  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.
