Annular Thin-Film Flows Driven by Azimuthal Variations in Interfacial Tension

Abstract

We consider a thin viscous film that lines a rigid cylindrical tube and surrounds a core of in-viscid fluid, and we model the flow that is driven by a prescribed azimuthally varying tension at the core-film interface, with dimensional form σ * m ― a * cos(nθ) (where constants n e ℕ and σ* m , a * ∈ ℝ). Neglecting axial variations, we seek steady two-dimensional solutions with the full symmetries of the evolution equation. For a * = 0 (constant interfacial tension), the fully symmetric steady solution is neutrally stable and there is a continuum of steady solutions, whereas for a* ≠ 0 and n = 2, 3, 4, ..., the fully symmetric steady solution is linearly unstable. For n = 2 and n = 3, we analyse the weakly nonlinear stability of the fully symmetric steady solution, assuming that 0 0 and n = 1, 2, 3,..., the film can evolve towards a steady solution featuring a drained region. We investigate the draining dynamics using matched asymptotic methods.