  The Boltzmann kinetic equation for dilute granular suspensions under simple
(or uniform) shear flow (USF) is considered to determine the non-Newtonian
transport properties of the system. In contrast to previous attempts based on a
coarse-grained description, our suspension model accounts for the real
collisions between grains and particles of the surrounding molecular gas. The
latter is modeled as a bath (or thermostat) of elastic hard spheres at a given
temperature. Two independent but complementary approaches are followed to reach
exact expressions for the rheological properties. First, the Boltzmann equation
for the so-called inelastic Maxwell models (IMM) is considered. The fact that
the collision rate of IMM is independent of the relative velocity of the
colliding spheres allows us to exactly compute the collisional moments of the
Boltzmann operator without the knowledge of the distribution function. Thanks
to this property the transport properties of the sheared granular suspension
can be \emph{exactly} determined. As a second approach, a
Bhatnagar--Gross--Krook (BGK)-type kinetic model adapted to granular
suspensions is solved to compute the velocity moments and the velocity
distribution function of the system. The theoretical results show in general a
good agreement with the approximate analytical results derived for inelastic
hard spheres (IHS) by means of Grad's moment method and with computer
simulations performed in the Brownian limiting case ($m/m_g\to \infty$, where
$m_g$ and $m$ are the masses of the particles of the molecular and granular
gases, respectively). In addition, as expected the IMM and BGK results show
that the temperature and non-Newtonian viscosity exhibit and $S$ shape in a
plane of stress-strain rate (discontinuous shear thickening, DST). The DST
effect becomes more pronounced as the mass ratio $m/m_g$ increases.
