In this work, we study the constitutive behavior of interacting colloidal suspensions for moderate and high concentrations. Specifically, using a lattice Boltzmann solver, we numerically examine suspensions flowing through narrow channels, and explore the significance of the interaction potential strength on the system's macroscopic response. When only a short-range interaction potential is considered, a Newtonian behavior is always recovered and the system's effective viscosity mostly depends on the suspension concentration. However, when using a Lennard-Jones potential we identify two rheological responses depending on the interaction strength, the volume fraction, and the pressure drop. Exploiting a model proposed in the literature we rationalize the simulation data and propose scaling relations to identify the relevant energy scales involved in these transport processes. Moreover, we find that the spatial distribution of colloids in layers parallel to the flow direction does not correlate with changes in the system macroscopic response; but, interestingly, the rheology changes do correlate with the spatial distribution of colloids within individual layers. Namely, suspensions characterized by a Newtonian response display a cubiclike structure of the colloids within individual layers, whereas for suspensions with non-Newtonian response colloids organize in a hexagonal structure.