A power-law model for nonlinear phonon hydrodynamics

The Guyer-Krumhansl equation for the heat flux is a phenomenological bridge between Fourier heat transport (for size of the system much bigger than the mean free path of heat carriers) and hydrodynamic heat transport (for size of the system comparable to the mean free path of heat carriers). The corresponding phonon hydrodynamics is analogous to Newtonian hydrodynamics, but with the velocity replaced by the heat flux, the pressure gradient replaced by the temperature gradient and the shear viscosity replaced by the square of the mean-free path divided by the thermal conductivity. In this paper, we propose a nonlinear generalization of the Guyer-Krumhansl equation and phonon hydrodynamics based on an analogy with the power-law model of non-Newtonian fluids leading to a non-diffusive behaviour of heat transport. On the basis of this model, we obtain the corresponding nonlinear effective thermal conductivity of the model, depending on the radius of the channel and on the temperature gradient. The present proposal could be useful in the light of recent analyses of Poiseuille phonon hydrodynamics which suggest a non-Newtonian behaviour.