The geometric structure of complex fluids

This paper develops the theory of affine Euler–Poincaré 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.