Two-dimensional slow viscous flows with time-dependent free boundaries driven by surface tension

Abstract

We consider the two-dimensional quasi-steady Stokes flow of an incompressible Newtonian fluid occupying a time-dependent simply-connected region bounded by a free surface. The motion is driven by a constant surface tension acting at the free boundary so that, with the effects of gravity ignored, one expects the boundary to approach a circular form as time evolves. It is shown that, if at some initial instant the region occupied by the fluid is given by a rational conformal map of the unit disc, then it must retain this property as long as the region remains simply-connected. Moreover, its evolution may be described analytically; in simple cases this description is explicit, but in more complicated problems the numerical integration of a system of first order differential equations may be required.