  While various phase-field models have recently appeared for two-phase fluids
with different densities, only some are known to be thermodynamically
consistent, and practical stable schemes for their numerical simulation are
lacking. In this paper, we derive a new form of thermodynamically-consistent
quasi-incompressible diffuse-interface Navier-Stokes Cahn-Hilliard model for a
two-phase flow of incompressible fluids with different densities. The
derivation is based on mixture theory by invoking the second law of
thermodynamics and Coleman-Noll procedure. We also demonstrate that our model
and some of the existing models are equivalent and we provide a unification
between them. In addition, we develop a linear and energy-stable
time-integration scheme for the derived model. Such a linearly-implicit scheme
is nontrivial, because it has to suitably deal with all nonlinear terms, in
particular those involving the density. Our proposed scheme is the first linear
method for quasi-incompressible two-phase flows with nonsolenoidal velocity
that satisfies discrete energy dissipation independent of the time-step size,
provided that the mixture density remains positive. The scheme also preserves
mass. Numerical experiments verify the suitability of the scheme for two-phase
flow applications with high density ratios using large time steps by
considering the coalescence and break-up dynamics of droplets including
pinching due to gravity.
