  The Enskog kinetic theory for moderately dense inertial suspensions under
simple shear flow is considered as a model to analyze the rheological
properties of the system. The influence of the background fluid on suspended
particles is modeled via a viscous drag force plus a Langevin-like term defined
in terms of the background temperature. In a previous paper [Hayakawa et al.,
Phys. Rev. E 96, 042903 (2017)], Grad's moment method with the aid of a linear
shear-rate expansion was employed to obtain a theory which gave good agreement
with the results of event-driven Langevin simulations of hard spheres for low
densities and/or small shear rates. Nevertheless, the previous approach had a
limitation of applicability to the high shear-rate and high density regime.
Thus, in the present paper, we extend the previous work and develop Grad's
theory including higher order terms in the shear rate. This improves
significantly the theoretical predictions, a quantitative agreement between
theory and simulation being found in the high-density region (volume fractions
smaller than or equal to $0.4$).
