Dissipative magnetogasdynamic flow

Abstract

An analysis of the structure of the wakes and waves in steady compressible magnetohydrodynamics is presented. No restriction is made on the equation of state of the gas or on the ratios of the various dissipative parameters. An asymptotic solution is obtained which furnishes directly the flow far from a body and which may be used in the construction of the entire flow field. The non-dissipative solutions are obtained as a non-uniform limit for vanishing dissipation; no matter how small the dissipation, one can go far enough from the origin that the flow is essentially dissipative. For non-aligned fields the wave pattern consists of a downstream wake and either two or four standing waves, depending on the flow regime. For aligned fields, two of these waves become wakes, so that the wake is a superposition of three structured layers, with either all downstream or two downstream and one up-stream. It is found that the nondissipative limit of the wake is non-unique for the aligned fields case. Different limits are obtained depending on how the various dissipative parameters vanish. In this paper we consider steady magnetohydrodynamic flow in the Oseen approximation. No restriction is placed on the equation of state of the gas or on the various dissipative parameters (viscosity, thermal and electrical conductivity). The method of approach is to obtain the fundamental solution. It was shown in an earlier paper (Salathe & Sirovich 1967) how this could be used to obtain the solution for arbitrary boundary-value problems. In 5 2 of the present paper we demonstrate that the fundamental solutions themselves provide the far field flow past a finite body. This is obtained simply in terms of such quantities as the total force on the body, heat added, etc. In 53 we obtain the fundamental solutions for the non-aligned fields case. An asymptotic solution is obtained applicable for distance from the origin large compared to the mean free path. It is well known that there exist two distinct flow regimes in magnetohydrodynamics, the doubly hyperbolic and the hyperliptic. In each of these we obtain a downstream wake which is a pure entropy wake. That is, it carries only density and temperature disturbances and is structured by thermal conductivity. In the doubly hyperbolic regime, the flow exhibits, in addition, four structured waves, while in the hyperliptic case, t Present address: Center for the Application of Mathematics, Lehigh University, Bethlehem, Pennsylvania.