Computational modeling of the steady capillary Poiseuille flow of flow-aligning discotic nematic liquid crystals (DNLCs) using the Leslie-Ericksen (LE) equations predicts solution multiplicity And multistability. The phenomena are independent of boundary conditions. The steady state solutions are classified into: (a) primary, (b) secondary, and (c) hybrid. Primary solutions exist for all orientation boundary conditions and all flow rates, and are characterized by a flow-alignment angle that is closest to the anchoring angle at the bounding surface. Secondary solutions exist for all orientation boundary conditions and flow rates above a certain critical value,. The secondary solutions are characterized by a flow-alignment angle which can be either the nearest neighbor below the primary solution or any multiple of pi above. Hybrid solutions interpolate between the primary and the nearest secondary solutions, and hence exhibit two alignment angles. All solutions are stable to planar, finite amplitude perturbations. Hybrid solutions are unstable to front propagation and lead to primary or secondary solutions. The non-Newtonian rheology of the primary and secondary solutions is characterized by non-classical shear thinning and thickening apparent viscosity behavior. Well-aligned monodomains can lead to shear thickening, thinning, or a sequence of both. The degree of rheological uncertainty is present for planar and homeotropic anchoring conditions. The non-Newtonian rheology of non-aligned samples leads to shear thinning and lack the uncertainty of well-aligned samples, since the apparent viscosity becomes insensitive to orientation. (C) 2003 Elsevier Science B.V. All rights reserved.