On the Hamiltonian structure of the intrinsic evolution of a closed vortex sheet

TL;DR

This paper investigates the Hamiltonian structure of the intrinsic evolution equations of a closed vortex sheet in a plane, with potential applications to fluid interface dynamics, and introduces a new boundary Poisson bracket related to the KdV bracket.

Contribution

It derives the Hamiltonian form of vortex sheet evolution equations, introduces a novel boundary Poisson bracket involving the curve-tangential derivative, and identifies Lagrangian invariants of the sheet motion.

Findings

Derived the Hamiltonian form of vortex sheet evolution.

Introduced a new boundary Poisson bracket related to the KdV bracket.

Identified Lagrangian invariants of the vortex sheet motion.

Abstract

Motivated by the work of previous authors on vortex sheets and their applications, the intrinsic inviscid evolution equations of a closed vortex sheet in a plane, separating two piecewise constant density fluids, and their Hamiltonian form are investigated. The model has potential applications to problems involving the dynamics of interfaces of two immiscible fluids. A boundary Poisson bracket, which appears to be new and related to the KdV bracket, is obtained containing the curve-tangential derivative $\partial / \partial s$. Lagrangian invariants of the sheet motion by its self-induced velocity--the Cauchy principal value of the Biot-Savart integral--are also derived.