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Steady vortex sheets in presence of surface tension

Federico Murgante, Emeric Roulley, Stefano Scrobogna

Abstract

We prove a bifurcation result of uniformly-rotating/stationary non-trivial vortex sheets near the circular distribution for a model of two irrotational fluids with same density taking into account surface tension effects. As bifurcation parameters, we play with either the speed of rotation, the surface tension coefficient or the mean vorticity.

Steady vortex sheets in presence of surface tension

Abstract

We prove a bifurcation result of uniformly-rotating/stationary non-trivial vortex sheets near the circular distribution for a model of two irrotational fluids with same density taking into account surface tension effects. As bifurcation parameters, we play with either the speed of rotation, the surface tension coefficient or the mean vorticity.

Paper Structure

This paper contains 12 sections, 8 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Known results
  4. Proof of the results
  5. Functional of interest and its linear operator
  6. Bifurcation from $c$
  7. Bifurcation from $\sigma$
  8. Bifurcation from $\gamma$ for stationary vortex sheets
  9. Appendix
  10. Derivation of \ref{['VS system']} from \ref{['Euler']}
  11. Derivation of \ref{['eq:KH2']} from \ref{['VS system']}
  12. Crandall-Rabinowitz Theorem

Key Result

Theorem 1.1

(Local bifurcation of vortex sheets from the circular distribution)

Theorems & Definitions (19)

  • Theorem 1.1
  • Remark 1.1
  • Lemma 2.1
  • proof
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 9 more