On the representation of incompressible Newtownian free-surface CFD flows

This paper presents an attempt to develop a comprehensive representation of two components incompressible Newtonian flows. The primary target is to have insight on how to improve the CFD simulations of free-surface flows so as to deal with very high density ratios. First, the Navier-Stokes equations of a fluid with two immiscible components is derived from the single component equations. In this case, the surface tension does not come from the equations and must be added externally. Second we shortly review the Cahn-Hilliard and Allen-Cahn equations describing a phase separation process, and the phenomenological relation with surface-tension phenomena. Third, we build phase momentum equations which are consistent with the classical Navier-Stokes equations, but this time for miscible fluids. The concept of energy conservation applied separately to the phases is systematically used to build the phase momentum forces, with a particular regard to the surface tension. Fourth, by use of a dynamical equilibrium assumption and a specifi c splitting of the energies and their derived force, we retrieve the basic properties of the phase transport equations. The method is extended to the surface tension and to the baro-di ffusion. The relation between the phase forces and diffusion fluxes is clarified. Finally, a complete and consistent set of equations is provided and discussed