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Variational formulation of a general dissipative fluid system with differential forms

Bastien Manach-Pérennou, François Gay-Balmaz

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.

Variational formulation of a general dissipative fluid system with differential forms

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.

Paper Structure

This paper contains 59 sections, 1 theorem, 152 equations.

Table of Contents

  1. Introduction
  2. Variational principles in the dissipationless case.
  3. Geometric formulation.
  4. Extension to non-equilibrium thermodynamics, closures, symmetries, and Curie's principle.
  5. Outline of the article.
  6. Physical description and notations
  7. Diffeomorphism group and vector fields.
  8. Differential forms.
  9. Reversible case
  10. Lagrangian formulation
  11. Reduction and Eulerian formulation
  12. Reduction of the Lagrangian.
  13. Reduction of the constraint.
  14. Resulting equations.
  15. MHD equations.
  16. ...and 44 more sections

Key Result

Lemma 5.1

Let $V_1$ and $V_2$ be two irreducible representations, then either Besides, according to Frobenius theorem, the only finite-dimensional associative division algebras over $\mathbb{R}$, up to isomorphism, are $\mathbb{R}$ itself, the complex numbers $\mathbb{C}$ and the space of quaternions $\mathbb{H}$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (20)

  • Remark 2.1
  • Remark 2.2
  • Remark 3.1: Formulation with $n$-forms
  • Remark 3.2: Form of the constraint term
  • Remark 3.3: Several advected variables
  • Remark 3.4: Formulation with $n$-forms
  • Remark 3.5
  • Remark 4.1: Formulation with $n$-forms
  • Remark 4.2
  • Remark 4.3: Formulation with $n$-forms
  • ...and 10 more