On Stratified Flows in a Gravitational Field

Abstract

This paper, which is restricted to the flow of inviscid fluids, is divided into three parts. In the first part existing formulas by A. R. Richardson, Hans Lewy, and K. N. Tong for constructing free-surface flows are identified with one another so that potential flows of one fluid in contact with a stagnant layer of another fluid can be constructed exclusively from one general formula, with proper modification of the gravitational acceleration. Further, a method for constructing potential flows of two fluids having a common interface (which may or may not be prescribed) is devised. It is hoped that this method will be of some use in the investigation of internal gravity waves. In the second part, it is shown that, for flows of a fluid system of discrete layers, Long's equation of motion for two-dimensional flow of an inhomogeneous fluid reduces to the Laplace equation for each layer (if the flow is irrotational) and the usual boundary conditions at the interfaces, so that flows with discontinuous density variations can be properly considered as limiting cases of flows with continuous ones. For the sake of completeness, equations of motion of an inhomogeneous fluid in axisymmetric motion in cylindrical and spherical coordinates are also given. In the third part the stability of a periodic disturbance present in a parallel flow with continuous density variation is discussed. Sufficient conditions for stability are found, and an upper bound for the amplification factor is given (if instability occurs) for a general class of flows.