High Spots for the Ice-Fishing Problem with Surface Tension

In the ice-fishing problem, a half-space of fluid lies below an infinite rigid plate (``the ice'') with a hole. In this paper, we investigate the ice-fishing problem including the effects of surface tension on the free surface. The dimensionless number that describes the effect of surface tension is called the Bond number. For holes that are infinite parallel strips or circular holes, we transform the problem to an equivalent eigenvalue integro-differential equation on an interval and expand in the appropriate basis (Legendre and radial polynomials, respectively). We use computational methods to demonstrate that the high spot, i.e., the maximal elevation of the fundamental sloshing profile, for the IFP is in the interior of the free surface for large Bond numbers, but for sufficiently small Bond number the high spot is on the boundary of the free surface. While several papers have proven high spot results in the absence of surface tension as it depends on the shape of the container, as far as we are aware, this is the first study investigating the effects of surface tension on the location of the high spot.