Surface-tension-driven flow outside a slender wedge with an application to the inviscid coalescence of drops

Abstract

We consider the two-dimensional inviscid flow that occurs when a fluid, initially at rest around a slender wedge-shaped void, is allowed to recoil under the action of surface tension. As noted by Keller & Miksis (1983), a similarity scaling is available, with lengths scaling like in the presence of viscosity is simply connected (Billingham 2005), the free surface must first pinch off at some finite time and then continue to do so at a sequence of later times. We investigate this using boundary integral solutions of the full inviscid initial value problem, with smooth initial conditions close to those of the original problem. In addition, we show that the inner asymptotic scalings that we developed for the steady problem can also be used in this time-dependent problem. The unsteady inner equations reduce to those for steady unidirectional flow outside a region of constant pressure, and can be solved numerically.We also show how the slender wedge solution can be related to the small-time behaviour of two coalescing drops, and describe the relationship between our solutions and those found by Duchemin, Eggers & Josserand (2003), for which a similar unsteady inner region exists. In each case, the free surface pinches off repeatedly, and no similarity solution exists.