A new method for solving of vector problems for kinetic equations with Maxwell boundary conditions

TL;DR

This paper introduces a novel method for solving vector kinetic equations with Maxwell boundary conditions, improving accuracy in temperature and concentration jump calculations in rarefied gases.

Contribution

It develops a new approach using vector Fredholm integral equations and Neumann polynomials to solve kinetic problems with Maxwell boundary conditions.

Findings

Achieves 10-fold increase in accuracy over previous methods

Reduces the problem to a solvable vector Fredholm integral equation

Provides numerical results for temperature and density jumps

Abstract

In the present work the classical problem of the kinetic theory of gases (the Smoluchowsky' problem about temperature jump in rarefied gas) is considered. The rarefied gas fills half-space over a flat firm surface. logarithmic gradient of temperature is set far from surface. The kinetic equation with modelling integral of collisions in the form of BGK-model (Bhatnagar, Gross and Krook) is used. The general mirror-diffuse boundary conditions of molecules reflexions of gas from a wall on border of half-space (Maxwell conditions) are considered. Expanding distribution function on two orthogonal directions in space of velocities, the Smoluchowsky' problem to the solution of the homogeneous vector one-dimensional and one-velocity kinetic equation with a matrix kernel is reduced. Then generalization of source-method is used and boundary conditions include in non-homogeneous vector kinetic equation. The solution in the form of Fourier integral is searched. The problem is reduced to the solution of vector Fredholm integral equation of the second sort with matrix kernel. The solution of Fredholm equation in the form of Neumann's polynoms with vector coefficients is searched. The system vector algebraic interengaged equations turns out. The solution of this system is under construction in the form of Neumann's polynoms. Comparison with well-known Barichello - Siewert' high-exact results is made. Zero and the first approach of jumps of temperature and numerical density are received. It is shown, that transition from the zero to the first approach raises 10 times accuracy in calculation coefficients of temperature and concentration jump.