Traveling waves for a model of gravity-driven film flows in cylindrical domains

Traveling wave solutions are studied for a recently-derived model of a falling viscous film on the interior of a vertical rigid tube. By identifying a Hopf bifurcation and using numerical continuation software, families of non-trivial traveling wave solutions may be traced out in parameter space. These families all contain a single solution at a 'turnaround point' with larger film thickness than all others in the family. In an earlier paper, it was conjectured that this turnaround point may represent a critical thickness separating two distinct flow regimes observed in physical experiments as well as two distinct types of behavior in transient solutions to the model. Here, these hypotheses are verified over a range of parameter values using a combination of numerical and analytical techniques. The linear stability of these solutions is also discussed; both large- and small-amplitude solutions are shown to be unstable, though the instability mechanisms are different for each wave type. Specifically, for small-amplitude waves, the region of relatively flat film away from the localized wave crest is subject to the same instability that makes the trivial flat-film solution unstable; for large-amplitude waves, this mechanism is present but dwarfed by a much stronger tendency to relax to a regime close to that followed by small-amplitude waves.