Dynamics of wetting: local contact angles

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Abstract
We discuss the motion of a triple line for a fluid spreading on a flat solid surface in conditions of partial wetting: the equilibrium contact angle θ e is assumed to be finite but small: 0 < θ e [Lt ] 1. We distinguish four regions: (1) a molecular domain of size a (≈ a few Ångströms) very near the triple line, where the continuum description breaks down; (2) a proximal region (of length a/θ2 e and height a/θ e ) where the long-range Van der Waals forces dominate; (3) a central region, where capillary forces and Poiseuille friction are the only important ingredients; (4) a distal region where macroscopic features (related to the size of the droplet, or to gravitational forces) come into play. In regions (2, 3, 4) the flow may be described in a lubrication approximation, and with a linearized form of the capillary forces. We restrict our attention to low capillary numbers Ca and expand the profiles to first order in Ca near the static solution. The main results are: (a) the logarithmic singularity which would have occurred in a simple wedge picture is truncated by the long-range forces, at a fluid thickness a/θ e . This effect is more important, at small θ e , than the effects of slippage which have often been proposed to remove the singularity, and which would lead to a truncation thickness comparable with the molecular size a; (b) in the central region, the local slope θ(x) grows logarithmically with the distance x from the triple line; (c) one can match explicitly the solutions in the central and distal region: we do this for one specific example: a plate plunging into a fluid with an incidence angle exactly equal to θ e . In this case we show that, far inside the distal region, the perturbation of the slope decays like 1/x2.

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