Internal tensorial variables and a heat transport equation with inertial, thermal viscosity and vorticity terms

Taking into account some results obtained within the framework of non-equilibrium thermodynamics with internal variables (NET-IV) in a previous paper, where generalized Guyer-Krumhansl evolution equations for the heat flux in heat rigid conductors were derived, in this paper we obtain a heat transport equation describing conductive, viscous and vortical motions of phonons. To do so, we take as independent variables the internal energy, the heat flux, and a tensorial internal variable, with a symmetric part and an antisymmetric part, which turns out to be related to the rotational terms in the final equation. Besides the shear phonon viscosity arising in usual phonon hydrodynamics, we propose a rotational phonon viscosity, which would describe a transfer from ordered rotational motion of phonon vortices to rotational microscopic motions of diatomic particles constituting complex polar crystals, in analogy to the hydrodynamics of classical micropolar fluids. This possibility emphasizes the interest of exploring the interactions between the average heat flow and the heat vortices found in some models of phonon hydrodynamics.