Irregular flow of a liquid film down a vertical column

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