A phase field model for the mixture of two incompressible fluids is
 presented in this paper.  The model consists of a Navier-Stokes
 system (linear momentum equation) coupled with a Cahn-Hilliard
 equation (phase field equation) through an extra stress term and the
 transport term.  The extra stress represents the (phase induced)
 capillary effect for the mixture due to the surface tension.  A
 Fourier-spectral method for the numerical approximation of this
 system is proposed and analyzed. Numerical results illustrating the
 robustness and versatility of the model are presented.