In this work, we propose a new method for studying dynamics in a liquid crystal arising from torques due to an externally imposed shear flow or other sources of torque. Here a spatially homogeneous liquid crystal system governed by a Smoluchowski equation for the orientational probability density function is investigated. To analyze the PDE model, we develop a novel direct computational algorithm based on a Voronoi cell-based finite volume scheme. This method has the capability of describing abrupt changes in the density function, using fluxes through cell boundaries. We first validate our approach by capturing prior results from Maier-Saupe theory illustrating a phase transition in the system due to nematic interactions as well as the effects of an externally imposed electric field. We then investigate the coupling of an external flow field with the Smoluchowski equation describing the full orientational dynamics of the system.