Analytical solution of the velocity-slip and diffuslon-slip problems by a cauchy integral method

Abstract

Some one-dimensional boundary value problems of the kinetic theory of gases can be solved analytically in closed form when the Boltzmann collision operator is replaced a simple kinetic model such as the linear BGK (Bhatnagar-Gross-Krook) model. Such are the slip-flow and the diffusion slip problems. Analytical solutions were obtained by Wiener-Hopf or singular eigenfunctions expansions methods in the sixties and seventies. Here, a Cauchy integral method of solution, developed for radiative transfer problems, is applied to the slip-flow problem and to the diffusion slip problem for a binary gas mixture (with A. Latyshev¹). In the CI method, which can handle Wiener-Hopf integral equations with exponentially or non-exponentially decreasing kernels, the Wiener-Hopf equation is recast into a singular integral equation of the Cauchy type, by an inverse Laplace transformation. The latter equation is solved by a classical method of reduction to a boundary value problem in the complex plane (i.e. a Riemann-Hilbert problem). The equivalence between our solutions and other other analytical representations is discussed.