Determination of Flows Behind Stationary and Pseudo-Stationary Shocks

Abstract

It is the purpose of this paper to discuss two dimensional stationary and pseudo-stationary flows behind shocks. We shall study the relations between geometric properties of the shock and the values of the flow variables behind the shock at points removed from the shock. The arguments which will be used will be local in character. Thus we shall attempt to describe the value of a flow variable at a point removed from a shock by the values of the derivatives of this variable at the shock. In particular we shall give a simple derivation of Thomas' results [4] on the relations between the geometric properties of the shock front and the geometric properties of a streak-line behind the shock front; that is, a locus of particles which have crossed a fixed point of the shock front at times earlier than the time under consideration. The discussion will first assume that the flow ahead of the shock is known and that the shock itself is a known analytic curve. The integration of the various conservation equations will then be shown to be equivalent to the solution of a Cauchy problem of rather simple form, with the shock front as the curve on which the initial data is given. It then follows that in a neighborhood of the shock the flow behind the shock exists and is uniquely determined. We next consider the problem of determining the geometric properties of the shock front from the geometric properties of the streak-lines, that is, of solving some of the relations referred to above. In view of the existence theorem quoted, it is evident that the existence and uniqueness of solutions of the hydrodynamic equations when shocks are present, subject to the usual boundary condition, namely the specification of a streak-line, is intimately related to the existence and uniqueness of the solutions of the equations determining the geometric properties of the shocks from those of the streak-lines. Finally we consider the case of shocks with singularities, that is, shocks which contain points where the tangent, the curvature or some derivative of the curvature is discontinuous. In a previous paper [3] it was shown that if one assumes that at time t every point in the region behind the shock is occupied by a single fluid particle then the geometric nature of the shock is determined in terms of one parameter. In this paper we examine this result in greater detail than previously.