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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: hrogrosky@vcu.edu
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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

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
Bassom, A. P., Blyth, M. G. & Papageorgiou, D. T. 2010 Nonlinear development of two-layer Couette-Poiseuille flow in the presence of surfactant. Phys. Fluids 22, 102102.Google Scholar
Bassom, A. P., Blyth, M. G. & Papageorgiou, D. T. 2012 Using surfactants to stabilize two-phase pipe flows of core-annular type. J. Fluid Mech. 704, 333–359.CrossRefGoogle Scholar
Blyth, M. G. & Bassom, A. P. 2013 Stability of surfactant-laden core-annular flow and rod-annular flow to non-axisymmetric modes. J. Fluid Mech. 716, R13-1–12.CrossRefGoogle Scholar
Blyth, M. G., Luo, H. & Pozrikidis, C. 2006 Stability of axisymmetric core–annular flow in the presence of an insoluble surfactant. J. Fluid Mech. 548, 207–235.CrossRefGoogle Scholar
Bonn, D., Eggers, J., Indekeu, J., Meunier, J. & Rolley, E. 2009 Wetting and spreading. Rev. Mod. Phys. 81, 739–805.CrossRefGoogle 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, 066305-1–11.CrossRefGoogle ScholarPubMed
Camassa, R., Marzuola, J., Ogrosky, H. R. & Swygert, S. 2021 On the stability of traveling wave solutions to thin-film and long-wave models for film flows inside a tube. Physica D 415, 132750.CrossRefGoogle 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.CrossRefGoogle Scholar
Camassa, R., Ogrosky, H. R. & Olander, J. 2014 Viscous film flow coating the interior of a vertical tube: part I. Gravity-driven flow. J. Fluid Mech. 745, 682–715.Google Scholar
Camassa, R., Ogrosky, H. R. & Olander, J. 2017 Viscous film flow coating the interior of a vertical tube. Part II. Air-driven flow. J. Fluid Mech. 825, 1056–1090.CrossRefGoogle Scholar
Cassidy, K. J., Halpern, D., Ressler, B. G. & Grotberg, J. B. 1999 Surfactant effects in model airway closure experiments. J. Appl. Physiol. 87, 415–427.CrossRefGoogle ScholarPubMed
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.CrossRefGoogle Scholar
Dietze, G. F., Lavalle, G. & Ruyer-Quil, C. 2020 Falling liquid films in narrow tubes: occlusion scenarios. J. Fluid Mech. 894, A17.CrossRefGoogle Scholar
Dietze, G. F. & Ruyer-Quil, C. 2015 Films in narrow tubes. J. Fluid Mech. 762, 68–109.CrossRefGoogle Scholar
Ding, Z., Liu, Z., Liu, R. & Yang, C. 2019 Thermocapillary effects on the dynamics of liquid films coating the interior surface of a tube. Intl J. Heat Mass Transfer 138, 524–533.Google Scholar
Doedel, E. J., Champneys, A. R., Dercole, F., Fairgrieve, T., Kuznetsov, Y., Oldeman, B., Paffenroth, R., Sandstede, B. J., Wang, X. & Zhang, C. 2008 AUTO-07P: continuation and bifurcation software for ordinary differential equations.Google Scholar
Duprat, C., Ruyer-Quil, C., Kalliadasis, S. & Giorgiutti-Dauphine, F. 2007 Absolute and convective instabilities of a viscous film flowing down a vertical fiber. Phys. Rev. Lett. 98, 244502.Google Scholar
Frenkel, A. L. 1992 Nonlinear theory of strongly undulating thin films flowing down vertical cylinders. Europhys. Lett. 18, 583–588.Google Scholar
