Notes on a differential equation which occurs in the two-dimensional motion of a compressible fluid and the associated variational problems

Abstract

In two recent papers* Prof. G. I. Taylor and Dr. C. F. Sharman have discussed the flow of a compressible fluid by an experimental method and have formulated tentatively the criterion that no irrotational solution of the equations of motion exists when the velocity of the fluid at any point of the field of flow exceeds the local velocity of sound. They remark, however, that there appears at present to be no theoretical ground for supposing that this hypothesis is true in general. In this connection it may be pointed out that there is a change in type of the governing partial differential equation when the local velocity of sound is exceeded and the question naturally arises whether the problem is properly put when such a change occurs and the usual boundary conditions are retained. In order to present this question more definitely we take the case of steady motion in two dimensions and denote the velocity components at the point ( x , y ) by u , v , respectively and the density of the fluid by ρ. The equation of continuity is then ∂/∂ x (ρ u ) + ∂/∂ y (ρ v ) = 0. (1) When the pressure, p , is a function of the density and gravity is the only external force, Bernoulli’s equation takes the form ∫ dp /ρ + ½ ( u 2 + v 2 ) + Ω = C (2) where Ω. is the potential energy per unit mass and C is constant along a streamline. Since the motion is steady we may write u ∂C/∂ x + v ∂C/∂ y = 0 (3)