Fluid flows obtained by geometrical characterizations

Abstract

A problem in fluid mechanics is considered solved in the usual sense, if we can describe fairly explicitly certain functions such as pressure, density, velocity, etc. over a suitable domain. From such a complete description of a “fluid flow” we can pick out certain purely geometrical attributes such as the family of constant pressure surfaces, or the family of curves determined by the streamlines. In actual fact, of course, we don’t often come up with complete explicit solutions of fluid mechanics problems, and often start our consideration of a problem by focusing our attention on the geometry-for instance, we might begin by assuming the flow is “two dimensional.” Such an assumption is usually motivated by a given physical situation. If we are more concerned with obtaining a general understanding of the nature of solutions in fluid mechanics rather than solving a specific problem, then any one of an endless number of geometrical assumptions might be imposed. Thus we might look for flows whose streamlines form normal congruences of curves, or whose principal normal congruences form normal congruences [I], etc. Rather than pick specific geometrical properties we might ask more general questions of the type “to what extent does the specification of the geometry of a flow determine the flow ?” Or “to what extent do the equations governing the flow restrict the possible shapes of the streamlines, constant pressure surfaces, or other intrinsic geometrical configurations ?” The study of the differential geometry of fluids is old. The spirit is represented (and most results presented) in Refs. [l-9] and references therein. The present contribution is in that spirit. We shall be concerned with steady nondissipative compressible fluid flow