New conserved vorticity integrals for moving surfaces in multi-dimensional fluid flow

TL;DR

This paper introduces new conserved vorticity integrals for moving surfaces in multi-dimensional inviscid fluid flow, generalizing classical invariants like helicity and Kelvin's theorem to higher dimensions and surfaces.

Contribution

It derives novel conserved vorticity integrals applicable to lower-dimensional moving surfaces in multi-dimensional fluid flows, extending classical invariants to higher dimensions.

Findings

New vorticity integrals are conserved for moving surfaces in n-dimensional flows.

Conditions are identified under which these integrals are constants of motion.

Generalization of Kelvin's circulation theorem to higher-dimensional surfaces in isentropic flows.

Abstract

For inviscid fluid flow in any n-dimensional Riemannian manifold, new conserved vorticity integrals generalizing helicity, enstrophy, and entropy circulation are derived for lower-dimensional surfaces that move along fluid streamlines. Conditions are determined for which the integrals yield constants of motion for the fluid. In the case when an inviscid fluid is isentropic, these new constants of motion generalize Kelvin's circulation theorem from closed loops to closed surfaces of any dimension.