RESUMEN

We consider an inviscid fluid, initially at rest inside a wedge, bounded by one free surface and one solid surface. When $t=0$, we allow the contact angle to change discontinuously, which leads the free surface to recoil under the action of surface tension. As noted by Keller and Miksis [SIAM J. Appl. Math., 43 (1983), pp. 268–277], a similarity scaling is available, with lengths scaling like $t^{2/3}$. We consider the situation when the wedge is slender, with angle $\epsilon \ll 1$, and the contact angle changes from to . The leading order asymptotic problem for $\lambda = O(1)$, a pair of nonlinear ordinary differential equations, was considered by King [Quart. J. Mech. Appl. Math., 44 (1991), pp. 173–192], numerically for $\lambda = O(1)$ and asymptotically for $|\lambda-1| \ll 1$. In this paper, we begin by considering this system when $1 \ll \lambda \ll \epsilon^{-1}$, and use Kuzmak's method to construct the asymptotic solution. When $\lambda =O(\epsilon^{-1})$, the slope of the free surface becomes of $O(1)$, and it is no longer possible to reduce the problem to ordinary differential equations alone. However, we can approach this problem in a similar manner, even though the underlying oscillator is the solution of a nonlinear boundary value problem for Laplace's equation, and construct an asymptotic solution. In fact, the solution takes the form of a modulated set of waves on fluid of finite depth, with the underlying analytical solution given by Kinnersley [J. Fluid Mech., 77 (1976), pp. 229–241]. The case $\lambda\epsilon = 90^\circ$ is the solution for the inviscid recoil of a wedge of fluid with two free surfaces and semiangle $\epsilon \ll 1$, which was discussed by Billingham and King [J. Fluid Mech., 533 (2005), pp. 193–221]. We also show that no non-self-intersecting solution is available for $\lambda\epsilon > 90^\circ$ as $\epsilon \to 0$, and compare our asymptotic solutions with numerical, boundary integral solutions of the full, nonlinear free boundary problem.

RESUMEN

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 $\alpha \to \pi$.