I shall present an energetic variational phase field model for
multiphase incompressible flows which leads to a set of coupled
nonlinear system consisting a phase equation and the Navier-Stokes
equations. We shall pay particular attention to situations with large
density ratios as they lead to formidable challenges in both analysis
and simulation.

I shall present efficient and accurate numerical schemes for solving
this coupled nonlinear system, and show ample numerical results (air
bubble rising in water, Newtonian bubble rising in a polymeric fluid, defect
motion in a liquid crystal flow, etc.) which not only demonstrate the
effectiveness of the numerical schemes, but also validate the
flexibility and robustness of the phase-field model.