Nonlinear phonon hydrodynamics: A Guyer–Krumhansl equation for the conjugate of the heat flux

The increasing interest in nanoscale heat transport sciences led modern nonequilibrium thermodynamic theories to depart from classical theory. One of the strategies to find thermodynamically consistent nonlinear transport equations is to search for linear evolution equations of the thermodynamic quantities conjugate to the respective fluxes. The starting point is the so-called Guyer-Krumhansl equation, which is able to capture nonlocal effects in the heat transport. In nanosystems, also nonlinear effects may strongly influence the properties of the material at hand. Furthermore, a number of experiments on heat transfer in the phonon hydrodynamic domain appear to indicate that the phonon fluid behaves as non-Newtonian. Since the conjugates to the fluxes are well-defined nonlinear functions of the fluxes, the final equations are nonlinear. In this paper we apply this tentative technique to formulate a nonlinear version of phonon hydrodynamics and to explore its consequences on the profile of the heat flux in cylindrical nanowires. We compare the result with that obtained from a power-law non-Newtonian phonon hydrodynamics.