Nine-moment phonon hydrodynamics based on the modified Grad-type approach: formulation

In this paper, we present the derivation of a new set of heat transport equations, called the equations of nine-moment phonon hydrodynamics, which are expected to describe transient processes under high thermal loads. The nine-moment model introduces the energy density, the heat flux and the flux of the heat flux as basic gas-state variables. The evolution equations for these variables are derived from the Grad-type expansion method applied to the Boltzmann-Peierls equation with Callaway's collisional terms. The basic idea is to expand the phase density about an anisotropic Planck distribution. The advantage of using this distribution is that the heat flux is incorporated into the model in a non-perturbative manner, thereby allowing virtually arbitrarily large values for the components of the heat flux. Special emphasis is placed on finding explicit closed-form expressions for the moment flux and collisional quantities in terms of independent gas-state variables. Our model involves two relaxation times and it seems particularly suited for describing phonon flows in the regime where the relaxation time for normal processes is much smaller than the relaxation time for resistive processes.