  Shear flow of dense, non-Brownian suspensions is simulated using the discrete
element method, taking particle contact and hydrodynamic lubrication into
account. The resulting flow regimes are mapped in the parametric space of solid
volume fraction, shear rate, fluid viscosity and particle stiffness. Below a
critical volume fraction $\phi_c$, the rheology is governed by the Stokes
number, which distinguishes between viscous and inertial flow regimes. Above
$\phi_c$, a quasistatic regime exists for low and moderate shear rates. At very
high shear rates, the $\phi$ dependence is lost and soft particle rheology is
explored. The transitions between rheological regimes are associated with the
evolving contribution of lubrication to the suspension stress. Transitions in
microscopic phenomena such as inter-particle force distribution, fabric and
correlation length are found to correspond to those in the macroscopic flow.
Motivated by the bulk rheology, a constitutive model is proposed combining a
viscous pressure term with a dry granular model presented by Chialvo, Sun and
Sundaresan [Phys. Rev. E. \textbf{85}, 021305 (2012)]. The model is shown to
successfully capture the flow regime transitions.
