Title: Energy stable schemes and numerical simulations of two phase
complex fluids by the phase-field method

Abstact: 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. Two efficient and easy-to-implement numerical schemes are
presented for solving the coupled nonlinear system. We use the schemes
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.