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

L. R. Band*, D. S. Riley, P. C. Matthews, J. M. Oliver & S. L. Waters

We consider a thin viscous film that lines a rigid cylindrical tube and surrounds a core of inviscid fluid, and we model the flow that is driven by a prescribed azimuthally varying tension at the core–film interface, with dimensional form σm* – a* cos(nθ) (where constants n ∈ 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 < ϵ2a*/σm* ≪ 1(where ϵ denotes the ratio between the typical film thickness and the tube radius); for n = 3, this analysis leads us to additional linearly unstable steady solutions. Solving the full nonlinear system numerically, we confirm the stability analysis and furthermore find that for a* gt 0 and n = 1, 2, 3, hellip, the film can evolve towards a steady solution featuring a drained region. We investigate the draining dynamics using matched asymptotic methods.

The Quarterly Journal of Mechanics and Applied Mathematics 62 (4), 403-430

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