  We present a systematic derivation of thermodynamically consistent
hydrodynamic phase field models for compressible viscous fluid mixtures using
the generalized Onsager principle. By maintaining momentum conservation while
enforcing mass conservation at different levels, we obtain two compressible
models. When the fluid components in the mixture are incompressible, we show
the compressible model reduces to the quasi-incompressible model via a Lagrange
multiplier approach. Several equivalent approaches to arrive at the
quasi-incompressible model are discussed. Finally, we conduct a linear
stability analysis on all binary models and show the differences of the models
in near equilibrium dynamics.
