Using method of matched asymptotic expansions, we derive the sharp interface limit for the diffusive interface model with the generalized Navier boundary condition recently proposed by Qian, Wang and Sheng in "Molecular scale contact line hydrodynamics of immiscible flows," and "Power-law slip profile of the moving contact line in two-phase immiscible flows," for the moving contact line problem. We show that the leading order problem satisfies a boundary value problem for a coupled Hale-Shaw and Navier-Stokes equations with the interface being a free boundary, and the leading order dynamic contact angle is the same as the static one satisfying the Young’s equation.
"The Sharp Interface Limit of a Phase Field Model for Moving Contact Line Problem." Methods Appl. Anal. 14 (3) 287 - 294, September 2007.