Applying some results of nonequilibrium statistical mechanics obtained in
the framework of Grad’s theory we evaluate nonequilibrium corrections
to the
entropy
of
resting incompressible continua in terms of the nonequilibrium density distribution function,
. To find
corrections
to
the energy
or
kinetic potential
we apply a relationship that links energy and entropy representations of
thermodynamics. We also evaluate the coefficients of the wave model of heat conduction,
such as relaxation time, propagation speed, and thermal inertia. With corrections to
we
then formulate a quadratic Lagrangian and a variational principle of Hamilton’s
(least action) type for a fluid with heat flux, or other random-type effect, in
the field or Eulerian representation of the fluid motion. Results that are
significant in the hydrodynamics of real incompressible fluids at rest and their
practical applications are discussed. In particular, we discuss canonical and
generalized conservation laws and show the satisfaction of the second law of
thermodynamics under the constraint of canonical conservation laws. We also show
the significance of thermal inertia and so-called thermal momentum in the variational
formulation.
