On the dynamics of a free surface of an ideal fluid in a bounded domain in the presence of surface tension

We derive a set of equations in conformal variables that describe a potential flow of an ideal inviscid fluid with free surface in a bounded domain. This formulation is free of numerical instabilities present in the equations for the surface elevation and potential derived by A. I. Dyachenko et al in 1996 with some of the restrictions on analyticity relieved. We illustrate with the results of a comparison of the numerical simulations with the exact solution, the Dirichlet ellipse. In presence of surface tension, we demonstrate the oscillations of the free surface of a unit disc droplet about its equilibrium, the disc shape.