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 sigma(m)* - a* cos(n theta) (where constants n is an element of N and sigma(m)*, a* is an element of 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* not equal 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 < epsilon(2)a*/sigma(m)* << 1 (where epsilon 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* > 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.