations reduce to a system of algebraic equations for determining of the mass diffusion fluxes, the heat fluxes of the components and the partial viscous stress tensors are analysed. The advantage in writing the transport equation in a “forces in terms of fluxes” representation for solving actual problems of the flows of multicomponent mixtures as compared with the classical “fluxes in terms of forces” representation in the standard Chapman-Enskog method [1–3] is demonstrated. Different ways of representing the transport equations and expressions for the transport coefficients are considered in an arbitrary order of approximation with respect to the number of Sonine polynomials which are retained in the expansion of the distribution function (the Chapman-Cowling method). This enables one, in particular, to establish a direct link between results obtained by different methods and to track more clearly the constraints which are actually used when employing classical method [1–3] and the modified method [4, 5] of deriving the transport equations and calculating the transport coefficients in the Chapman-Enskog scheme