Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-24T20:25:03.469Z Has data issue: false hasContentIssue false
  • English
  • Français

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: hrogrosky@vcu.edu
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© 2017 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aul, R. W. & Olbricht, W. L. 1990 Stability of a thin annular film in pressure-driven, low-Reynolds-number flow through a capillary. J. Fluid Mech. 215, 585–599.Google Scholar
Bailey, S. C. C., Vallikivi, M., Hultmark, M. & Smits, A. J. 2014 J. Fluid Mech. 749, 79–98.Google Scholar
Benjamin, T. B. 1959 Shearing flow over a wavy boundary. J. Fluid Mech. 6, 161–205.Google Scholar
Buckles, J., Hanratty, T. J. & Adrian, R. J. 1984 Turbulent flow over large-amplitude wavy surfaces. J. Fluid Mech. 140, 27–44.Google Scholar
Camassa, R., Forest, M. G., Lee, L., Ogrosky, H. R. & Olander, J. 2012 Ring waves as a mass transport mechanism in air-driven core-annular flows. Phys. Rev. E 86, 066305.Google Scholar
Camassa, R. & Lee, L. 2006 Advances in Engineering Mechanics – Reflections and Outlooks (ed. Chwang, A., Teng, M. & Valentine, D.), pp. 222–238. World Scientific.Google Scholar
Camassa, R., Marzuola, J., Ogrosky, H. R. & Vaughn, N. 2016 Traveling waves for a model of gravity-driven film flows in cylindrical domains. Physica D 333, 254–265.Google Scholar
Camassa, R. & Ogrosky, H. R. 2015 On viscous film flows coating the interior of a tube: thin-film and long-wave models. J. Fluid Mech. 772, 569–599.Google Scholar
Camassa, R., Ogrosky, H. R. & Olander, J. 2014 Viscous film flow coating the interior of a vertical tube: Part 1. Gravity-driven flow. J. Fluid Mech. 745, 682–715.Google Scholar
Chauve, M. P. & Schiestel, R. 1985 Influence of weak wall undulations on the structure of turbulent pipe flow: an experimental and numerical study. J. Fluid Mech. 160, 47–75.Google Scholar
Cherukat, P., Na, Y., Hanratty, T. J. & McLaughlin, J. B. 1998 Direct numerical simulation of a fully developed turbulent flow over a wavy wall. Theor. Comput. Fluid Dyn. 11, 109–134.Google Scholar
Clarke, S. W., Jones, J. G. & Oliver, D. R. 1970 Resistance to two-phase gas-liquid flow in airways. J. Appl. Phys. 29, 464–471.Google Scholar
Craster, R. V. & Matar, O. K. 2006 On viscous beads flowing down a vertical fibre. J. Fluid Mech. 553, 85–105.Google Scholar
Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 1131–1198.Google Scholar
Dietze, G. F. & Ruyer-Quil, C. 2015 Films in narrow tubes. J. Fluid Mech. 762, 68–109.Google Scholar
Frenkel, A. L. 1992 Nonlinear theory of strongly undulating thin films flowing down vertical cylinders. Europhys. Lett. 18, 583–588.Google Scholar
Goren, S. L. 1962 The instability of an annular thread of fluid. J. Fluid Mech. 27, 309–319.Google Scholar
Halpern, D. & Grotberg, J. B. 1992 Fluid-elastic instabilities of liquid-lined flexible tubes. J. Fluid Mech. 244, 615–632.Google Scholar
Hamed, A. M., Kamdar, A., Castillo, L. & Chamorro, L. P. 2015 Turbulent boundary layer over 2D and 3D large-scale wavy walls. Phys. Fluids 27, 106601.Google Scholar
Hammond, P. S. 1983 Nonlinear adjustment of a thin annular film of viscous fluid surrounding a thread of another within a circular cylindrical pipe. J. Fluid Mech. 137, 363–384.Google Scholar
Hickox, C. 1971 Instability due to viscosity and density stratification in axisymmetric pipe flow. Phys. Fluids 14, 251–262.Google Scholar
Hsu, S. T.1968 Turbulent flow in wavy pipes. PhD thesis, University of Iowa Iowa City, Iowa.Google Scholar
Hsu, S.-T. & Kennedy, J. F. 1971 Turbulent flow in wavy pipes. J. Fluid Mech. 47, 481–502.Google Scholar
Hudson, J. D., Dykhno, L. & Hanratty, T. J. 1996 Turbulence production in flow over a wavy wall. Exp. Fluids 20, 257–265.Google Scholar
