The alignment properties and distribution of flow speed during Poiseuille flow through a microchannel of a nematic liquid crystal in a cell with homeotropic surface alignment has been measured using a combination of conoscopy, fluorescence confocal polarizing microscopy, and time-lapse imaging. Two topologically distinct director profiles, with associated fluid velocity fields, are found to exist with the preferred state dictated by the volumetric flow rate of the liquid crystal. The results show excellent agreement with model data produced using the Ericksen-Leslie nematodynamics theory.