A boundary condition for fluid/fluid flow at the solid interface

When using classical hydrodynamics, the imposition of no-slip boundary condition on a solid surface results in a logarithmically singular shear force at the fluid/fluid/solid line of contact. Boundary conditions that admit some form of slip at the solid interface, such as the linear slip-shear relation also known as Navier slip boundary condition, have been used to regularize this singularity. The widely used Cox's theoretical work is also derived under the assumption of slip at the solid interface. Here we present a boundary condition, based on Cox's analysis, that resolves the contact line singularity in the numerical simulation of contact line flows using a volume of fluid method. This boundary condition provides a universal formula that determines the contact angle as a function of the gird size. The model can be thought of as a subgrid-scale model for computations, where the resolution is not sufficient to accurately describe the contact line inner region frictional drag. We provide numerical examples for a wide range of steady flow problems.