Hydromagnetic Plane Steady Flow in Compressible Ionized Gases

Abstract

The magnetohydrodynamics in compressible gases of infinite conductivity is discussed under the assumptions that the flow is isentropic and steady and that the directions of flow velocity and of magnetic field are in the same plane. The hyperbolicity conditions for the starting equations, under which the discontinuity surfaces can appear, are investigated. The special case in which the flow is parallel to the direction of magnetic field is analyzed in detail. has been investigated in connection with the hydromagnetic shock waves appearing in various branches of plasma physics, and it has been shown that there exist many remarkable features essentially due to the effect of magnetic field. It has especially been pointed out by a few authorsl)2) that the equation for steady hydromagnetic flow can be hyperbolic even for the subsonic fluid velocity if the magnetic field is of moderate strength. In this paper, we shall deal with the plane isentropic steady flow, which is specified by the condition that the flow and the magnetic field are in the same plane, and discuss the conditions for the discontinuity surfaces to occur, or under which the characteristic equation of the original system has real roots. In the next section, we shall write down the characteristic equation explicitly and show that in our case this equation is the fourth order algebraic equation for the local characteristic direction and that the above conditions can be expressed not only by the usual sonic Mach number but also by the magnetic Mach number, the flow speed versus the Alfven velocity. VVe shall also give a graphical method convenient for the calculation of the real roots of this equation under arbitrary configurations of flow and magnetic field and it will then be shown that there exist, at least, two real roots when one double root is counted as two single roots, more­ over, that the stream lines cannot, in general, be the characteristics unless the flow is parallel to the magnetic field. * In the parallel flow, the stream line corresponds to one double root of the characteristic equation and may be a separation or a free surface, as is usual in hydrodynamics. It seems interesting to note that as the