Plane Stokes flow driven by capillarity on a free surface

Abstract

The free creeping viscous incompressible plane flow of a finite region, bounded by a simple smooth closed curve and driven solely by surface tension, is analysed. The shape evolution is 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 Ω (ζ, t ) is derived. In practice, it has been necessary to guess a parametric form, i.e. Ω (ζ, t ) = Ω[ζ; a 1 ( t ), a 2 ( t ), …], whose validity must be verified using the shape-evolution equation. Polynomial and proper rational mappings with no repeated factors are apparently always valid in principle. Solutions are given for (i) regions bounded initially by a regular epitrochoid, (ii) the limiting case of a half-plane bounded by a trochoid, and (iii) a class of rosettes whose mapping is rational. The two-lobed rosette gives the exact solution of the coalescence of equal cylinders. All these mappings involve limiting initial shapes having inward-pointing cusps. Useful parameterizations providing regions whose limiting shapes possess corners or outward-pointing cusps have not been found.