Variational setting for reversible and irreversible fluids with heat flow

This paper derives and discusses variational formulations for heat flows subject to physical constraints that involve the (generally) non-conserved balance of internal energy and the entropy representation kinetics in the form of the Cattaneo equation of heat. Another approach is also outlined which uses the (generally) non-conserved balance of the entropy and the energy-representation counterpart of the Cattaneo equation called Kaliski’s equation. Results of nonequilibrium statistical mechanics (Grad’s theory) lead to nonequilibrium corrections to entropy and energy of the fluid in terms of the nonequilibrium density distribution function, f. These results also yield coefficients of the wave model of heat such as: relaxation time, propagation speed and thermal inertia. With these data a quadratic Lagrangian and a variational principle of Hamilton’s type follows for a fluid with heat flux in the field representation of fluid motion. For an irreversible heat transfer we show that despite of generally non-canonical form of the matter tensor the coefficients in source terms of the variational conservation laws can be suitably adjusted, so that physical (source-less and canonical) conservation laws are obtained for the energy and momentum. We discuss canonical and generalized conservation laws and show the satisfaction of the second law under the constraint of canonical conservation laws.