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On viscous film flows coating the interior of a tube: thin-film and long-wave models

Published online by Cambridge University Press:  07 May 2015

Roberto Camassa  and
H. Reed Ogrosky
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, WI 53706-1325, USA
*
†Email address for correspondence: ogrosky@math.wisc.edu
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Abstract

A theoretical and numerical investigation of two classes of models for pressure-driven core–annular flow is presented. Both classes, referred to as ‘long-wave’ and ‘thin-film’ models, may be derived from a unified perspective using long-wave asymptotics, but are distinct from one another in the role played by the curved tube geometry with respect to the planar (limiting) case. Analytical and numerical techniques are used to show and quantify the significant differences between the behaviour of solutions to both model types. Temporal linear stability analysis of the constant solution is carried out first to pinpoint with closed-form mathematical expressions the different dynamical regimes associated with absolute or convective instabilities. Numerical simulations for the models are then performed and qualitative differences in the evolution of the free surface are explored. Mathematically, different levels of asymptotic accuracy are found to result in different regularizing properties affecting the long-time behaviour of generic numerical solutions. Travelling wave solutions are also studied, and qualitative differences in the topology of streamline patterns describing the flow of the film in a moving reference frame are discussed. These topological differences allow for further classification of the models. In particular, a transition from a regime in which waves trap a fluid core to one where waves travel faster than any parcel of the underlying fluid is documented for a variant of the primary model. In the corresponding thin-film model, no such transition is found to occur. The source of these differences is examined, and a comparison of the results with those of related models in the literature is given. A brief discussion of the merits of each class of models concludes this study.

JFM classification

Instability: Instability Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 772 , 10 June 2015 , pp. 569 - 599
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Copyright
© 2015 Cambridge University Press 

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