  We present a control-volume approach for deriving a simplified model for the
gravity-driven flow of an axisymmetric liquid film along a vertical fiber. The
model accounts for gravitational, viscous, inertial and surface tension effects
and results in a pair of coupled one-dimensional nonlinear partial differential
equations for the film profile and average downward velocity as functions of
time and axial distance along the fiber. Two versions of the model are
obtained, one assuming a plug-flow velocity profile and a constant thin
boundary layer thickness to model the drag force on the fluid, the other
approximating the drag using the fully-developed laminar velocity profile for a
locally uniform film. A linear stability analysis shows both models to be
unstable to long waves or short wavenumbers, with a specific wavenumber in that
range having a maximal growth rate. Numerical simulations confirm this
instability and lead to nonlinear periodic traveling wave solutions which can
be thought of as chains of identical droplets falling down the fiber. Physical
experiments are also carried out on such a system using safflower oil as the
working liquid and a taut fishing line as the fiber. A machine learning scheme
is used to find the best set of parameters in the laminar flow model to match
the experimental results to the simulations. Good agreement is found between
the two, with parameter values that are quite close to their original estimates
based on the approximate values of the physical parameters.
