Simple shear flow in inelastic Maxwell models

TL;DR

This paper analyzes the behavior of inelastic Maxwell models under simple shear flow, providing exact rheological properties and moments up to fourth degree, revealing divergence at high shear rates depending on dissipation.

Contribution

It offers the first exact evaluation of velocity moments in inelastic Maxwell models under shear flow, including the shear-rate dependence and divergence criteria.

Findings

Rheological properties decrease with shear rate for fixed inelasticity.

Third and asymmetric fourth-degree moments vanish over time.

Symmetric fourth-degree moments diverge above a critical shear rate.

Abstract

The Boltzmann equation for inelastic Maxwell models is considered to determine the velocity moments through fourth degree in the simple shear flow state. First, the rheological properties (which are related to the second-degree velocity moments) are {\em exactly} evaluated in terms of the coefficient of restitution $\alpha$ and the (reduced) shear rate $a^*$. For a given value of $\alpha$, the above transport properties decrease with increasing shear rate. Moreover, as expected, the third-degree and the asymmetric fourth-degree moments vanish in the long time limit when they are scaled with the thermal speed. On the other hand, as in the case of elastic collisions, our results show that, for a given value of $\alpha$, the scaled symmetric fourth-degree moments diverge in time for shear rates larger than a certain critical value $a_c^*(\alpha)$ which decreases with increasing dissipation. The explicit shear-rate dependence of the fourth-degree moments below this critical value is also obtained.