2D capillary liquid drops with constant vorticity: rotating waves existence and a conditional energetic stability result for rotating circles

TL;DR

This paper investigates the existence and stability of rotating wave solutions in a two-dimensional capillary liquid drop with constant vorticity, using Hamiltonian and bifurcation methods.

Contribution

It introduces a novel analysis of rotating waves in capillary drops with vorticity, combining Hamiltonian structure, symmetry, and bifurcation theory.

Findings

Existence of rotating wave solutions via bifurcation theory.

Linear stability of rotating circles.

Conditional energetic stability under fixed volume and barycenter.

Abstract

We consider a two-dimensional, pure capillary drop of nearly-circular shape, having constant vorticity. We write the Craig-Sulem equations on the unit circle, then on the flat torus. We show their Hamiltonian structure and we then observe symmetries and we derive constants of motions. After showing linear stability for rotating circles, we prove the existence of rotating waves by combining a bifurcation-theoretical approach together with critical point theory. Finally, by exploiting the Hamiltonian structure, we show that whenever volume and barycenter are fixed to be the same as those of rotating circle, this solution is also conditionally energetically stable. This holds in the irrotational case as well, in agreement with the stability analysis of rotating cylinder jets in Rayleigh [25].