Local heat flux and energy loss in a two-dimensional vibrated granular gas

We performed event-driven simulations of a two-dimensional granular gas between two vibrating walls and directly measured the local heat flux and local energy dissipation in the stationary state. Describing the local heat flux as a function of the coordinate in the direction perpendicular to the driving walls, we test a generalization of Fourier's law, q(x)=-kappa inverted delta T(x)+mu inverted delta rho(x), by relating the local heat flux to the local gradients of the temperature and density. This ansatz accounts for the fact that heat flux can also be generated by density gradients, not only by temperature gradients. Assuming the transport coefficients kappa and mu to be independent of x, we check the validity of this assumption and test the generalized Fourier law in the simulations. Both kappa and mu are determined for different system parameters, in particular, for a wide range of coefficients of restitution. We also compare our numerical results to existing hydrodynamic theories. Agreement is found for kappa for very small inelasticities only, i.e., when the gradients are small. Beyond this region, kappa and mu exhibit a striking nonmonotonic behavior. This may hint that hydrodynamics to Navier-Stokes order cannot be applied to moderately inelastic vibrated systems.