Generalized transport coefficients for inelastic Maxwell mixtures under shear flow

Abstract

The Boltzmann equation framework for inelastic Maxwell models is considered to determine the transport coefficients associated with the mass, momentum and heat fluxes of a granular binary mixture in spatially inhomogeneous states close to the simple shear flow. The Boltzmann equation is solved by means of a Chapman-Enskog-like expansion around the (local) shear flow distributions $f_r^{(0)}$ for each species that retain all the hydrodynamic orders in the shear rate. Due to the anisotropy induced by the shear flow, tensorial quantities are required to describe the transport processes instead of the conventional scalar coefficients. These tensors are given in terms of the solutions of a set of coupled equations, which can be analytically solved as functions of the shear rate $a$, the coefficients of restitution $α_{rs}$ and the parameters of the mixture (masses, diameters and composition). Since the reference distribution functions $f_r^{(0)}$ apply for arbitrary values of the shear rate and are not restricted to weak dissipation, the corresponding generalized coefficients turn out to be nonlinear functions of both $a$ and $α_{rs}$. The dependence of the relevant elements of the three diffusion tensors on both the shear rate and dissipation is illustrated in the tracer limit case, the results showing that the deviation of the generalized transport coefficients from their forms for vanishing shear rates is in general significant. A comparison with the previous results obtained analytically for inelastic hard spheres by using Grad's moment method is carried out showing a good agreement over a wide range of values for the coefficients of restitution. Finally, as an application of the theoretical expressions derived here for the transport coefficients, thermal diffusion segregation of an intruder immersed in a granular gas is also studied.