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

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 $t^{2/3}$. We find that an asymptotic balance is possible when the wedge semi-angle, $\alpha$, is small, in an inner region of $O(\alpha^{4/9})$, a distance of $O(\alpha^{-2/9})$ from the origin, which leads to a simpler boundary value problem at leading order. Although we are able to reformulate the inner problem in terms of a complex potential and reduce it to a single nonlinear integral equation, we are unable to find a solution numerically. This is because, as noted by Vanden-Broeck, Keller & Milewski (2000), and reproduced here numerically using a boundary integral method, the free surface is self-intersecting for $\alpha\,{<}\, \alpha_0\,{\approx}\,2.87^\circ$. Since disconnected solutions are only possible when there is a void inside the initial wedge, we consider the effect of an inviscid low-density fluid inside the wedge. In this case, a solution is available for a slightly smaller range of wedge semi-angles, since the flow of the interior fluid sucks the free surfaces together, with the exterior flow seeing pinch-off at a finite angle. We conclude that for $\alpha\,{<}\,\alpha_0$, we must introduce the effect of viscosity at small times in order to regularize the initial value problem. Since the solution for $t\,{\ll}\,1$ 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.