Dynamics of $2$D fluid in bounded domain via conformal variables

TL;DR

This paper numerically investigates traveling wave solutions of a 2D ideal fluid boundary problem with surface tension, revealing behaviors like approaching Crapper waves and elongation with nonlinearity, using high-resolution Fourier methods.

Contribution

It introduces a high-precision numerical approach to analyze boundary wave solutions, exploring their limiting behaviors and properties in a 2D fluid model.

Findings

Solutions with multiple lobes approach Crapper capillary waves.

Few-lobe solutions become elongated with increased nonlinearity.

High Fourier resolution enables detailed solution analysis.

Abstract

In the present work we compute numerical solutions of an integro-differential equation for traveling waves on the boundary of a $2$D blob of an ideal fluid in the presence of surface tension. We find that solutions with multiple lobes tend to approach Crapper capillary waves in the limit of many lobes. Solutions with a few lobes become elongated as they become more nonlinear. It is unclear whether there is a limiting solution for small number of lobes, and what are its properties. Solutions are found from solving a nonlinear pseudo--differential equation by means of the Newton-Conjugate Residual method. We use Fourier basis to approximate the solution with the number of Fourier modes up to $N = 65536$.