Impurity in a granular gas under nonlinear Couette flow

Abstract

We study in this work the transport properties of an impurity immersed in a granular gasunder stationary nonlinear Couette flow. The starting point is a kinetic model forlow-density granular mixtures recently proposed by the authors (Vega Reyes et al 2007 Phys. Rev. E 75 061306). Two routes have been considered. First, a hydrodynamic ornormal solution is found by exploiting a formal mapping between the kinetic equations forthe gas particles and for the impurity. We show that the transport properties of theimpurity are characterized by the ratio between the temperatures of the impurity and gasparticles and by five generalized transport coefficients: three related to the momentum flux(a nonlinear shear viscosity and two normal stress differences) and two related to the heatflux (a nonlinear thermal conductivity and a cross-coefficient measuring a component of theheat flux orthogonal to the thermal gradient). Second, by means of a Monte Carlosimulation method we numerically solve the kinetic equations and show that ourhydrodynamic solution is valid in the bulk of the fluid when realistic boundary conditionsare used. Furthermore, the hydrodynamic solution applies to arbitrarily (inside thecontinuum regime) large values of the shear rate, of the inelasticity, and of therest of the parameters of the system. Preliminary simulation results of the trueBoltzmann description show the reliability of the nonlinear hydrodynamic solution ofthe kinetic model. This shows again the validity of a hydrodynamic descriptionfor granular flows, even under extreme conditions, beyond the Navier–Stokesdomain.