Coating flows down a vertical fibre: Towards the full Navier-Stokes problem

We revisit the dynamics of a thick liquid film flowing down a vertical fibre. Instead of deriving a long-wave model, we directly solve the Navier-Stokes equations using a domain mapping technique and the exact steady travelling wave solutions are explored using a dynamical systems theory approach. Three distinct flow regimes, labelled 'a', 'b' and 'c', observed in previous experiments (Kliakhandler et al., J. Fluid Mech., vol. 429, 2001, pp. 381-390) are investigated. Flow regime 'a' refers to a steady flow state in which large droplets are separated by a long thin film. Flow regime 'b' is a necklace-like flow. In flow regime 'c', a cyclic process of droplet coalescence and breakup was observed. By matching the mean flow rates of the travelling wave solutions and experimental data, our travelling wave solutions show an excellent agreement with flow regimes 'a' and 'b'. The time-periodic flow regime 'c' is compared with direct numerical simulation of the Navier-Stokes equations. A snapshot of the simulation shows a remarkable similarity to an experimental image and the discrepancy of mean wave speed and maximal wave height between our numerical simulation and experimental data is negligible.