Joseph, D. D. & Renardy, Y. 1993 Fundamentals of Two-Fluid Dynamics, Part 2: Lubricated Transport, Drops, and Miscible Liquids. Springer.Google Scholar
Kalliadasis, S. & Chang, H.-C. 1994 Drop formation during coating of vertical fibers. J. Fluid Mech. 261, 135–168.Google Scholar
Kerchman, V. I. 1995 Strongly nonlinear interfacial dynamics in core-annular flows. J. Fluid Mech. 290, 131–166.Google Scholar
Kerchman, V. I. & Frenkel, A. L. 1994 Interactions of coherent structures in a film flow: simulations of a highly nonlinear evolution equation. Theor. Comput. Fluid Dyn. 6, 235–254.Google Scholar
Kim, C. S. et al. 1986 Criteria for mucus transport in the airways by two-phase gas-liquid flow mechanism. J. Appl. Phys. 60, 901–907.Google Scholar
Kim, C. S. et al. 1986 Mucus transport in the airways by two-phase gas-liquid flow mechanism: continuous flow model. J. Appl. Phys. 60, 908–917.Google Scholar
Kim, C. S., Iglesias, A. J. & Sackner, M. A. 1987 Mucus clearance by two-phase gas-liquid flow mechanism: asymmetric periodic flow model. J. Appl. Phys. 62, 959–971.Google Scholar
Kliakhandler, I. L., Davis, S. H. & Bankoff, S. G. 2001 Viscous beads on vertical fibre. J. Fluid Mech. 429, 381–390.Google Scholar
Lin, S. P. & Liu, W. C. 1975 Instability of film coating of wires and tubes. AIChE J. 24, 775–782.Google Scholar
Lister, J. R. et al. 2006 Capillary drainage of an annular film: the dynamics of collars and lobes. J. Fluid Mech. 552, 311–343.Google Scholar
Miles, J. W. 1957 On generation of surface waves by shear flows. J. Fluid Mech. 3, 185–204.Google Scholar
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 932–980.Google Scholar
Rosenberg, B. J., Van Buren, T., Fu, M. K. & Smits, A. J. 2016 Turbulent drag reduction over air- and liquid- impregnated surfaces. Phys. Fluids 28, 015103.Google Scholar
Schlichting, H. & Gersten, K. 2017 Boundary-Layer Theory. Springer.Google Scholar
Solomon, B. R., Khalil, K. S. & Varanasi, K. K. 2014 Drag reduction using lubricant-impregnated surfaces in viscous laminar flow. Langmuir 30, 10970–10976.Google Scholar
Suresh, V. & Grotberg, J. B. 2005 The effect of gravity on liquid plug propagation in a two-dimensional channel. Phys. Fluids 17, 031507.Google Scholar
Thorsness, C. B., Morrisroe, P. E. & Hanratty, T. J. 1978 A comparison of linear theory with measurements of the variation of shear stress along a solid wave. Chem. Engng Sci. 33, 579–592.Google Scholar
Trifonov, Y. Y. 2010 Flooding in two-phase counter-current flows: numerical investigation of the gas–liquid wavy interface using the Navier–Stokes equations. Intl J. Multiphase Flow 36, 549–557.Google Scholar
Trifonov, Y. Y. 2010 Counter-current gas-liquid wavy film flow between the vertical plates analyzed using the Navier–Stokes equations. AIChE J. 56, 1975–1987.Google Scholar
Trifonov, Y. Y. 2014 Stability of a film flowing down an inclined corrugated plate: the direct Navier–Stokes computations and Floquet theory. Phys. Fluids 26, 114101.Google Scholar
Tseluiko, D. & Kalliadasis, S. 2011 Nonlinear waves in counter-current gas-liquid film flow. J. Fluid Mech. 673, 19–59.Google Scholar
van Driest, E. R. 1956 On turbulent flow near a wall. J. Aero. Sci. 23, 1007–1011.Google Scholar
Vellingiri, R., Tseluiko, D. & Kalliadasis, S. 2015 Absolute and convective instabilities in counter-current gas-liquid film flows. J. Fluid Mech. 763, 166–201.Google Scholar
White, F. M. 1991 Viscous Fluid Flow. McGraw-Hill.Google Scholar
Yih, C.-S. 1967 Instability due to viscosity stratification. J. Fluid Mech. 27, 337–352.Google Scholar
Zilker, D. P., Cook, G. W. & Hanratty, T. J. 1977 Influence of the amplitude of a solid wavy wall on a turbulent flow. Part 1. Non-separated flows. J. Fluid Mech. 82, 29–51.Google Scholar

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