Tracer diffusion coefficients in a sheared inelastic Maxwell gas

Abstract

We study the transport properties of an impurity in a sheared granular gas, in the framework of the Boltzmann equation for inelastic Maxwell models. We investigate here the impact of a nonequilibrium phase transition found in such systems, where the tracer species carries a finite fraction of the total kinetic energy (ordered phase). To this end, the diffusion coefficients are first obtained for a granular binary mixture in spatially inhomogeneous states close to the simple shear flow. In this situation, the set of coupled Boltzmann equations are solved by means of a Chapman–Enskog-like expansion around the (local) shear flow distributions for each species, thereby retaining all the hydrodynamic orders in the shear rate a. Due to the anisotropy induced by the shear flow, three tensorial quantities Dij, Dp,ij, and DT,ij are required to describe the mass transport process instead of the conventional scalar coefficients. These tensors are given in terms of the solutions of a set of coupled algebraic equations, which can be exactly solved as functions of the shear rate a, the coefficients of restitution and the parameters of the mixture (masses and composition). Once the forms of Dij, Dp,ij, and DT,ij are obtained for arbitrary mole fraction (where nr is the number density of species r), the tracer limit () is carefully considered for the above three diffusion tensors. Explicit forms for these coefficients are derived showing that their shear rate dependence is significantly affected by the order-disorder transition.