  We present a numerical study of noncolloidal spherical and rigid particles
suspended in Newtonian, shear thinning and shear thickening fluids employing an
Immersed Boundary Method. We consider a linear Couette configuration to explore
a wide range of solid volume fractions ($0.1\le \Phi \le 0.4$) and particle
Reynolds Numbers ($0.1\le Re_p \le 10$). We report the distribution of solid
and fluid phase velocity and solid volume fraction and show that close to the
boundaries inertial effects result in a significant slip velocity between the
solid and fluid phase. The local solid volume fraction profiles indicate
particle layering close to the walls, which increases with the nominal $\Phi$.
This feature is associated with the confinement effects. We calculate the
probability density function of local strain rates and compare their mean value
with the values estimated from the homogenization theory of \cite{Chateau08},
indicating a reasonable agreement in the Stokesian regimes. Both the mean value
and standard deviation of the local strain rates increase primarily with the
solid volume fraction and secondarily with the $Re_p$. The wide spectrum of the
local shear rate and its dependency on $\Phi$ and $Re_p$ points to the
deficiencies of the mean value of the local shear rates in estimating the
rheology of these noncolloidal complex suspensions. Finally, we show that in
the presence of inertia, the effective viscosity of these noncolloidal
suspensions deviates from that of Stokesian suspensions. We discuss how inertia
affects the microstructure and provide a scaling argument to give a closure for
the suspension shear stress for both Newtonian and power-law suspending fluids.
The stress closure is valid for moderate particle Reynolds numbers, $
O(Re_p)\sim 10$.
