Analytical solutions to the Leslie-Ericksen equations are discussed for flow through a circular capillary and a rectangular channel. The analysis allows for a director component out of the flow-gradient plane. The resulting orientation profiles match the commonly observed textures of mesophase pitch-based carbon fibres, specifically the radial and line-origin textures. Thus, this formulation would appear to be correct for the flow of discotic mesophase pitches. Furthermore, the velocity profile for capillary flow suggests the possibility of measuring the Leslie coefficient α4for a mesophase pitch using a capillary rheometer.