  Over the last decades, many diffuse-interface Navier-Stokes Cahn-Hilliard
models with non-matching densities have appeared in the literature. These
models claim to describe the same physical phenomena, yet they are distinct
from one another. The overarching objective of this work is to bring all of
these models together by laying down a unified framework of Navier-Stokes
Cahn-Hilliard models with non-zero mass fluxes. Our development is based on
three unifying principles: (1) there is only one system of balance laws based
on continuum mixture theory that describes the physical model, (2) there is
only one natural energy-dissipation law that leads to quasi-incompressible
Navier-Stokes Cahn-Hilliard models, (3) variations between the models only
appear in the constitutive choices. The framework presented in this work now
completes the fundamental exploration of alternate non-matching density
Navier-Stokes Cahn-Hilliard models that utilize a single momentum equation for
the mixture velocity, but leaves open room for further sophistication in the
energy functional and constitutive dependence.
