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

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