J. Phys. France 43, 459-466 (1982)
DOI: 10.1051/jphys:01982004303045900

Irregular flow of a liquid film down a vertical column

T. Shlang et G.I. Sivashinsky

Department of Applied Mathematics, Tel-Aviv University, Tel-Aviv, Israel

Abstract
Using the strong surface-tension approximation, an asymptotic equation is derived which describes the evolution of the disturbed surface of a film ζ = Φ(ξ, η, τ) flowing down an infinite vertical column. In non-dimensional scaled variables this equation is Φτ + ΦΦξ + Φ ξξ + (1/μ2) ∇2Φ + ∇ 4Φ = 0, where (ξ, η) are cartesian coordinates on the surface of the cylinder, - ∞ < ξ < ∞, 0 ≤ η ≤ 2 πμ; μ is the scaled radius of the column. For μ ≤ μ c = 1, the steady flow of the film is a one-dimensional train of rings flowing irregularly downward. At μ > μc the one-dimensional nature of the flow disappears, and at μ >> μc the film surface is expected to assume the form of down-flowing drops in a state of irregular splitting and merging.

Résumé
En utilisant une approximation de tension superficielle forte, on établit l'équation d'évolution de la surface d'un film ζ = Φ(ξ,η, τ) tombant le long de la surface d'un cylindre vertical infini. Celle-ci s'ecrit en utilisant des variables sans dimension Φτ + ΦΦξ + Φξξ + (1/μ 2) ∇2Φ + ∇4Φ = 0, ou (ξ, η) sont les coordonnées cartésiennes sur la surface du cylindre, - ∞ < ξ < ∞, 0 ≤ η ≤ 2 πμ; μ est le rayon du cylindre. Pour μ ≤ μc = 1, l'écoulement stationnaire du film est formé d'un train d'anneaux s'écoulant vers le bas de façon irreguliere. A μ > μc, la nature unidimensionnelle de l'écoulement disparait, et pour μ >> μc la surface du film se festonne.

PACS
4720 - Hydrodynamic stability.
4727N - Boundary layer and shear turbulence.
4760 - Flows in ducts, channels, nozzles, and conduits.
6815 - Liquid thin films.

Key words
boundary layer turbulence -- flow instability -- liquid films -- pipe flow -- surface tension