contact line. Numerous methods have been proposed to resolve this discontinuity or work around it. @hocking1977 showed that, using domain perturbation method in cylindrical coordinates, any slip-velocity model [@navier1823; @thompson1997] resolves the velocity singularity. Another method to circumvent the contact line singularity has been to use the Cahn-Hilliard-van der Waals model [@cahn1977] as the basis for either diffusive interface models [@seppecher1996; @jacqmin2000], or for precursor film models [@hervet1984; @degennes1985]. Precursor film models use a disjoining pressure [@derjaguin1955; @teletzke1987; @herring2010] to model the van der Waals forces that cause the formation of a precursor film ahead of the interface, removing the singularity. More exotic models have suggested local shear-thinning [@cox1998] and non-constant surface tension [@shikhmurzaev1993; @shikhmurzaev1997] as possible solutions.

For the question of how the contact angle depends on the contact line velocity there are also various models. Typically a distinction is made between the local microscopic contact angle at the contact line and the macroscopic apparent contact angle, which is observed in experiments. Due to experimental limitations to access sufficiently small length scales, the apparent contact angle is measured away from the contact line, and curvature of the interface causes this angle to be different from the microscopic contact angle 