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Inertially induced cyclic solutions in thin-film free-surface flows

Published online by Cambridge University Press:  22 August 2014

C. M. Groh  and
M. A. Kelmanson
C. M. Groh
Affiliation:
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK Department of Radiation Oncology, University of Würzburg, D-97080 Würzburg, Germany
M. A. Kelmanson*
Affiliation:
Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK
*
†Email address for correspondence: [email protected]
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Abstract

New mechanisms are discovered regarding the effects of inertia in the transient Moffatt–Pukhnachov problem (J. Méc., vol. 187, 1977, pp. 651–673) on the evolution of the free surface of a viscous film coating the exterior of a rotating horizontal cylinder. Assuming two-dimensional evolution of the film thickness (i.e. neglecting variation in the axial direction), a multiple-timescale procedure is used to obtain explicitly parameterized high-order asymptotic approximations of solutions of the spatio-temporal evolution equation. Novel, hitherto-unexplained transitions from stability to instability are observed as inertia is increased. In particular, a critical Reynolds number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Re}_c$ is predicted at which occurs a supercritical pitchfork bifurcation in wave amplitude that is fully explained by the new asymptotic theory. For $\mathit{Re}<\mathit{Re}_c$, free-surface profiles converge algebraically-cum-exponentially to a steady state and, for $\mathit{Re}> \mathit{Re}_c$, stable temporally periodic solutions with leading-order amplitudes proportional to $(\mathit{Re}-\mathit{Re}_c)^{1/2}$ are found, i.e. in the régime in which previous related literature predicts exponentially divergent instability. For $\mathit{Re}=\mathit{Re}_c$, stable solutions are found that decay algebraically to a steady state. A model solution is proposed that not only captures qualitatively the interaction between fundamental and higher-order wave modes but also offers an explanation for the formation of the lobes observed in Moffatt’s original experiments. All asymptotic theory is convincingly corroborated by numerical integrations that are spectrally accurate in space and eighth/ninth-order accurate in time.

JFM classification

Low-Reynolds-number flows: Lubrication theory Instability: Parametric instability Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 755 , 25 September 2014 , pp. 628 - 653
Copyright
© 2014 Cambridge University Press 

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