Cox-Voinov theory with slip

Most of our understanding of moving contact lines relies on the limit of small capillary number Ca. This means the contact line speed is small compared to the capillary speed gamma/eta, where gamma is the surface tension and eta the viscosity, so that the interface is only weakly curved. The majority of recent analytical work has assumed in addition that the angle between the free surface and the substrate is also small, so that lubrication theory can be used. Here, we calculate the shape of the interface near a slip surface for arbitrary angles, and for two phases of arbitrary viscosities, thereby removing a key restriction in being able to apply small capillary number theory. Comparing with full numerical simulations of the viscous flow equation, we show that the resulting theory provides an accurate description up to Ca approximate to 0.1 in the dip coating geometry, and a major improvement over theories proposed previously.