Steady Ring-Shaped Vortex Sheets

TL;DR

This paper constructs and analyzes steady, ring-shaped vortex sheet solutions in two-phase Euler flows, extending classical formulas to include surface tension effects and establishing local uniqueness via the implicit function theorem.

Contribution

It introduces a method to construct traveling wave solutions with hollow vortex rings, generalizing Kelvin--Hicks formula to account for surface tension effects.

Findings

Existence of steady ring-shaped vortex sheet solutions.

Extension of Kelvin--Hicks formula with surface tension.

Local uniqueness of the constructed solutions.

Abstract

In this work, we construct traveling wave solutions to the two-phase Euler equations, featuring a vortex sheet at the interface between the two phases. The inner phase exhibits a uniform vorticity distribution and may represent a vacuum, forming what is known as a hollow vortex. These traveling waves take the form of ring-shaped vortices with a small cross-sectional radius, referred to as thin rings. Our construction is based on the implicit function theorem, which also guarantees local uniqueness of the solutions. Additionally, we derive asymptotics for the speed of the ring, generalizing the well-known Kelvin--Hicks formula to cases that include surface tension.