  We characterize the flow of a viscous suspension in an inclined channel where
the flow is maintained in a steady state under the competing influences of
gravity and an applied pressure drop. The basic model relies on a
diffusive-flux formalism. Such models are common in the literature, yet many of
them possess an unphysical singularity at the channel centreline where the
shear rate vanishes. We therefore present a regularization of the basic
diffusive-flux model that removes this singularity. This introduces an explicit
(physical) dependence on the particle size into the model equations. This
approach enables us to carry out a detailed parameter study showing in
particular the opposing effects of the pressure drop and gravity. Conditions
for counter-current flow and complete flow reversal are obtained from numerical
solutions of the model equations. These are supplemented by an analytic lower
bound on the ratio of the gravitational force to the applied pressure drop
necessary to bring about complete flow reversal.
