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Linear stability and nonlinear dynamics in a long-wave model of film flows inside a tube in the presence of surfactant

Published online by Cambridge University Press:  08 December 2020

H. Reed Ogrosky Open the ORCID record for H. Reed Ogrosky [Opens in a new window]
H. Reed Ogrosky*
Affiliation:
Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA23284, USA
*
†Email address for correspondence: [email protected]
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Abstract

A long-wave model based on lubrication theory is developed for the flow of a viscous liquid film lining the interior of a tube in the presence of an insoluble surfactant on the interface; no thin-film assumption is made. Linear stability analysis identifies two modes; in the absence of base flow, the ‘interface’ mode is the only unstable mode. The growth rates of this mode serve as an accurate predictor of how surfactant concentration increases plug formation time, and the effects of film thickness on this increase are quantified. In the presence of base flow, both the interface mode and ‘surfactant’ mode may be unstable, resulting in a richer variety of free-surface dynamics. In previous work, turning points in families of travelling wave solutions for a falling viscous film lining the interior of a vertical tube with a clean interface have been shown to be a good indicator of $h_c$, the critical thickness past which plugs may form, and this approach is adapted here for flow with surfactant. It is found that turning points in branches of travelling waves that arise from an unstable surfactant mode give an estimate of $h_c$, provided the interface mode is linearly stable. When both modes are unstable, interpretation of these turning points as they relate to plug formation is more complicated. The study concludes by examining the impact of film thickness on growth rates and travelling wave solutions for core–annular flow with surfactant.

JFM classification

Interfacial Flows (free surface): Liquid bridges Interfacial Flows (free surface): Thin films Low-Reynolds-number flows: Lubrication theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 908 , 10 February 2021 , A23
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

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