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On the multiple solutions of coating and rimming flows on rotating cylinders

Published online by Cambridge University Press:  27 November 2017

André v. B. Lopes ,
Uwe Thiele  and
Andrew L. Hazel Open the ORCID record for Andrew L. Hazel [Opens in a new window]
André v. B. Lopes
Affiliation:
School of Mathematics and Manchester Centre for Nonlinear Dynamics (MCND), University of Manchester, Oxford Road, Manchester M13 9PL, UK
Uwe Thiele
Affiliation:
Institut für Theoretische Physik, Westfälische Wilhelms-Universität Münster, Wilhelm Klemm Str. 9, 48149 Münster, Germany Center of Nonlinear Science (CeNoS), Westfälische Wilhelms-Universität Münster, Corrensstr. 2, 48149 Münster, Germany
Andrew L. Hazel*
Affiliation:
School of Mathematics and Manchester Centre for Nonlinear Dynamics (MCND), University of Manchester, Oxford Road, Manchester M13 9PL, UK
*
†Email address for correspondence: [email protected]
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Abstract

We consider steady solutions of the Stokes equations for the flow of a film of fluid on the outer or inner surface of a cylinder that rotates with its axis perpendicular to the direction of gravity. We find that previously unobserved stable and unstable steady solutions coexist over an intermediate range of rotation rates for sufficiently high values of the Bond number (ratio of gravitational forces relative to surface tension). Furthermore, we compare the results of the Stokes calculations to the classic lubrication models of Pukhnachev (J. Appl. Mech. Tech. Phys., vol 18, 1977, pp. 344–351) and Reisfeld & Bankoff (J. Fluid Mech., vol. 236, 1992, pp. 167–196); an extended lubrication model of Benilov & O’Brien (Phys. Fluids, vol. 17, 2005, 052106) and Evans et al. (Phys. Fluids, vol. 16, 2004, pp. 2742–2756); and a new lubrication approximation formulated using gradient dynamics. We quantify the range of validity of each model and confirm that the gradient-dynamics model is most accurate over the widest range of parameters, but that the new steady solutions are not captured using any of the simplified models because they contain features that can only be described by the full Stokes equations.

JFM classification

Interfacial Flows (free surface): Interfacial Flows (free surface) Low-Reynolds-number flows: Lubrication theory Interfacial Flows (free surface): Thin films
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 835 , 25 January 2018 , pp. 540 - 574
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Copyright
© 2017 Cambridge University Press 

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