Frenkel, A. L., Babchin, A. J., Levich, B. G., Shlang, T. & Sivashinsky, G. I. 1987 Annular flows can keep unstable films from breakup: nonlinear saturation of capillary instability. J. Colloid Interface Sci. 115, 225–233.CrossRefGoogle Scholar
Frenkel, A. L. & Halpern, D. 2002 Stokes-flow instability due to interfacial surfactant. Phys. Fluids 14, L45–L48.CrossRefGoogle Scholar
Frenkel, A. L. & Halpern, D. 2017 Surfactant and gravity dependent instability of two-layer Couette flows and its nonlinear saturation. J. Fluid Mech. 826, 158–204.Google Scholar
Frenkel, A. L., Halpern, D. & Schweiger, A. J. 2019 a Surfactant- and gravity-dependent instability of two-layer channel flows: linear theory covering all wavelengths. Part 1. ‘Long-wave’ regimes. J. Fluid Mech. 863, 150–184.Google Scholar
Frenkel, A. L., Halpern, D. & Schweiger, A. J. 2019 b Surfactant- and gravity-dependent instability of two-layer channel flows: linear theory covering all wavelengths. Part 2. Mid-wave regimes. J. Fluid Mech. 863, 185–214.CrossRefGoogle Scholar
Gauglitz, P. A. & Radke, C. J. 1988 An extended evolution equation for liquid film breakup in cylindrical capillaries. Chem. Engng Sci. 43, 1457–1465.CrossRefGoogle Scholar
Gauglitz, P. A. & Radke, C. J. 1990 The dynamics of liquid film breakup in constricted cylindrical capillaries. J. Colloid Interface Sci. 134, 14–40.CrossRefGoogle Scholar
Georgiou, E. C., Maldarelli, C., Papageorgiou, D. T. & Rumschitzki, D. S. 1992 An asymptotic theory for the linear stability of a core-annular flow in the thin annular limit. J. Fluid Mech. 243, 653–677.CrossRefGoogle Scholar
Goren, S. L. 1962 The instability of an annular thread of fluid. J. Fluid Mech. 27, 309–319.CrossRefGoogle Scholar
Halpern, D. & Frenkel, A. L. 2003 Destabilization of a creeping flow by interfacial surfactant: linear theory extended to all wavenumbers. J. Fluid Mech. 485, 191–220.CrossRefGoogle Scholar
Halpern, D. & Frenkel, A. L. 2008 Nonlinear evolution, travelling waves, and secondary instability of sheared-film flows with insoluble surfactants. J. Fluid Mech. 594, 125–156.CrossRefGoogle Scholar
Halpern, D. & Grotberg, J. B. 1992 Fluid-elastic instabilities of liquid-lined flexible tubes. J. Fluid Mech. 244, 615–632.CrossRefGoogle Scholar
Halpern, D. & Grotberg, J. B. 1993 Surfactant effects on fluid-elastic instabilities of liquid-lined flexible tubes: a model of airway closure. Trans. ASME: J. Biomech Engng 115, 271–277.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.CrossRefGoogle Scholar
Hu, H. H. & Patankar, N. 1995 Non-axisymmetric instability of core-annular flow. J. Fluid Mech. 290, 213–234.Google Scholar
Indireshkumar, K. & Frenkel, A. L. 1996 Math Modeling and Simulation in Hydrodynamic Stability (ed. D. N. Riahi), pp. 35–81. World Scientific.Google Scholar
Jensen, O. E. 2000 Draining collars and lenses in liquid-lined vertical tubes. J. Colloid Interface Sci. 221, 38–49.CrossRefGoogle ScholarPubMed
Joseph, D. D., Bai, R., Chen, K. & Renardy, Y. Y. 1997 Core-annular flows. Annu. Rev. Fluid Mech. 29, 65–90.CrossRefGoogle Scholar
Joseph, D. D. & Renardy, Y. 1993 Fundamentals of Two-Fluid Dynamics, Part 2: Lubricated Transport, Drops, and Miscible Liquids. Springer Verlag.Google Scholar
