Numerical approximations to a new phase field model for two phase flows of complex fluids

We derive a new phase field theory for immiscible mixtures of nematic liquid crystals and viscous fluids using the variational principle coupled with the generalized Onsager principle. A novel phase transition mechanism is implemented to couple the nematic liquid crystal phase with the viscous fluid phase to arrive at the dissipative hydrodynamic model for incompressible fluid mixtures. Through a delicate explicit–implicit numerical discretization, we develop a decoupled, linear scheme for a simplified version of the phase field model, as well as a coupled, nonlinear scheme for the full model. Both schemes are shown as unconditionally energy stable with consistent, discrete dissipative energy laws. Several numerical examples are presented to show the effectiveness of the new model and the new numerical schemes developed for it.