  The Boltzmann kinetic equation for low-density granular suspensions under
simple shear flow is considered to determine the velocity moments through the
fourth degree. The influence of the interstitial gas on solid particles is
modeled by a viscous drag force term plus a stochastic Langevin-like term. Two
independent but complementary approaches are followed to achieve exact results.
First, to keep the structure of the Boltzmann collision operator, the so-called
inelastic Maxwell models (IMM) are considered. In this model, since the
collision rate is independent of the relative velocity of the two colliding
particles, the forms of the collisional moments can be obtained without the
knowledge of the velocity distribution function. As a complement of the
previous effort, a BGK-type kinetic model adapted to granular gases is solved
to get the velocity moments of the velocity distribution function. The
analytical predictions of the rheological properties (which are \emph{exactly}
obtained in terms of the coefficient of restitution $\alpha$ and the reduced
shear rate $a^*$) show in general an excellent agreement with event-driven
simulations performed for inelastic hard spheres. In particular, both
theoretical approaches show clearly that the temperature and non-Newtonian
viscosity exhibit an $S$ shape in a plane of stress-strain rate (discontinuous
shear thickening effect). With respect to the fourth-degree velocity moments,
we find that while those moments have unphysical values for IMM in a certain
region of the parameter space of the system, they are well defined functions of
both $\alpha$ and $a^*$ in the case of the BGK kinetic model. The explicit
shear-rate dependence of the fourth-degree moments beyond this critical region
is also obtained and compared against available computer simulations.
