High-degree collisional moments of inelastic Maxwell mixtures. Application to the homogeneous cooling and uniform shear flow states

TL;DR

This paper derives exact collisional moments for inelastic Maxwell mixtures and analyzes their time evolution in homogeneous cooling and shear flow states, revealing conditions for divergence of higher moments.

Contribution

It provides explicit formulas for collisional moments of second to fourth degree in inelastic Maxwell mixtures and studies their behavior in nonequilibrium states.

Findings

Third and fourth degree moments may diverge in the homogeneous cooling state.

Second degree moments always converge in the uniform shear flow.

Third degree moments of tracer species can diverge over time.

Abstract

The Boltzmann equation for $d$-dimensional inelastic Maxwell models is considered to determine the collisional moments of second, third and fourth degree in a granular binary mixture. These collisional moments are exactly evaluated in terms of the velocity moments of the distribution function of each species when diffusion is absent (mass flux of each species vanishes). The corresponding associated eigenvalues as well as cross coefficients are obtained as functions of the coefficients of normal restitution and the parameters of the mixture (masses, diameters and composition). The results are applied to the analysis of the time evolution of the moments (scaled with a thermal speed) in two different nonequilibrium situations: the homogeneous cooling state (HCS) and the uniform (or simple) shear flow (USF) state. In the case of the HCS, in contrast to what happens for simple granular gases, it is shown that the third and fourth degree moments could diverge in time for given values of the parameters of the system. An exhaustive study on the influence of the parameter space of the mixture on the time behavior of these moments is carried out. Then, the time evolution of the second- and third-degree velocity moments in the USF is studied in the tracer limit (namely, when the concentration of one of the species is negligible). As expected, while the second-degree moments are always convergent, the third-degree moments of the tracer species can be also divergent in the long time limit.