Poiseuille Flow in a Heated Granular Gas

Abstract

The planar Poiseuille flow induced by a constant external field (e.g., gravity) has been the subject of recent interest in the case of molecular gases. One of the predictions from kinetic theory (confirmed by computer simulations) has been that the temperature profile exhibits a bimodal shape with a local minimum in the middle of the slab surrounded by two symmetric maxima, in contrast to the unimodal shape expected from the Navier–Stokes (NS) equations. However, from a practical point of view, the interest of this non-Newtonian behavior in molecular gases is rather academic since it requires values of gravity extremely higher than the terrestrial one. On the other hand, gravity plays a relevant role in the case of granular gases due to the mesoscopic nature of the grains. In this paper we consider a dilute gas of inelastic hard spheres enclosed in a slab under the action of gravity along the longitudinal direction. In addition, the gas is subject to a white-noise stochastic force that mimics the effect of external vibrations customarily used in experiments to compensate for the collisional cooling. The system is described by means of a kinetic model of the inelastic Boltzmann equation and its steady-state solution is derived through second order in gravity. This solution differs from the NS description in that the hydrostatic pressure is not uniform, normal stress differences are present, a component of the heat flux normal to the thermal gradient exists, and the temperature profile includes a positive quadratic term. As in the elastic case, this new term is responsible for the bimodal shape of the temperature profile. The results show that, except for high inelasticities, the effect of inelasticity on the profiles is to slightly decrease the quantitative deviations from the NS results.