A differential geometric description of thermodynamics in continuum mechanics with application to Fourier-Navier-Stokes fluids

TL;DR

This paper develops a coordinate-free, differential geometric framework for thermodynamics in continuum mechanics, applying exterior calculus to describe conservation laws, entropy creation, and Fourier-Navier-Stokes fluids.

Contribution

It introduces a novel geometric formulation of thermodynamics using exterior calculus, providing a rigorous, coordinate-free approach for continuum systems including fluid dynamics.

Findings

Formulation of thermodynamics using exterior calculus on manifolds

Axiomatic derivation of entropy creation mechanisms

Application to Fourier-Navier-Stokes fluid models

Abstract

A description of thermodynamics for continuum mechanical systems is presented in the coordinate-free language of exterior calculus. First, a careful description of the mathematical tools that are needed to formulate the relevant conservation laws is given. Second, following an axiomatic approach, the two thermodynamic principles will be described, leading to a consistent description of entropy creation mechanisms on manifolds. Third, a specialisation to Fourier-Navier-Stokes fluids will be carried through.