Shear-rate dependent transport coefficients for inelastic Maxwell models

Abstract

The Boltzmann equation for d-dimensional inelastic Maxwell models is considered to analyze transport properties in spatially inhomogeneous states close to the simple shear flow. A normal solution is obtained via a Chapman--Enskog--like expansion around a local shear flow distribution f⁽⁰⁾ that retains all the hydrodynamic orders in the shear rate. The constitutive equations for the heat and momentum fluxes are obtained to first order in the deviations of the hydrodynamic field gradients from their values in the reference state and the corresponding generalized transport coefficients are exactly determined in terms of the coefficient of restitution αand the shear rate a. Since f⁽⁰⁾ applies for arbitrary values of the shear rate and is not restricted to weak dissipation, the transport coefficients turn out to be nonlinear functions of both parameters a and α. A comparison with previous results obtained for inelastic hard spheres from a kinetic model of the Boltzmann equation is also carried out.