  We consider axisymmetric traveling waves propagating on the gravity-driven
flow of a liquid down a vertical fibre. Our starting point is the two-equation
model for the flow derived in the study by Ruyer-Quil \emph{et al.} [\emph{J.
Fluid Mech.} {\bf 603}, 431 (2008)]. The speed, amplitude and shape of the
traveling waves are obtained for a wide range of parameters by using asymptotic
analysis and elements from dynamical systems theory. Four different regimes are
identified corresponding to the predominance of four different physical
effects: Advection by the flow, azimuthal curvature, inertia and viscous
dispersion. Construction of the traveling-wave branches of solutions reveals
complex transitions from one regime to another. A phase diagram of the
different regimes in the parameter space is oferred.

Fluid Dynamics