Abstract
In this paper, we consider the two-dimensional motion of a viscous, incompressible fluid with a free surface, initially lying inside a wedge. The fluid flows under the action of surface tension, and we analyze its small time motion using the method of matched asymptotic expansions. We show that, in contrast to the case where there is a surrounding fluid with viscosity [M. J. Miksis and J.-M. Vanden-Broeck, Phys. Fluids, 11 (1999), pp. 3227-3231], the initial motion is not self-similar but develops over two asymptotic regions: an inner, nonlinear, surface tension--driven Stokes flow region near the tip of the wedge, and an outer, linear, unsteady Stokes flow region, where inertia is important but surface tension is not. The initial velocity of the tip of the wedge is singular, of $O(\log t)$ as $t \to 0$. We calculate numerical solutions of both the inner and outer problem for a general wedge semiangle, $\alpha$, and also construct asymptotic solutions in the limits $\alpha \to 0$ and $\alpha \to \pi$.
Published Version
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