The Existence and Limit Behavior of the One-Dimensional Shock Layer

Abstract

Introduction. We consider steady, one-dimensional flows of a viscous, heat-conducting fluid which approach finite limit values at x = +oo and x = -oo. Such flows display the character of a shock wave (for small viscosity, ,u, and heat conductivity, A), in that they differ sensibly from their end states at x + oo only in a small interval of rapid transition. In analogy with the classical boundary layer, and also to distinguish these flows from the shock waves which belong properly to the theory of ideal fluids, we follow Weyl [1] in naming such a flow a shoclk layer. The one-dimensional shock layer is in certain respects the prototype of all shock phenomena and has therefore been studied widely, with particular emphasis on the problemn of thickness of the shock front [2, 3, 5, 6, 8]. I-Towever, basic problems concerning these flows, such as those of existence, and limit behavior for small A, ,u, remain open. Their solution, which we consider here, is a step towards placing on a sound basis the relation between the theories of real and ideal fluids. The general problem of existence of the shock layer for a fluid with given A, j, and with the preassigned end states, has been studied inconclusively by Rayleigh [4] and Weyl [1]. Until now, the existence of the shock layer seems to have been definitely proved only for an exceptional set of ideal gases for which a postulated relation between A, ,u, and the specific heat at constant pressure,' permits explicit integration of the equations of motion; (Becker [2], also [5, 6]). We succeed here in obtaining an essentially complete solution of the existence problem by proving the existence andI uniqueness of the shock layer for the general class of fluids considered by Weyl, with A, ,u arbitrary functions of the state, and for arbitrary end states satisfying the shock relations (Theorem 1). This result, therefore, establishes for gene.ral fluids an exact correspondence between the steady one-dimensional shock waves and the shock layers.