Exact Rheology of Uniform Shear Flow in a Gas of Inelastic and Rough Maxwell Particles

Abstract

We investigate the steady uniform shear flow of a granular gas composed of inelastic and rough Maxwell particles. Exploiting the mean-field character of the model, we derive exact expressions for the collisional production rates of the second-degree moments and obtain a closed nonlinear solution for the stress and spin-spin tensors. The rotational-to-translational temperature ratio and the proportionality between the spin-spin and stress tensors are shown to be independent of the coefficient of normal restitution and determined solely by roughness and moment of inertia. The reduced normal stresses, shear stress, and shear rate are obtained explicitly in terms of two effective parameters generalizing the cooling and stress relaxation rates of the smooth model. From these results we derive exact expressions for the non-Newtonian shear viscosity, the first viscometric function, and the friction coefficient. The dependence of the rheological properties on the normal and tangential restitution coefficients is analyzed in detail, revealing strong non-Newtonian behavior and nonmonotonic effects of roughness. The results reduce, in the appropriate limits, to those of the inelastic Maxwell model for smooth particles and to the Pidduck gas in the elastic perfectly rough case.

Figures (3)

• Figure 1: Rotational-to-translational temperature ratio $\theta=T_r/T_t$ and proportionality factor $\lambda$ relating the spin-spin tensor to the stress tensor, $\Omega_{ij}^*=-\lambda \Pi_{ij}^*$, as functions of the tangential restitution coefficient $\beta$ for $\kappa=\frac{2}{5}$.
• Figure 2: Top panels: reduced normal stress $\Pi_{xx}^*$ (a), reduced shear stress $-\Pi_{xy}^*$ (b), and reduced shear rate $\dot{\gamma}^*$ (c) as functions of the normal restitution coefficient $\alpha$ and the tangential restitution coefficient $\beta$ for $\kappa=\frac{2}{5}$. The displayed planar surfaces correspond to $\alpha=0.4$, $0.6$, $0.8$, and $1$. Bottom panels (d--f): corresponding cross-sections at fixed $\alpha$.
• Figure 3: Top panels: reduced shear viscosity $\eta^*$ (a), first viscometric function $\Psi_{1}$ (b), and friction coefficient $\mu$ (c) as functions of the normal restitution coefficient $\alpha$ and the tangential restitution coefficient $\beta$ for $\kappa=\frac{2}{5}$. The displayed planar surfaces correspond to $\alpha=0.4$, $0.6$, $0.8$, and $1$. Bottom panels (d--f): corresponding cross-sections at fixed $\alpha$.