  In this paper, a statistical physical derivation of thermodynamically
consistent fluid mechanical equations is presented for non-isothermal viscous
molecular fluids. The coarse-graining process is based on (i) the adiabatic
expansion of the one-particle probability density function around Local
Thermodynamic Equilibrium, (ii) the assumption of decoupled particle positions
and momenta, and (iii) the variational principle. It is shown that there exists
a class of free energy functionals for which the conventional thermodynamic
formalism can be naturally adopted for non-equilibrium scenarios, and describes
entropy monotonic fluid flow in isolated systems. Furthermore, the analysis of
the general continuum equations revealed the possibility of a non-local
transport mode of energy in highly compressible dense fluids.
