On the stability of superposed streams of Fluids of different densities

Abstract

1.1. The gliding of warm air over cold air, and of fresh water over salt water, without appreciable turbulent mixing, is a known phenomenon. Theoretically we know that, with a sudden transition in velocity and density, the laminar motion of superposed streams of fluid is unstable for any relative velocity, no matter how small, when viscosity is neglected. In reality, how­ ever, such sudden transitions cannot occur, or at any rate cannot be permanent, and when the width of the layer of transition is comparable with the wave­ length of the disturbance assumed, the analysis for an abrupt transition has no physical significance. The stability of such a system with a finite width of the layer of transition is therefore investigated below by the method of small oscillations. The investigation is a generalisation to heterogeneous stratified fluids of Rayleigh’s discussion of the homogeneous case. 1.2. The effect of a stratification of density in tending to prevent turbulence has lately been investigated by Prandtl, who, by considerations of energy, has arrived at the following criterion for the continuance of turbulence.