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

摘要

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 σ_m* – 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*/σ_m* ≪ 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.