  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.
