The Quarterly Journal of Mechanics and Applied Mathematics Advance Access originally published online on June 29, 2009
The Quarterly Journal of Mechanics and Applied Mathematics 2009 62(4):403-430; doi:10.1093/qjmam/hbp015
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Annular Thin-Film Flows Driven by Azimuthal Variations in Interfacial Tension
( Centre for Plant Integrative Biology, University of Nottingham, Sutton Bonington, LE12 5RD and School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD )
( Division of Applied Mathematics, School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD )
( Mathematical Institute, University of Oxford, 24–29 St. Giles’, Oxford, OX1 3LB )
leah.band{at}nottingham.ac.uk
Received 26 September 2008. Revise 24 February 2009. Accepted 18 May 2009.
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Abstract |
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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 
m* – a* cos(n
) (where constants n 
and *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 < 
2a*/m* << 1(where 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.