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The rheology of a dilute binary mixture of inertial suspension under simpleshear flow is analyzed in the context of the Boltzmann kinetic equation. Theeffect of the surrounding viscous gas on the solid particles is accounted forby means of a deterministic viscous drag force plus a stochastic Langevin-liketerm defined in terms of the environmental temperature $T_\text{env}$. Grad'smoment method is employed to determine the temperature ratio and the pressuretensor in terms of the coefficients of restitution, concentration, the massesand diameters of the components of the mixture, and the environmentaltemperature. Analytical results are compared against event-driven Langevinsimulations 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 themass and diameter, respectively, of the species $i$. It is confirmed that thetheoretical predictions agree with simulations of various size ratios$\sigma^{(1)}/\sigma^{(2)}$ and for elastic and inelastic collisions in thewide 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 partialcontribution of the species $i$ to the total shear viscosity$\eta=\eta_1+\eta_2$) discontinuously change at a certain shear rate as thesize ratio increases; this feature (which is expected to occur in thethermodynamic limit) cannot be completely captured by simulations due to smallsystem size. In addition, a Bhatnagar--Gross--Krook (BGK)-type kinetic modeladapted to mixtures of inelastic hard spheres is exactly solved when$T_\text{env}$ is much smaller than the kinetic temperature $T$. A comparisonbetween the velocity distribution functions obtained from Grad's method, BGKmodel, and simulations is carried out.

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