  This paper address the approximation of the dynamic of two fluids with non
matching densities and viscosities modeled by the Allen-Cahn equation coupled
with the time dependent Navier-Stokes equations. Existence, uniqueness and a
maximum principle are obtained for a totally implicit semi-discrete in time
formulation. These results are based on an original stabilized fixed point
algorithm for which well posedness and convergence is analyzed. Numerical
experiments are performed to show the influence of the iterative process.
