Plane Stokes flow driven by capillarity on a free surface. Part 2. Further developments

Abstract

For the free creeping viscous incompressible plane flow of a finite region, bounded by a simple smooth closed curve and driven solely by surface tension, analyzed previously, the shape evolution was described in terms of a time-dependent mapping function z = Ω(ζ,t) of the unit circle, conformal on |ζ| [les ] 1. An equation giving the time evolution of the map, typically in parametric form, was derived. In this article, the flow of the infinite region exterior to a hypotrochoid is given. This includes the elliptic hole, which shrinks at a constant rate with a constant aspect ratio. The theory is extended to a class of semi-infinite regions, mapped from Im ζ [les ] 0, and used to solve the flow in a half-space bounded by a certain groove. The depth of the groove ultimately decays inversely with time.