Stability of a liquid layer draining around a horizontal cylinder: Interplay of capillary and gravity forces

We study the drainage of a viscous liquid film coating the outside of a solid horizontal cylinder, where gravity acts vertically. We focus on the limit of large Ohnesorge numbers Oh, where inertia is negligible compared to viscous effects. We first study the evolution of the axially invariant draining flow, initiated at rest with uniform film thickness 8. Nonlinear simulations indicate that for each 8, there is a threshold in the Bond number (Bo), which compares the gravitational effects with surface tension, above which the draining liquid bulk ruptures. This critical Bo is found to scale inversely with 8, and defines the existence of a quasistationary pendant liquid curtain remaining sustained below the cylinder by surface tension. The interface of the pendant curtain is unconditionally linearly unstable and is prone to Rayleigh-Plateau-like, capillarity-driven, and Rayleigh-Taylor, gravity-driven, instabilities. The linear stability of the quasistatic state along with an energy analysis of the unstable mode illustrates that while the Rayleigh-Taylor instability is always present, capillary effects dominate the instability at small Bo, which promotes the formation of pearls enveloping the cylinder. In contrast, at large Bo, capillarity acts in a stabilizing way and the instability is purely gravity driven, forming underside modulations. We present the asymptotic energy repartition representing the different physical mechanisms at play in the instability of the saturated curtains for a wide range of {Bo, 8}. The results of the linear analysis agree with the preexisting experiments of de Bruyn [Phys. Fluids 9, 1599 (1997)] and nonlinear simulations of Weidner et al. [J. Colloid Interface Sci. 187, 243 (1997)] in the limit of a thin film and extend the results for thick films. Additionally, based on the volume made available for droplet growth by the development of the most linearly amplified wavelength, we build a tentative regime diagram that predicts the final patterns emerging from the pendant curtain, namely, an array of saturated pearls or pendant drops or the onset of three-dimensional droplet pinch-off. Furthermore, a transient growth analysis accounting for the time evolution of the base state towards a saturated curtain conclusively demonstrates that the initial flow evolution does not result in altering the most amplified wavelength, thus rationalizing a posteriori the asymptotic analysis to predict the fate of the three-dimensional patterns.