Surface Tension-driven Flow in A

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 t2/3. We consider the situation when the wedge is slender, with angle 1, and the contact angle changes from to λ . The leading order asymptotic problem for λ = 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 λ = O(1) and asymptotically for |λ − 1| 1. In this paper, we begin by considering this system when 1 λ −1, and use Kuzmak’s method to construct the asymptotic solution. When λ = O( −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 λ = 90◦ is the solution for the inviscid recoil of a wedge of fluid with two free surfaces and semiangle 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 λ > 90◦ as → 0, and compare our asymptotic solutions with numerical, boundary integral solutions of the full, nonlinear free boundary problem.