The thesis contains two parts. In the first part, for time-dependent Navier-Stokes equations, a full discrete version of the velocity-correction method and consistent splitting method are introduced and analyzed rigorously. Optimal time and spatial error estimates are derived for the standard and rotational versions of both methods.

In the second part, an energetic variational phase field model which describes the motions of mixtures of two incompressible flows are used to study the retraction and pinch-off of a liquid filament. An efficient and accurate numerical scheme is presented and implemented for the coupled nonlinear systems consisting of Navier-Stokes type linear momentum equations and a volume preserving Allen-Cahn type phase equation.

The annihilation of a hedgehog-antihedgehog pair in hydrodynamics of the (elastically) isotropic nematic liquid crystal materials is modeled using the Ericksen-Leslie theory which results in a nonlinear system for the flow velocity field and liquid crystal director field coupled through the transport of the directional order parameter and the induced elastic stress. An efficient and accurate numerical scheme is presented and implemented for this coupled nonlinear system in an axisymmetric cylindrical domain.