Applied Mathematics, University of Waterloo
Problems in Capillarity
A capillary surface represents the interface between fluids, which can be determined by the Young-Laplace equation. In this talk, we consider several problems raised in capillarity. The problem of a two-dimensional cylinder on an unbounded bath was first studied in a groundbreaking paper of Bhatnagar and Finn in 2006. We reconsider the problem and give a full discussion of the number of equilibrium configurations and their stability. For the floating ball problem on an unbounded bath, the radially symmetric fluid interface can be obtained numerically by the shooting method. We give an example of two different configurations with the same height, where one of them has lower energy. A further discussion of the configurations and their stability is expected. We are also interested in a capillary surface on a domain with a corner or with exotic containers. For the wedge problem, with a restriction on the opening and the contact angles, it is well known that the fluid height is unbounded at the corner. There also exists an asymptotic expansion near the corner, which was obtained by Miersemann in 1993. If the wedge is tilted with a small angle, we wonder whether the unboundedness still holds and whether a nice asymptotic expansion can be obtained. Exotic containers are rotationally symmetric containers that admit an entire continuum of distinct equilibrium capillary surfaces. In 1991, Concus and Finn showed that there exists such an exotic container with free surfaces all enclosing the same liquid volume and having the same energy and contact angle. They also showed it is possible to find a stable configuration that is not symmetric. The exterior problem of exotic containers is also worthwhile to study.