A free energy Lagrangian variational formulation of the Navier-Stokes-Fourier system

TL;DR

This paper develops a variational formulation for the Navier-Stokes-Fourier system using a free energy Lagrangian, extending thermodynamic variational principles to continuum fluid dynamics on Riemannian manifolds.

Contribution

It introduces a systematic infinite-dimensional variational approach based on free energy, complementing previous formulations using internal energy and employing differential geometry.

Findings

Formulates Navier-Stokes-Fourier system variationally with free energy

Derives Eulerian and Lagrangian representations within the framework

Applicable to systems on Riemannian manifolds

Abstract

We present a variational formulation for the Navier-Stokes-Fourier system based on a free energy Lagrangian. This formulation is a systematic infinite dimensional extension of the variational approach to the thermodynamics of discrete systems using the free energy, which complements the Lagrangian variational formulation using the internal energy developed in \cite{GBYo2016b} as one employs temperature, rather than entropy, as an independent variable. The variational derivation is first expressed in the material (or Lagrangian) representation, from which the spatial (or Eulerian) representation is deduced. The variational framework is intrinsically written in a differential-geometric form that allows the treatment of the Navier-Stokes-Fourier system on Riemannian manifolds.