In this article, we investigate the application of an operator-splitting/finite element method to the numerical simulation of a liquid crystal  flow. The operator-splitting is achieved through three stages, so that each stage is simpler and easier to deal with than the step of any un-split implicit scheme. The first stage deals with the system coupling a Stokes equation for velocity with an equation modeling the diffusion of the liquid crystal director field. The second stage deals with the convection of both the velocity and director field; a wave-like equation approach is used to treat this advection part and proves being quite efficient. Finally, the third stage deals with the nonlinear terms; a (quasi) closed form solution can be derived for this stage. Overall, with this type of splitting, the nonlinear terms in the liquid crystal model can be treated quite easily. The results of several numerical experiments show the good performances of the three-stage splitting method discussed in this article.