Asymptotic theory of fluid entrainment in dip coating

When a contact line moves with a sufficiently large speed, liquid or gas films can be entrained on a solid depending on the direction of contact-line movement. In this work, the contact-line dynamics in the situation of a generic two-fluid system is investigated. We demonstrate that the hydrodynamics of a contact line, no matter whether advancing or receding, can formally reduce to that of a receding one with small interfacial slopes. Since the latter can be well treated under the classical lubrication approximation, this analogy allows us to derive an asymptotic solution of the interfacial profiles for arbitrary values of contact angle and viscosity ratio. For the dip-coating geometry, we obtain, with no adjustable parameters, an analytical formula for the critical speed of wetting transition, which in particular predicts the onset of both liquid and gas entrainment. Moreover, the present analysis also builds a novel connection between the Cox-Voinov law and classical lubrication theory for moving contact lines.