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Viscous film flow coating the interior of a vertical tube. Part 1. Gravity-driven flow

Published online by Cambridge University Press:  25 March 2014

Roberto Camassa ,
H. Reed Ogrosky  and
Jeffrey Olander
Roberto Camassa
Affiliation:
Carolina Center for Interdisciplinary Applied Mathematics, Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599-3250, USA
H. Reed Ogrosky*
Affiliation:
Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706-1325, USA
Jeffrey Olander
Affiliation:
Department of Physics, University of North Carolina, Chapel Hill, NC 27599-3255, USA
*
†Email address for correspondence: [email protected]
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Abstract

The gravity-driven flow of a viscous liquid film coating the inside of a tube is studied both theoretically and experimentally. As the film moves downward, small perturbations to the free surface grow due to surface tension effects and can form liquid plugs. A first-principles strongly nonlinear model based on long-wave asymptotics is developed to provide simplified governing equations for the motion of the film flow. Linear stability analysis on the basic solution of the model predicts the speed and wavelength of the most unstable mode, and whether the film is convectively or absolutely unstable. These results are found to be in remarkable agreement with the experiments. The model is also solved numerically to follow the time evolution of instabilities. For relatively thin films, these instabilities saturate as a series of small-amplitude travelling waves, while thicker films lead to solutions whose amplitude becomes large enough for the liquid surface to approach the centre of the tube in finite time, suggesting liquid plug formation. Next, the model’s periodic travelling wave solutions are determined by a continuation algorithm using the results from the time evolution code as initial seed. It is found that bifurcation branches for these solutions exist, and the critical turning points where branches merge determine film mean thicknesses beyond which no travelling wave solutions exist. These critical thickness values are in good agreement with those for liquid plug formations determined experimentally and numerically by the time-evolution code.

JFM classification

Instability: Instability Interfacial Flows (free surface): Interfacial Flows (free surface) Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 745 , 25 April 2014 , pp. 682 - 715
Copyright
© 2014 Cambridge University Press 

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