On the Instability of Superposed Fluids in a Gravitational Field.

In this paper an approximate analytic solution is obtained for the following problem: A perfect, incompressible fluid occupies the upper half of a vertical tube, being supported against gravity by a rigid diaphragm. The lower half of the tube is empty. At time to the diaphragm is removed, and an infinitesimal disturbance of a simple kind is impressed on the free surface of the fluid. The problem is to describe the subsequent flow, on the assumption that the fluid at sufficiently great heights above the free surface is permanently at rest. The initial disturbance is so chosen that the fluid rises in the center of the tube and runs down at the sides. The range of validity of the approximate solution obtained is discussed. It is shown that the vertex height increases exponentially, in agreement with the linearized theory, until 0.2 X/2p0, where Xis the diameter of the tube and Pi is the first zero of the Bessel function J1(r). When >1.5 X/2 Pi, it increases at a nearly constant rate. The method of solution described is applied to an analogous problem involving two-dimensional flow between parallel plane walls and also to spatially periodic flows.