A Compact Eulerian Representation of Axisymmetric Inviscid Vortex Sheet Dynamics

Adriana I. Pesci, Raymond E. Goldstein, Michael J. Shelley

Research output: Contribution to journalArticle

Abstract

A classical problem in fluid mechanics is the motion of an axisymmetric vortex sheet evolving under the action of surface tension, surrounded by an inviscid fluid. Lagrangian descriptions of these dynamics are well-known, involving complex nonlocal expressions for the radial and longitudinal velocities in terms of elliptic integrals. Here we use these prior results to arrive at a remarkably compact and exact Eulerian evolution equation for the sheet radius r(z, t) in an explicit flux form associated with the conservation of enclosed volume. The flux appears as an integral involving the pairwise mutual induction formula for vortex loop pairs first derived by Helmholtz and Maxwell. We show how the well-known linear stability results for cylindrical vortex sheets in the presence of surface tension and streaming flows [A. M. Sterling and C. A. Sleicher, J. Fluid Mech. 68, 477 (1975)] can be obtained directly from this formulation. Furthermore, the inviscid limit of the empirical model of Eggers and Dupont [J. Fluid Mech. 262 205 (1994); SIAM J. Appl. Math. 60, 1997 (2000)], which has served as the basis for understanding singularity formation in droplet pinchoff, is derived within the present formalism as the leading-order term in an asymptotic analysis for long slender axisymmetric vortex sheets and should provide the starting point for a rigorous analysis of singularity formation.

Original languageEnglish (US)
Pages (from-to)239-256
Number of pages18
JournalCommunications on Pure and Applied Mathematics
Volume73
Issue number2
DOIs
StatePublished - Feb 1 2020

Fingerprint

Vortex Sheet
Vortex flow
Surface Tension
Fluid
Singularity
Inviscid Limit
Fluids
Surface tension
Elliptic integral
Empirical Model
Fluid Mechanics
Hermann Von Helmholtz
Linear Stability
Fluxes
Streaming
Asymptotic Analysis
Droplet
Evolution Equation
Asymptotic analysis
Conservation

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

A Compact Eulerian Representation of Axisymmetric Inviscid Vortex Sheet Dynamics. / Pesci, Adriana I.; Goldstein, Raymond E.; Shelley, Michael J.

In: Communications on Pure and Applied Mathematics, Vol. 73, No. 2, 01.02.2020, p. 239-256.

Research output: Contribution to journalArticle

Pesci, Adriana I. ; Goldstein, Raymond E. ; Shelley, Michael J. / A Compact Eulerian Representation of Axisymmetric Inviscid Vortex Sheet Dynamics. In: Communications on Pure and Applied Mathematics. 2020 ; Vol. 73, No. 2. pp. 239-256.
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