Spreading of rotating liquid annular ring with hysteresis effects

The spreading of a liquid, subjected to centrifugal forces or an air jet, leads to a lowering of the liquid height at the center. If the system is partially wetting, experiments show a break up of the liquid and the appearance of a dry patch at the center. In this article, the spreading of a liquid droplet or a liquid film, featuring a dry patch at the center, is investigated. Via a generalized Tanner’s law, we allow for a contact-angle hysteresis in partially wetting systems. By means of the lubrication approximation, an analytical quasi-steady solution can be derived in the limit of small capillary numbers. We find a power-law regime for the spreading, in which at least one of the contact lines follows a power law in time, almost independent of both static contact angles. The influence of the static contact angles remains restricted to the beginning of the spreading and to transition periods. Four different types of spreading can be identified, namely spreading (i) with a very thin central liquid layer, (ii) as annular ring with central dry patch, (iii) into a static equilibrium, and (iv) with closing of the central dry patch. In a flow map, these types of spreading are mapped as function of both static contact angles. Moreover, the dependency from gravitational and centrifugal forces is investigated and included in the flow map. For a perfectly wetting system, a disjoining-pressure correction in combination with Tanner’s law prohibits negative liquid heights at the center. Hereby, the magnitudes of the Hamaker constants have a negligible influence and arbitrary-small values can be used, since the dynamics of the contact line remains controlled by Tanner’s law.