  We construct a divergence-free HDG scheme for the Cahn-Hilliard-Navier-Stokes
phase field model. The scheme is robust in the convection-dominated regime,
produce a globally divergence-free velocity approximation, and can be
efficiently implemented via static condensation. Two numerical benchmark
problems, namely the rising bubble problem, and the Rayleigh-Taylor instability
problem are used to show the good performance of the proposed scheme.
