Draining and spreading along geometries that cause converging flows: Viscous gravity currents on a downward-pointing cone and a bowl-shaped hemisphere

When a viscous liquid is released on an inclined solvophilic substrate, it spreads downwards by gravity and tends to coat the substrate. Here we report the gravitational axisymmetric spreading of a viscous liquid with a fixed volume on inclined geometries that cause converging flows, for example, on the inside of a downward-pointing hollow cone (funnel) and the lower part of a sphere (bowl). For the limit that the effect of surface tension is negligible, we analytically determine the thickness profile as well as the time-dependent position of the front of the spreading liquid. In the typical scenario that a liquid spreads on a geometry such as an inclined plate, the thickness of the front of the spreading liquid monotonically decreases in time throughout the spreading and a fingering instability occurs while the liquid film thins. However, we show that on an inclined geometry, where convergence of the flow occurs, the thickness of the spreading front first decreases in time and then increases. We also predict a critical volume Vc such that a fingering instability occurs if the volume of the spreading liquid is less than Vc. Experiments of axisymmetric spreading on a downward-pointing cone (funnel) are then performed and the measured position of the front and the critical liquid volume of the fingering instability agree with our theoretical predictions. This study highlights the effect of a geometry that focuses and thickens a thin film.