Geometric, Variational, and Bracket Descriptions of Fluid Motion with Open Boundaries

TL;DR

This paper extends Arnold's geometric framework for fluid motion to include open boundaries, providing a unified variational and Hamiltonian approach suitable for diverse physical systems with permeable boundaries.

Contribution

It introduces a Lie group geometric framework for open fluid flows, extending classical models to incorporate boundary effects and bulk forces within a variational and bracket formulation.

Findings

Developed a geometric framework for fluids with permeable boundaries.

Extended Euler-Poincaré and Lie-Poisson structures to open fluids.

Demonstrated applicability to various physical systems like geophysical flows and porous media.

Abstract

We develop a Lie group geometric framework for the motion of fluids with permeable boundaries that extends Arnold's geometric description of fluid in closed domains. Our setting is based on the classical Hamilton principle applied to fluid trajectories, appropriately amended to incorporate bulk and boundary forces via a Lagrange-d'Alembert approach, and to take into account only the fluid particles present in the fluid domain at a given time. By applying a reduction by relabelling symmetries, we deduce a variational formulation in the Eulerian description that extends the Euler-Poincar\'e framework to open fluids. On the Hamiltonian side, our approach yields a bracket formulation that consistently extends the Lie-Poisson bracket of fluid dynamics and contains as a particular case a bulk+boundary bracket formulation proposed earlier. We illustrate the geometric framework with several examples, and also consider the extension of this setting to include multicomponent fluids, the consideration of general advected quantities, the analysis of higher-order fluids, and the incorporation of boundary stresses. Open fluids are found in a wide range of physical systems, including geophysical fluid dynamics, porous media, incompressible flow and magnetohydrodynamics. This new formulation permits a description based on geometric mechanics that can be applied to a broad class of models.