Internal tensorial variables and a heat transport equation with inertial, thermal viscosity and vorticity terms
Phonon hydrodynamics describes the motions of heat carriers (phonons) at sub-continuum scales: diffusive, ballistic, viscous, and vortical. In a previous paper, these behaviours were investigated within the framework of non-equilibrium thermodynamics with internal variables at the macroscopic scale, deriving generalizations of the Guyer-Krumhansl equation. In particular, a generalized heat conduction equation, containing not only the Fourier, Maxwell-Vernotte-Cattaneo, and Guyer-Krumhansl contributions, but also a term describing phonon vortices, was obtained. In this paper, we provide new insight and clarifications into the same model for rigid heat-conducting media. Then, we a posteriori identify two non-local macroscopic internal variables, $Q^s$ and $Q^a$ (the symmetric part and the antisymmetric part of a second order tensor $Q$) with the symmetric (changed in sign) and antisymmetric gradients of the heat flux, $-(\nabla J(q))^s$ and $(\nabla J(q))^{a}$. Also an identification of these two tensorial internal variables is obtained by an asymptotic approach. This generalizes the heat equation with additional terms containing the time derivative of the heat flux describing the viscous and vortical motions of phonons. These terms may describe the 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. Therefore, this paper fits into the currently explored area of phonon vorticity and its interactions with the heat flux itself.