Annular thin-film flows driven by azimuthal variations in interfacial tension

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 ∈ and σ*m, a* ∈ ). 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.