We present an energetic variational phase-field model for the two-phase Incompressible flow with one phase being the nematic liquid crystal. The model leads to a coupled nonlinear system satisfying an energy law. An efficient and easy-to-implement numerical scheme is presented for solving the coupled nonlinear system. We use this scheme to simulate two benchmark experiments: one is the formation of a bead-on-a-string phenomena, and the other is the dynamics of drop pinching-off. We investigate the detailed dynamical pinch-off behavior, as well as the formation of the consequent satellite droplets, by varying order parameters of liquid crystal bulk and interfacial anchoring energy constant. Qualitative agreements with experimental results are observed.