On the identification of two internal tensorial variables and a heat transport equation with inertial, thermal viscosity and vorticity terms

Abstract In a rigid crystal, phonons are quasiparticles that carry heat, and they can be seen at various levels of description like phonon kinetic theory, phonon hydrodynamics, Guyer–Krumhansl equation, or Fourier heat conduction. In a previous paper, we extended the Guyer–Krumhansl equation by adding two internal state variables, a symmetric tensor and an antisymmetric tensor. These internal variables express the viscous and vortical motion of phonons. In the present paper, we analyze the model by means of asymptotic expansion, and we find a geometric analogue of the model. While the zeroth-order expansion reduces the model to the Guyer–Krumhansl equation, the first-order expansion gives a more complicated heat flux evolution equation, of which the stability conditions of equilibrium solutions are studied. The geometric formulation of the model gives the heat flux as a functional of the entropy flux, in contrast to the usual setting of Extended Irreversible Thermodynamics, and also the two fluxes cease to be proportional via the temperature (in contrast to the Clausius formula for entropy flux).