On the evolution of disturbances at an inviscid interface

Abstract

The initial-value problem for the evolution of the interface η(x, t) separating two unbounded, inviscid streams is considered in the framework of linearized analysis. Given the initial shape y = εη₀(x) of the interface at t = 0 the objective is to calculate the interface shape η(x, t) for later times. First, it is shown that, if the vortex sheet is of infinite extent, if surface tension is absent and if the two streams are of the same density, the evolution is given by \[ \eta(x,t) = \epsilon(1-\alpha)^{-1}{\rm Re}[\{(1-\alpha)+(1+\alpha)i\}\eta_0\{x-{\textstyle\frac{1}{2}}((1+\alpha)+(1-\alpha)i)t\}], \] where α ([not equal] 1) is the ratio of the speeds of the streams, provided the initial interface shape εη₀(x) is analytic and its Fourier transform decays sufficiently rapidly. An interesting consequence is that it is possible, under certain circumstances, for the interface to develop singularities after a finite time. Next it is shown that when the two streams move at the same speed (α = 1) the growth of η is given by \[ \eta(x,t) = \epsilon\eta_0(x-t)+\epsilon t\,d\eta_0(x-t)/dx \] with mild restrictions on η₀(x). The major effect of surface tension, it is found, is to prevent the occurrence of singularities after a finite time, a distinct possibility in its absence. Finally the vortex sheet shed by a semi-infinite flat plate is considered. The unsteady mixed boundary-value problem is formally solved by using parabolic coordinates and Fourier-Laplace transforms.