The evolution of viscous flow on a cylinder

The temporal development of a thin viscous layer on a stationary cylinder is investigated both theoretically and experimentally. A constant volume of fluid is released instantaneously at the top of a cylinder, whose axis is horizontal. The resultant flow is confined laterally by vertical plates and propagates down a channel whose slope varies continuously along the flow. The structure of the current is shown using lubrication theory to initially have a uniform thickness. This thickness is independent of the fluid volume per cross-stream width. The thickness and length of the current after time $t$ from initiation are given by $(3\nu R/2g)^{1/2}t^{-1/2}$ and $A(2g/3\nu R)^{1/2}t^{1/2}$ respectively, where $g$ is gravity, $A$ the cross sectional area of fluid, $\nu$ its kinematic viscosity and $R$ the radius of the cylinder. These results are in good agreement with experimental data taken on a cylinder of radius $15\,\mathrm{cm}$ and width $11\,\mathrm{cm}$. The front of the current, which can produce a series of rivulets after it has propagated a distance proportional to $A^{1/2}$, ultimately detaches from the underside of the cylinder. Video clips of the laboratory experiments highlight some remarkable features at the contact line due to capillary action.