Spreading of a wetting film under the action of van der Waals forces

It has been long recognized that liquids can spread under the action of disjoining pressure. In the case of a liquid attracted to the solid by van der Waals forces, a precursor film is formed, and there is no well-defined contact line that would give rise to the infamous viscous stress singularity. Nevertheless, the problem remains difficult, due to the divergence of both viscous and disjoining pressure terms ~which have to compensate each other when the layer thickness tends to zero! and a wide separation between the nearly molecular scale of van der Waals forces and macroscopic scales, where gravity and other extrinsic factors become important. It turns out that the behavior on both scales is interrelated, so that weak forces acting at macroscopic distances strongly influence the shape of a moving film, even close to the transition region between the precursor and the macroscopic film, affecting the visible contact angle. This necessitates matching between computations carried out in the ‘‘outer’’ ~gravity! and ‘‘inner’’ ~van der Waals! regions and makes the results sensitive to the matching conditions. After formulating the basic equations in lubrication approximation in Sec. II, we shall stress in Sec. III the pitfalls of this sensitive problem: instability of the static solution and a rescaling symmetry that disallows using the drift velocity as a perturbation parameter. We shall further demonstrate that the familiar cubic-root dependence of the dynamic contact angle on the drift velocity is a direct consequence of the scaling invariance. On the other hand, attempts at computing the proportionality constant in this dependence by solving the stationary equation satisfying asymptotic conditions in the comoving frame are hindered by numerical instabilities, and the ‘‘outer’’ asymptotics converges too slowly to allow effective separation of the inner region, where the van der Waals interactions are essential, and the outer region, where macroscopic factors, like gravity, come into play. This persuades us to seek a numerical solution of the problem using a variable grid extending over widely separated scales and