VELOCITY INTEGRAL BRACKETS FOR A ONE-COMPONENT SYSTEM WITH SMALL INTERACTION

Abstract

The standard and widely used approach to the description of the hydrodynamic stage of the system evolution is the Chapman–Enskog method, in the framework of which the system distribution function is calculated in a perturbation theory in small gradients. The system kinetic coefficients are calculated on the basis of the obtained first-order-in-gradients distribution function, the temperature and velocity parts of which are the solutions of Fredholm integral equations of the first kind. In the literature it is considered that the standard approach to the solution of the corresponding integral equations is an approximate search for their solution with the help of the Galerkin method based on an artificially truncated Sonine polynomial expansion.

The so-called integral brackets are needed in order to calculate the coefficients multiplying the polynomials, and the calculation of the integral brackets is the most cumbersome stage of the kinetic coefficient calculation. In the literature it is assumed that the solution of a Fredholm equation of the first kind converges fast with increasing number of polynomials, that is why the corresponding analytical solutions are often restricted to the one- or two-polynomial approximations. However, for a number of systems the numerical investigation of the corresponding convergence is made on the basis of the corresponding numerical calculations for many-polynomial approximations. We do not know any works where the corresponding numerical investigation would be provided for a system with small interaction described by the Landau kinetic equation which contains the Landau collision integral. 

In our previous paper, we calculated the so-called temperature integral brackets for a one-component system with small interaction up to the thirteen-polynomial approximation; the corresponding integral brackets are needed in order to calculate the system thermal conductivity. In this paper, we calculate the so-called velocity integral brackets for a one-component system with small interaction up to the thirteen-polynomial approximation, the corresponding integral brackets are needed in order to calculate the system viscosity. The obtained results are important for a further numerical investigation of the convergence of the solutions for the system thermal conductivity and viscosity with increasing number of polynomials. The corresponding numerical investigation with a detailed calculation of the first-order-in-gradients distribution function may be given in another paper.

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