Contact-line instability of liquids spreading on top of rotating substrates

Abstract

A (film or) droplet of viscous liquid spreads isothermally on a smooth horizontal solid surface. The lubrication approximation is used to study the linear stability of thin (films or) droplets, subject to capillary, gravitational, and centrifugal forces, and a variety of contact-angle-versus-speed conditions. All equations are derived for plane spreading films and rotationally-symmetric spreading droplets, while the discussion of the results is carried out for the droplets. It is found that in general two types of two-dimensional base states develop. Early on there is a simple convex contour and later a contour with a pronounced capillary ridge near the contact line. While the convex contour remains stable, the capillary-ridge contour becomes unstable with regard to disturbances, which are periodic in the lateral direction. As the contact line advances in time, this instability involves a transition from two-dimensional to three-dimensional spreading, whereas modes with increasing wave numbers become successively unstable. The onset of the instability is controlled by gravitational, centrifugal, and capillary forces, whereas gravitational and capillary forces tend to stabilize and centrifugal forces tend to destabilize the system. For partially-wetting systems, the neutral stability appears to be not affected by the static advancing contact angle, though the growth rates are modified.