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Viscous film-flow coating the interior of a vertical tube. Part 2. Air-driven flow

Published online by Cambridge University Press:  27 July 2017

Roberto Camassa ,
H. Reed Ogrosky Open the ORCID record for H. Reed Ogrosky [Opens in a new window]  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 and Applied Mathematics, Virginia Commonwealth University, Richmond, VA 23284-2014, USA
Jeffrey Olander
Affiliation:
Carolina Center for Interdisciplinary Applied Mathematics, Department of Mathematics, University of North Carolina, Chapel Hill, NC 27599-3250, USA 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 flow of a viscous liquid film coating the interior of a vertical tube is studied for the case when the film is driven upwards against gravity by a constant volume flux of air through the centre of the tube. A nonlinear model exploiting the slowly varying liquid–air interface is first developed to estimate the interfacial stresses created by the airflow. A comparison of the model with both experiments and previously developed theoretical results is conducted for two geometrical settings: channel and pipe flow. In both geometries, the model compares reasonably well with previous experiments. A long-wave asymptotic theory is then developed for the air–liquid interface taking into account the estimated free-surface stresses created by the airflow. The stability of small interfacial disturbances is studied analytically, and it is shown that the modelled free-surface stresses contribute to both an increased upwards disturbance velocity and a more rapid instability growth than those of a previously developed ‘locally Poiseuille’ model. Numerical solutions to the long-wave model exhibit saturated waves, whose profiles and velocities show substantial improvement with respect to the previous model predictions. The theoretical results are compared with new experiments for a modified version of the set-up described in Part 1.

JFM classification

Interfacial Flows (free surface): Interfacial Flows (free surface) Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 825 , 25 August 2017 , pp. 1056 - 1090
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Copyright
© 2017 Cambridge University Press 

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Camassa et al. supplementary movie 1

Clip from experiment showing, in real time, the behavior of disturbances to a viscous liquid layer inside a 0.5 cm radius tube. High volume flux air flow (330 cm^3/s) is blown from bottom to top of the tube, forming a core flow which opposes the downward pull of gravity. No liquid is being added to the test section. Annular waves initially form as low-amplitude disturbances that grow into larger-amplitude crests which travel upward. Even at their largest amplitude, the wave crests do not extend all the way to the tube center, i.e., bridges or plugs do not form. This can be seen when waves are near the bottom of the tube where the wide-angle view offers a perspective slightly downward on the crest rings. Waves that grow to full amplitude can frequently slow down and then stop traveling. Often this is accompanied by a decrease in the wave amplitude. Sometimes these waves grow again and are carried up by the air flow shear. Liquid viscosity is 129 P; mean film thickness is 0.10 cm.

Download Camassa et al. supplementary movie 1(Video)
Video 66.5 MB

Camassa et al. supplementary movie 2

Movie of a model solution with airflow rate of 330 cm^3/s, tube radius 0.5 cm, mean film thickness 0.17 cm, liquid viscosity 600 P. The simulation has been sped up and shows a total elapsed time of approximately 120 s. Details of the numerical methods used to solve the model equation are given in the main text in Section 6.3.

Download Camassa et al. supplementary movie 2(Video)
Video 2.3 MB