The Classification of One-Dimensional Flows and the General Shock Problem of a Compressible, Viscous, Heat-Conducting Fluid

Abstract

Recently a great deal of attention has been focused on the problem of determining the structure of a shock wave in a on the basis of the Navier-Stokes equations. In this paper, we shall denote by perfect gas a viscous, heat-conducting fluid that satisfies the equation of state p — gRpT, whose specific heats are in general functions of temperature. Becker was the first to consider the exact structure for a case where the Prandtl Number is neither zero nor infinite. His analysis was restricted to constant specific heats, constant coefficients of heat conduction and viscosity, and constant Prandtl Number = 3/4. After him several authors ~ 10 have attempted to solve the problem under more general conditions. Since the first approximate discussion by Taylor, other approximate methods have been developed. n In some cases, the Navier-Stokes equations were reduced to an ordinary differential equation of higher than the first order; in others the reduction of Becker to a first-order equation was used. The approximate methods resort directly to the Navier-Stokes equations. I t was von Mises who reformulated Becker's approach and thus gave not only a much simpler presentation of the shock problem, but also a method of reviewing the general characteristics of all possible one-dimensional steady motions. I t is the purpose of this paper to show that , if the Navier-Stokes equations for the flow of a are taken as a basis, then the method of review given by von Mises enables a complete classification of the flows to be made. Moreover, a discussion is presented of the problem of the shock wave in more than one dimension, which shows that a fuller numerical treatment of von Mises' equations and not a new approximation is needed in this case.