Hydrodynamics of inelastic Maxwell models

TL;DR

This paper provides an exact hydrodynamic analysis of inelastic Maxwell models for granular gases, deriving transport coefficients and rheological properties for various flow regimes using the Chapman--Enskog method.

Contribution

It offers the first exact derivation of Navier--Stokes transport coefficients and rheological properties for inelastic Maxwell models of granular gases.

Findings

Explicit expressions for transport coefficients in inelastic Maxwell models.

Exact solutions for uniform shear flow and Couette flows with heat flux.

Analysis of small perturbations in shear flow using Chapman--Enskog-like methods.

Abstract

An overview of recent results pertaining to the hydrodynamic description (both Newtonian and non-Newtonian) of granular gases described by the Boltzmann equation for inelastic Maxwell models is presented. The use of this mathematical model allows us to get exact results for different problems. First, the Navier--Stokes constitutive equations with explicit expressions for the corresponding transport coefficients are derived by applying the Chapman--Enskog method to inelastic gases. Second, the non-Newtonian rheological properties in the uniform shear flow (USF) are obtained in the steady state as well as in the transient unsteady regime. Next, an exact solution for a special class of Couette flows characterized by a uniform heat flux is worked out. This solution shares the same rheological properties as the USF and, additionally, two generalized transport coefficients associated with the heat flux vector can be identified. Finally, the problem of small spatial perturbations of the USF is analyzed with a Chapman--Enskog-like method and generalized (tensorial) transport coefficients are obtained.