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.
"Generalized Navier Boundary Condition for the Moving Contact Line." Commun. Math. Sci. 1 (2) 333 - 341, June 2003.