  Two different kinetic theories [J. Solsvik and E. Manger (SM-theory), Phys.
Fluids \textbf{33}, 043321 (2021) and V. Garz\'o, J. W. Dufty, and C. M. Hrenya
(GDH-theory), Phys. Rev. E \textbf{76}, 031303 (2007)] are considered to
determine the shear viscosity $\eta$ for a moderately dense granular binary
mixture of smooth hard spheres. The mixture is subjected to a simple shear flow
and heated by the action of an external driving force (Gaussian thermostat)
that exactly compensates the energy dissipated in collisions. The set of Enskog
kinetic equations is the starting point to obtain the dependence of $\eta$ on
the control parameters of the mixture: solid fraction, concentration, mass and
diameter ratios, and coefficients of normal restitution. While the expression
of $\eta$ found in the SM-theory is based on the assumption of Maxwellian
distributions for the velocity distribution functions of each species, the
GDH-theory solves the Enskog equation by means of the Chapman--Enskog method to
first order in the shear rate. To assess the accuracy of both kinetic theories,
the Enskog equation is numerically solved by means of the direct simulation
Monte Carlo (DSMC) method. The simulation is carried out for a mixture under
simple shear flow, using the thermostat to control the cooling effects. Given
that the SM-theory predicts a vanishing kinetic contribution to the shear
viscosity, the comparison between theory and simulations is essentially made at
the level of the collisional contribution $\eta_c$ to the shear viscosity. The
results clearly show that the GDH-theory compares with simulations much better
than the SM-theory over a wide range of values of the coefficients of
restitution, the volume fraction, and the parameters of the mixture (masses,
diameters, and concentration).
