Year: 2024
Type: article
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 relabeling symmetries, we deduce a variational formulation in the Eulerian description that extends the Euler–Poincaré 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.
Source: Geometric Mechanics
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Sandia National Laboratories
OpenAlex ID: W4406808062