Rheology of a dilute binary mixture of inertial suspension under simple shear flow

TL;DR

This paper analyzes the rheology of a dilute binary inertial suspension under shear flow using kinetic theory, comparing analytical models with simulations to understand the effects of particle size, inelasticity, and shear rate.

Contribution

It introduces a theoretical framework combining Grad's moment method and a BGK-type model to predict suspension rheology, validated against Langevin simulations across various parameters.

Findings

Theoretical predictions agree with simulations for different size ratios and collision inelasticities.

Temperature and viscosity ratios exhibit discontinuous changes at certain shear rates as size ratio increases.

The BGK model accurately reproduces velocity distribution functions in the studied regime.

Abstract

The rheology of a dilute binary mixture of inertial suspension under simple shear flow is analyzed in the context of the Boltzmann kinetic equation. The effect of the surrounding viscous gas on the solid particles is accounted for by means of a deterministic viscous drag force plus a stochastic Langevin-like term defined in terms of the environmental temperature $T_\text{env}$. Grad's moment method is employed to determine the temperature ratio and the pressure tensor in terms of the coefficients of restitution, concentration, the masses and diameters of the components of the mixture, and the environmental temperature. Analytical results are compared against event-driven Langevin simulations for mixtures of hard spheres with the same mass density $m_1/m_2=(\sigma^{(1)}/\sigma^{(2)})^3$, $m_i$ and $\sigma^{(1)}$ being the mass and diameter, respectively, of the species $i$. It is confirmed that the theoretical predictions agree with simulations of various size ratios $\sigma^{(1)}/\sigma^{(2)}$ and for elastic and inelastic collisions in the wide range of parameters' space. It is remarkable that the temperature ratio $T_1/T_2$ and the viscosity ratio $\eta_1/\eta_2$ ($\eta_i$ being the partial contribution of the species $i$ to the total shear viscosity $\eta=\eta_1+\eta_2$) discontinuously change at a certain shear rate as the size ratio increases; this feature (which is expected to occur in the thermodynamic limit) cannot be completely captured by simulations due to small system size. In addition, a Bhatnagar--Gross--Krook (BGK)-type kinetic model adapted to mixtures of inelastic hard spheres is exactly solved when $T_\text{env}$ is much smaller than the kinetic temperature $T$. A comparison between the velocity distribution functions obtained from Grad's method, BGK model, and simulations is carried out.