  We study the drainage of a viscous liquid film coating outside a horizontal
cylinder. We first study the evolution of the axially invariant draining flow,
initiated at rest with uniform film thickness $\delta$. Non-linear simulations
indicate that for each $\delta$, 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 $\delta$, defining the existence of a quasi-stationary pendant
curtain 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 quasi-static 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 stabilising 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 $\{Bo,\delta\}$. The results of the linear analysis agree with the
pre-existing experiments of~\citeauthor{Bruyn1997} and non-linear simulations
of~\citeauthor{Weidner1997} on 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 predicting the final patterns emerging from the pendant curtain:
an array of pearls or pendant drops or three-dimensional droplet pinch-off.
