Nonlinear transport in inelastic Maxwell mixtures under simple shear flow

TL;DR

This paper uses the Boltzmann equation for inelastic Maxwell models to analyze nonlinear transport phenomena in granular mixtures under simple shear flow, providing explicit expressions for rheological and diffusion properties.

Contribution

It offers exact solutions for rheological properties and diffusion tensors in inelastic Maxwell mixtures under shear, comparing results with inelastic hard sphere models and simulations.

Findings

Good agreement between Maxwell and hard sphere models across parameter space

Explicit formulas for shear and normal stresses in inelastic Maxwell mixtures

Diffusion tensor determined from first-order perturbation in concentration gradient

Abstract

The Boltzmann equation for inelastic Maxwell models is used to analyze nonlinear transport in a granular binary mixture in the steady simple shear flow. Two different transport processes are studied. First, the rheological properties (shear and normal stresses) are obtained by solving exactly the velocity moment equations. Second, the diffusion tensor of impurities immersed in a sheared inelastic Maxwell gas is explicitly determined from a perturbation solution through first order in the concentration gradient. The corresponding reference state of this expansion corresponds to the solution derived in the (pure) shear flow problem. All these transport coefficients are given in terms of the restitution coefficients and the parameters of the mixture (ratios of masses, concentration, and sizes). The results are compared with those obtained analytically for inelastic hard spheres in the first Sonine approximation and by means of Monte Carlo simulations. The comparison between the results obtained for both interaction models shows a good agreement over a wide range values of the parameter space.