Geometry of generalized fluid flows

TL;DR

This paper explores the geometric structure of generalized fluid flows, showing they all relate to geodesics on groupoids of multiphase diffeomorphisms and are Hamiltonian systems, extending classical fluid dynamics concepts.

Contribution

It introduces a unified geometric framework for various fluid flow models using groupoids and Lie algebroids, generalizing the Hamiltonian structure of Euler's equations.

Findings

Generalized flows correspond to geodesics on groupoids of multiphase diffeomorphisms.

All these flow models are Hamiltonian systems with respect to a Poisson structure.

The framework unifies classical and multiphase fluid flow theories.

Abstract

The Euler equation of an ideal (i.e. inviscid incompressible) fluid can be regarded, following V.Arnold, as the geodesic flow of the right-invariant $L^2$-metric on the group of volume-preserving diffeomorphisms of the flow domain. In this paper we describe the common origin and symmetry of generalized flows, multiphase fluids (homogenized vortex sheets), and conventional vortex sheets: they all correspond to geodesics on certain groupoids of multiphase diffeomorphisms. Furthermore, we prove that all these problems are Hamiltonian with respect to a Poisson structure on a dual Lie algebroid, generalizing the Hamiltonian property of the Euler equation on a Lie algebra dual.