Effect of density variation on fluid flow

Abstract

The effect of density variation on the flow of an incompressible and inviscid fluid is twofold. On the one hand, the inertia of the fluid changes in direct proportion to the density. On the other hand, the body force acting on a fluid element also changes in direct proportion to the density. Since body force is not the only force acting on the fluid, the inertia effect and the gravity effect of density variation do not cancel each other, and many interesting phenomena occur in the flow of a heterogeneous fluid that do not occur in the flow of a homogeneous fluid. In this paper it is shown that the inertia effect can be simply evaluated for steady flows. If the velocity in the steady flow of a heterogeneous fluid in the absence of gravity is multiplied by the square root of the density, the result represents a dynamically possible flow of a homogeneous fluid. At the other extreme, when the gravitational effect dominates the flow, it has been shown both analytically and experimentally that the motion of a fluid is confined to the layer at which it originates. As usual, it is when the inertia effect and the gravity effect are comparable that the solutions of stratified flows become difficult, even if the flow is assumed to be steady and the fluid inviscid. From one series of such solutions and the supporting experiments one sees that, on the one hand, infinitely many modes of stationary internal waves of finite amplitude are dynamically possible (apart from the consideration of generation), and, on the other hand, physically significant solutions of stratified flows may involve velocity discontinuities.