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.