Global Bifurcation of Steady Surface Capillary Waves on a $2D$ Droplet

TL;DR

This paper constructs global bifurcation curves of steady rotational surface waves on 2D droplets, revealing symmetric wave solutions influenced by surface tension without gravity, using conformal and bifurcation theory.

Contribution

It introduces a novel global bifurcation analysis of rotational water waves on droplets with symmetry, focusing on surface tension effects and conformal formulation.

Findings

Existence of global bifurcation curves for steady surface waves.

Wave solutions exhibit discrete rotational and reflection symmetry.

The model describes tiny water droplets in breaking waves and white caps.

Abstract

We construct global curves of rotational traveling wave solutions to the $2D$ water wave equations on a compact domain. The real analytic interface is subject to surface tension, while gravitational effects are ignored. In contrast to the rotational surface waves, the fluid flow follows the incompressible, irrotational Euler equations. This model can provide a description for tiny water droplets in breaking waves and white caps. The primary tool we use is global bifurcation theory, via a conformal formulation of the problem. The obtained fluid domains have $m$-fold discrete rotational symmetry, as well as a reflection symmetry.