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Steady Ring-Shaped Vortex Sheets

David Meyer, Christian Seis

TL;DR

The paper proves the local existence and uniqueness of axisymmetric traveling vortex ring solutions with a vortex sheet at a two phase interface for the Euler equations in three dimensions. It employs a perturbative construction around thin rings using an implicit function theorem together with a Lyapunov Schmidt reduction to handle translation invariance. The work yields precise asymptotics for the ring speed that generalize the Kelvin Hicks law to include surface tension and hollow vortex configurations. The results provide a rigorous framework for thin ring vortex structures in two phase flows and illuminate how surface tension regularizes vortex sheet dynamics and influences the speed and geometry of the ring.

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.

Steady Ring-Shaped Vortex Sheets

TL;DR

The paper proves the local existence and uniqueness of axisymmetric traveling vortex ring solutions with a vortex sheet at a two phase interface for the Euler equations in three dimensions. It employs a perturbative construction around thin rings using an implicit function theorem together with a Lyapunov Schmidt reduction to handle translation invariance. The work yields precise asymptotics for the ring speed that generalize the Kelvin Hicks law to include surface tension and hollow vortex configurations. The results provide a rigorous framework for thin ring vortex structures in two phase flows and illuminate how surface tension regularizes vortex sheet dynamics and influences the speed and geometry of the ring.

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.
Paper Structure (24 sections, 28 theorems, 286 equations, 1 figure)

This paper contains 24 sections, 28 theorems, 286 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Original mathematical description
  3. An overdetermined boundary problem
  4. The perturbative setting
  5. Strategy of the proof
  6. Preliminaries and notation
  7. Notation
  8. Some Preliminaries on fractional spaces and Fréchet differentiability in them
  9. Geometric lemmata
  10. Regularity of geometric quantities
  11. The tangent space of $\mathcal{M}$
  12. The interior contribution
  13. The homogenous equation
  14. Reformulation as an integral equation
  15. Mapping properties of perturbed logarithmic kernels
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $\bar{b}_{\mathrm{int}}\in\mathbb{R}$, $\bar{b}_{\mathrm{ext}}\in\mathbb{R}_{>0}$, $\rho_{\mathrm{int}}\in \mathbb{R}_{\ge0}$, $\rho_{\mathrm{ext}}\in \mathbb{R}_{>0}$, $R\in\mathbb{R}_{>0}$, $\bar{\epsilon}\in\mathbb{R}_{>0}$, and $\bar{\sigma}\in\mathbb{R}_{\ge 0}$ be given in such a way that Then there exists an $\epsilon_0\in\mathbb{R}_{>0}$ such that for all $\epsilon\in(0, \epsilon_0)$,

Figures (1)

  • Figure 1: $\Omega$ in the original variables

Theorems & Definitions (52)

  • Theorem 1.1: First non-specific formulation
  • Theorem 1.2: Second non-specific formulation
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • ...and 42 more