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Stability of film flow over inclined topography based on a long-wave nonlinear model

Published online by Cambridge University Press:  24 July 2013

D. Tseluiko ,
M. G. Blyth  and
D. T. Papageorgiou
D. Tseluiko*
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK
M. G. Blyth
Affiliation:
School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK
D. T. Papageorgiou
Affiliation:
Department of Mathematics and Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK
*
†Email address for correspondence: [email protected]
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Abstract

The stability of a viscous liquid film flowing under gravity down an inclined wall with periodic corrugations is investigated. A long-wave model equation valid at near-critical Reynolds numbers is used to study the film dynamics, and calculations are performed for either sinusoidal or rectangular wall corrugations assuming either a fixed flow rate in the film or a fixed volume of fluid within each wall period. Under the two different flow assumptions, steady solution branches are delineated including subharmonic branches, for which the period of the free surface is an integer multiple of the wall period, and the existence of quasi-periodic branches is demonstrated. Floquet–Bloch theory is used to determine the linear stability of steady, periodic solutions and the nature of any instability is analysed using the method of exponentially weighted spaces. Under certain conditions, and depending on the wall period, the flow may be convectively unstable for small wall amplitudes but undergo transition to absolute instability as the wall amplitude increases, a novel theoretical finding for this class of flows; in other cases, the flow may be convectively unstable for small wall amplitudes but stable for larger wall amplitudes. Solutions with the same spatial period as the wall become unstable at a critical Reynolds number, which is strongly dependent on the period size. For sufficiently small wall periods, the corrugations have a destabilizing effect by lowering the critical Reynolds number above which instability occurs. For slightly larger wall periods, small-amplitude corrugations are destabilizing but sufficiently large-amplitude corrugations are stabilizing. For even larger wall periods, the opposite behaviour is found. For sufficiently large wall periods, the corrugations are destabilizing irrespective of their amplitude. The predictions of the linear theory are corroborated by time-dependent simulations of the model equation, and the presence of absolute instability under certain conditions is confirmed. Boundary element simulations on an inverted substrate reveal that wall corrugations can have a stabilizing effect at zero Reynolds number.

JFM classification

Instability: Absolute/convective instability Low-Reynolds-number flows: Boundary integral methods Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 729 , 25 August 2013 , pp. 638 - 671
Copyright
©2013 Cambridge University Press 

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