# Steady bubbles and drops in inviscid fluids

## TL;DR

This paper constructs steady non-spherical bubbles and drops in inviscid fluids using mathematical models involving surface tension and vorticity.

## Contribution

The novelty lies in constructing non-spherical steady solutions with surface tension and analyzing their bifurcation behavior.

## Key findings

- Steady non-spherical bubbles and drops are constructed with uniform vorticity and a vortex sheet.
- Bifurcation analysis is performed at critical Weber numbers using the Crandall–Rabinowitz theorem.
- The model with surface tension is richer than one-phase Euler equations in terms of vortex solutions.

## Abstract

We construct steady non-spherical bubbles and drops, which are traveling wave solutions to the axisymmetric two-phase Euler equations with surface tension, whose inner phase is a bounded connected domain. The solutions have a uniform vorticity distribution in this inner phase and they have a vortex sheet on its surface. Our construction relies on a perturbative approach around an explicit spherical solution, given by Hill’s vortex enclosed by a spherical vortex sheet. The construction is sensitive to the Weber numbers describing the flow. At critical Weber numbers, we perform a bifurcation analysis utilizing the Crandall–Rabinowitz theorem in Sobolev spaces on the 2-sphere. Away from these critical numbers, our construction relies on the implicit function theorem. Our results imply that the model containing surface tension is richer than the ordinary one-phase Euler equations, in the sense that for the latter, Hill’s spherical vortex is unique (modulo translations) among all axisymmetric simply connected uniform vortices of a given circulation.