Two- and Three-Dimensional Flows in Magnetogasdynamics

Abstract

In the analysis of one-dimensional flow (chapter IX), we actually considered only the average values of the velocity, pressure, etc., over certain sections. In order to study the details of the flow patterns, we have to find the velocity, pressure, etc., at every point in space. Hence we have to study the three dimensional flow and consider all three components of the velocity and magnetic field together with the pressure, density and temperature of the plasma as functions of the three spatial coordinates x, y, and z and time t. In general the three dimensional flow problem is too complicated to be analyzed. We have to make some reasonable simplifications in order to discuss some flow problems which are good approximations to the actual conditions. It is well known that the effects of viscosity µ and heat conductivity ϰ are negligibly small for high Reynolds number flow except in the boundary region near the solid wall or some other transition regions where the velocity and the temperature gradients are large. Hence in a majority of flow problems, the plasma may be considered as an inviscid and non-heat-conducting fluid. We shall make such an assumption in our treatments in §§ 2 to 5. For a plasma, besides viscosity and heat conductivity, there is a third diffusion property which may be characterized by the magnetic viscosity v H (or by electrical conductivity σ). If we assume v H = 0 in addition to µ = 0 and ϰ = 0, we have an ideal plasma (cf. chapter IV, § 5). The properties of an ideal plasma have some similarities to those of an inviscid and non-heat-conducting gas. Particularly the system of the fundamental equations for the three dimensional and unsteady flows of an ideal plasma is hyperbolic just as that for a similar flow of ordinary gasdynamics of an inviscid fluid. We may use the method of characteristics to solve such problems. In § 2, we shall discuss the three dimensional unsteady flows of an ideal plasma.