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

TL;DR

This paper develops a stable conformal variable formulation for modeling ideal fluid free surface dynamics in bounded domains, including effects of surface tension, and validates it through numerical simulations and exact solutions.

Contribution

It introduces a new stable set of equations in conformal variables that improves upon previous models by reducing numerical instabilities and relaxing analyticity restrictions.

Findings

Numerical simulations agree with the Dirichlet ellipse solution.

Surface tension causes oscillations of the droplet's free surface.

The formulation effectively models free surface dynamics with surface tension.

Abstract

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.