In this paper we present a finite difference method on a staggered grid for solving two-dimensional free surface flows of liquid crystalline polymers governed by the Ericksen-Leslie dynamic equations. The numerical technique is based on a projection method and employs Cartesian coordinates. The technique solves the governing equations using primitive variables for velocity, pressure, extra-stress tensor and the director. These equations are nonlinear partial differential equations consisting of the mass conservation equation and the balance laws of linear and angular momentum. Code verification and convergence estimates are effected by solving a flow problem on 5 different meshes. Two free surface problems are tackled: A jet impinging on a flat surface and injection molding. In the first case the buckling phenomenon is examined and shown to be highly dependent on the elasticity of the fluid. In the second case, injection molding of two differently shaped containers is carried out and the director is shown to be strongly dependent on its orientation at the boundaries.