Structure of weak shock wave discontinuities in magnetohydrodynamics

Abstract

In this paper, we study the propagation of a weak shock wave in a medium of initially constant fluid velocity, magnetic field and thermodynamic parameters. The structure of discontinuities for such a shock in real cases will be analyzed. By examining the change in variables inside the relaxation transition region, the length of the latter, i.e. of the disturbed region will be obtained. In order to derive the physical model explaining the finite shock length, several assumptions have been made: the medium has been treated as a very large layer of non-negligible viscosity and thermal conductivity. Starting from basic MHD relations, the invariants on the shock fronts, taking into consideration the process inside the disturbed region, have been calculated. Modified Rankine-Hugoniot equation discussing the process inside the relaxation region has been derived therefrom. Finally, the dependence of pressure upon distance has been examined under the assumptions: the fluid is considered as polytropic. Hence, by approximate integration of an obtained transcendental function, we get the length of relaxation region and discuss the result obtained.