The Geometric Structure of Complex Fluids

TL;DR

This paper develops a geometric framework for complex fluids using affine Euler-Poincaré and Lie-Poisson reductions, providing variational formulations, Poisson brackets, and circulation theorems for various fluid models.

Contribution

It introduces a unified geometric approach to complex fluids, extending reduction techniques to diverse fluid types and deriving their variational and Hamiltonian structures.

Findings

Derived variational formulations for complex fluid equations

Established Poisson brackets via reduction of cotangent bundles

Presented a Kelvin-Noether circulation theorem for these fluids

Abstract

This paper develops the theory of affine Euler-Poincar\'e and affine Lie-Poisson reductions and applies these processes to various examples of complex fluids, including Yang-Mills and Hall magnetohydrodynamics for fluids and superfluids, spin glasses, microfluids, and liquid crystals. As a consequence of the Lagrangian approach, the variational formulation of the equations is determined. On the Hamiltonian side, the associated Poisson brackets are obtained by reduction of a canonical cotangent bundle. A Kelvin-Noether circulation theorem is presented and is applied to these examples.