Reductions of the Navier-Stokes-Allen-Cahn and the Navier-Stokes-Cahn-Hilliard equations

This paper studies two well-known models for two-phase fluid flow at constant temperature, the isothermal Navier-Stokes-Allen-Cahn and the isothermal Navier-Stokes-Cahn-Hilliard equations, both of which consist of equations for the (total) fluid density rho, the (mass-averaged)velocity u and the concentration (of one of the phases,) c. Assuming in either case that both phases are incompressible with different densities, each of the models is shown to reduce to a system of evolution equations in rho and u alone. In the case of the Navier-Stokes-Allen-Cahn model, this reduced system is the classical Navier-Stokes-Korteweg model. The reduced system resulting from the Navier-Stokes-Cahn-Hilliard equations is a novel `integro'-differential system in which a non-local operator acts on the divergence of the velocity.