From molecular dynamics simulations on immiscible flows, we find the
relative slipping between the fluids and the solid wall everywhere to
follow the generalized Navier boundary condition, in which the amounnt
of slipping is proportional to the sum of the tangential viscous stress
and the uncompensated Young stress. The latter arises fron deviation of
the fluid-fluid interface from its static configuration. We give a
continuum formulaiton of the immiscible flow hydrodynamics, comprising
the generalized Navier boundary condition, the Navier-Strokes equation,
and the Cahn-Hilliard interfacial and velocity profiles matching
quantitatively with those from the molecular dynamics simulations.