Temperature inversion in granular fluids under gravity

We study, via hydrodynamic equations, the granular temperature profile of a granular fluid under gravity and subjected to energy injection from a base. It is found that there exists a turn-up in the granular temperature and that, far from the base, it increases linearly with height. We show that this phenomenon, observed previously in experiments and computer simulations, is a direct consequence of the heat flux law, different form Fourierʹs, in granular fluids. The positive granular temperature gradient is proportional to gravity and a transport coefficient μ0, relating the heat flux to the density gradients, that is characteristic of granular systems. Our results provide a method to compute the value μ0 for different restitution coefficients. The theoretical predictions are verified by means of molecular dynamics simulations, and the value of μ0 is computed for the two-dimensional inelastic hard sphere model. We provide, also, a boundary condition for the temperature field that is consistent with the modified Fourierʹs law