Variational formulation of compressible hydrodynamics in curved spacetime and symmetry of stress tensor

TL;DR

This paper develops a variational approach to compressible fluid dynamics in curved spacetime, revealing an asymmetric stress tensor and extending the Navier-Stokes-Fourier equation to curved geometries, with implications for gauge interactions.

Contribution

It introduces a stochastic variational formulation for compressible hydrodynamics in curved spacetime, highlighting an asymmetric stress tensor and extending classical equations to include gauge interactions.

Findings

Stress tensor becomes asymmetric in curved spacetime.

Viscous term represented with Bochner Laplacian in incompressible limit.

Modified Navier-Stokes-Fourier equation applicable in curved spacetime.

Abstract

Hydrodynamics of the non-relativistic compressible fluid in the curved spacetime is derived using the generalized framework of the stochastic variational method (SVM) for continuum medium. The fluid-stress tensor of the resultant equation becomes asymmetric for the exchange of the indices, differently from the standard Euclidean one. Its incompressible limit suggests that the viscous term should be represented with the Bochner Laplacian. Moreover the modified Navier-Stokes-Fourier (NSF) equation proposed by Brenner can be considered even in the curved spacetime. To confirm the compatibility with the symmetry principle, SVM is applied to the gauge-invariant Lagrangian of a charged compressible fluid and then the Lorentz force is reproduced as the interaction between the Abelian gauge fields and the viscous charged fluid.