Variational formulation of a general dissipative fluid system with differential forms

TL;DR

This paper develops a geometric variational framework for dissipative fluid systems using differential forms, ensuring thermodynamic consistency and encompassing complex models like multi-species MHD.

Contribution

It introduces a novel differential-form based variational formulation for dissipative fluids that unifies thermodynamic principles with geometric methods.

Findings

Framework aligns with conservation of energy and entropy production.

Simplifies Onsager's principle and revisits Curie's principle geometrically.

Includes models like multi-species magnetohydrodynamics with dissipation.

Abstract

This work is devoted to the study of dissipative fluid systems, through the lens of a geometric variational formulation. Building upon previous works extending Hamilton's principle to non-equilibrium thermodynamics, the present method incorporates an arbitrary number of additional variables expressed as differential forms. Dissipation sources, thermodynamic flux closures, and their associated boundary conditions are also all expressed in this differential-form framework. The resulting equations are consistent with the fundamental laws of thermodynamics, namely conservation of total energy and positive entropy production. Onsager's principle is also given a simple formulation, while Curie's principle is revisited within this geometric setting through the lens of representation theory. It is shown that this general framework encompasses physically relevant models, such as multi-species magnetohydrodynamics (MHD) equations with intricate dissipation mechanisms.