Two-dimensional subsonic flows of a compressible fluid and their singularities

Abstract

1. Methods for generating stream functions of two-dimensional flows of a compressible fluid. The consideration of physical phenomena, in particular the study of electric and magnetic fields, was one of the starting points from which Riemann developed his approach to the theory of integrals of algebraic functions. In particular the consideration of two-dimensional electric and magnetic fields without singularities or with such singularities as vortices, sinks, sources, doublets, and so on, suggests the introduction of integrals of the first, second, and third kinds. The investigation of certain phenomena in fluid dynamics, namely the consideration of two-dimensional, irrotational, steady flow patterns of an incompressible fluid, leads to the same mathematical notions as those mentioned above since these flows are, from an abstract mathematical point of view, not essentially different from electric and magnetic fields. Generalizing this approach, one can introduce flow patterns of a compressible fluid with corresponding singularities and investigate relations between potentials and stream functions of these flows. The compressibility equation is, however, much more complicated than Laplace's equation, and it is very questionable whether such an immediate generalization would lead to results comparable with those in the theory of functions of a complex variable. It seems that it is preferable in this case to use the hodograph method (see below) and to link this approach with the theory of operators which transform solutions of one partial differential equation into solutions of another one. A two-dimensional steady irrotational flow of a perfect fluid can be described either by its potential (in the following denoted by 0) or by the stream function, 41. In the case of an incompressible fluid 0 and 41 are connected by the Cauchy-Riemann equations, so that O+i4l is an analytic function, f, of a complex variable. Taking the real and imaginary part of f we obtain 0 and 41, respectively. This process can obviously be interpreted as an operation