We present some efficient modeling approach for the two phase complex fluid system, that is the mixture of nematic liquid crystal drop and viscous ambient fluids.

Similar approach has been applied to the two phase fluid system with variable density and viscosity, The models, derived by employing an energetic variational phase field approach, are thermodynamically consistent that admits an energy dissipation law.

Some decoupled, linear, energy stable schemes are developed to solve the models efficiently.

Qualitative agreements with experimental results are observed.