Joseph, D. D., Renardy, M. & Renardy, Y. 1984 Instability of the flow of two immiscible liquids with different viscosities in a pipe. J. Fluid Mech. 141, 309–317.Google Scholar
Kalliadasis, S. & Chang, H.-C. 1994 Drop formation during coating of vertical fibers. J. Fluid Mech. 261, 135–168.Google Scholar
Kas-Danouche, S. A., Papageorgiou, D. T. & Siegel, M. 2009 Nonlinear dynamics of core-annular film flows in the presence of surfactant. J. Fluid Mech. 626, 415–448.CrossRefGoogle Scholar
Kerchman, V. I. 1995 Strongly nonlinear interfacial dynamics in core-annular flows. J. Fluid Mech. 290, 131–166.CrossRefGoogle 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.CrossRefGoogle Scholar
Kliakhandler, I. L., Davis, S. H. & Bankoff, S. G. 2001 Viscous beads on vertical fibre. J. Fluid Mech. 429, 381–390.CrossRefGoogle Scholar
Levy, R., Shearer, M. & Witelski, T. P. 2007 Gravity-driven thin liquid films with insoluble surfactant: smooth traveling waves. Eur. J. Appl. Math. 18, 679–708.CrossRefGoogle Scholar
Lin, S. P. & Liu, W. C. 1975 Instability of film coating of wires and tubes. AIChE J. 24, 775–782.CrossRefGoogle Scholar
Lister, J. R., Rallison, J. M., King, A. A., Cummings, L. J. & Jensen, O. E., 2006 Capillary drainage of an annular film: the dynamics of collars and lobes. J. Fluid Mech. 552, 311–343.CrossRefGoogle 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
Otis, D. R. Jr., Johnson, M., Pedley, T. J. & Kamm, R. D. 1993 Role of pulmonary surfactant in airway closure: a computational study. J. Appl. Physiol. 75, 1323–1333.CrossRefGoogle ScholarPubMed
Papageorgiou, D. T., Maldarelli, C. & Rumschitzki, D. S. 1990 Nonlinear interfacial stability of core-annular film flows. Phys. Fluids A 2 (3), 340–352.Google Scholar
Preziosi, L., Chen, K. & Joseph, D. D. 1989 Lubricated pipelining: stability of core-annular flow. J. Fluid Mech. 201, 323–356.CrossRefGoogle Scholar
Smolka, L., North, J. & Guerra, B. 2008 Dynamics of free surface perturbations along an annular viscous film. Phys. Rev. E 77, 036301.CrossRefGoogle ScholarPubMed
Tseluiko, D. & Kalliadasis, S. 2011 Nonlinear waves in counter-current gas-liquid film flow. J. Fluid Mech. 673, 19–59.CrossRefGoogle Scholar
Wei, H.-H. 2005 a On the flow-induced Marangoni instability due to the presence of surfactant. J. Fluid Mech. 544, 173–200.CrossRefGoogle Scholar
Wei, H.-H. 2005 b Marangoni destabilization on a core-annular film flow due to the presence of surfactant. Phys. Fluids 17, 027101.CrossRefGoogle Scholar
Wei, H.-H. 2007 Role of base flows on surfactant-driven interfacial instabilities. Phys. Rev. E 75, 036306.CrossRefGoogle ScholarPubMed
Wei, H.-H. & Rumschitzki, D. S. 2005 The effects of insoluble surfactants on the linear stability of a core–annular flow. J. Fluid Mech. 541, 115–142.CrossRefGoogle Scholar
Yih, C.-S. 1967 Instability due to viscosity stratification. J. Fluid Mech. 27, 337–352.CrossRefGoogle Scholar
Zhou, Z.-Q., Peng, J., Zhang, Y.-J. & Zhuge, W.-L. 2014 Instabilities of viscoelastic liquid film coating tube in the presence of surfactant. J. Non-Newtonian Fluid Mech. 204, 94–103.CrossRefGoogle Scholar