Asymptotic theory of wetting transition in dip coating*

When a plate is withdrawn from a liquid bath up to a sufficiently large speed, the capillary force fails to compete with the viscous drag, leading to the entrainment of liquid films. This wetting transition has been theoretically understood at small equilibrium contact angles. For a more general case where the viscosity ratio of two fluids and the contact angle are almost arbitrary, we demonstrate that this problem can always be formally reduced to the simpler above-mentioned problem through an appropriate mathematical transformation. Both the dragging and pushing problems can be treated under this approach. By this means, we obtain the critical plate speed for the onset of wetting transition. Moreover, we build a connection between the Cox-Voinov law and the classical lubrication theory for moving contact lines.