Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-25T04:36:18.971Z Has data issue: false hasContentIssue false
  • English
  • Français

Films in narrow tubes

Published online by Cambridge University Press:  27 November 2014

Georg F. Dietze Open the ORCID record for Georg F. Dietze [Opens in a new window]  and
Christian Ruyer-Quil
Georg F. Dietze*
Affiliation:
Univ. Paris-Sud, CNRS, Lab. FAST, Bât. 502, Campus Univ., Orsay, F-91405, France
Christian Ruyer-Quil
Affiliation:
Univ. Savoie, CNRS, LOCIE, F-73000 Chambéry, France Institut Universitaire de France, France
*
†Email address for correspondence: dietze@fast.u-psud.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the axisymmetric arrangement of an annular liquid film, coating the inner surface of a narrow cylindrical tube, in interaction with an active core fluid. We introduce a low-dimensional model based on the two-phase weighted residual integral boundary layer (WRIBL) formalism (Dietze & Ruyer-Quil, J. Fluid Mech., vol. 722, 2013, pp. 348–393) which is able to capture the long-wave instabilities characterizing such flows. Our model improves upon existing works by fully representing interfacial coupling and accounting for inertia as well as streamwise viscous diffusion in both phases. We apply this model to gravity-free liquid-film/core-fluid arrangements in narrow capillaries with specific attention to the dynamics leading to flooding, i.e. when the liquid film drains into large-amplitude collars that occlude the tube cross-section. We do this against the background of linear stability calculations and nonlinear two-phase direct numerical simulations (DNS). Due to the improvements of our model, we have found a number of novel/salient physical features of these flows. First, we show that it is essential to account for inertia and full interphase coupling to capture the temporal evolution of flooding for fluid combinations that are not dominated by viscosity, e.g. water/air and water/silicone oil. Second, we elucidate a viscous-blocking mechanism which drastically delays flooding in thin films that are too thick to form unduloids. This mechanism involves buckling of the residual film between two liquid collars, generating two very pronounced film troughs where viscous dissipation is drastically increased and growth effectively arrested. Only at very long times does breaking of symmetry in this region (due to small perturbations) initiate a sliding motion of the liquid film similar to observations by Lister et al. (J. Fluid Mech., vol. 552, 2006, pp. 311–343) in thin non-flooding films. This kickstarts the growth of liquid collars anew and ultimately leads to flooding. We show that streamwise viscous diffusion is essential to this mechanism. Low-frequency core-flow oscillations, such as occur in human pulmonary capillaries, are found to set off this sliding-induced flooding mechanism much earlier.

JFM classification

Interfacial Flows (free surface): Interfacial Flows (free surface) Nonlinear Dynamical Systems: Low-dimensional models Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 762 , 10 January 2015 , pp. 68 - 109
Check for updates
Copyright
© 2014 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

Alekseenko, S. V., Nakoryakov, V. E. & Pokusaev, B. G. 1994 Wave Flow of Liquid Films. Begell House.Google Scholar
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
Bai, R., Kelkar, K. & Joseph, D. D. 1996 Direct simulation of interfacial waves in a high-viscosity-ratio and axisymmetric core–annular flow. J. Fluid Mech. 327, 1–34.Google Scholar
Bian, S., Tai, C.-F., Halpern, D., Zheng, Y. & Grotberg, J. B. 2010 Experimental study of flow fields in an airway closure model. J. Fluid Mech. 647, 391–402.Google Scholar
Brooke Benjamin, T. 1957 Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2, 554–574.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 (6), 066305.Google Scholar
Chen, K. P. & Joseph, D. D. 1991 Long wave and lubrication theories for core–annular flow. Phys. Fluids 3 (11), 2627–2679.CrossRefGoogle Scholar
Dao, E. K. & Balakotaiah, V. 2000 Experimental study of wave occlusion on falling films in a vertical pipe. AIChE J. 46 (7), 1300.Google Scholar
Delaunay, C. 1841 Sur la surface de révolution dont la courbure moyenne est constante. J. Math. Pures Appl. 6, 309–320.Google Scholar
Dietze, G. F. & Ruyer-Quil, C. 2013 Wavy liquid films in interaction with a confined laminar gas flow. J. Fluid Mech. 722, 348–393.Google Scholar
Drosos, E. I. P., Paras, S. V. & Karabelas, A. J. 2006 Counter-current gas–liquid flow in a vertical narrow channel – liquid film characteristics and flooding phenomena. Intl J. Multiphase Flow 32, 51–81.Google Scholar
Eggers, J. & Dupont, T. F. 1994 Drop formation in a one-dimensional approximation of the Navier–Stokes equation. J. Fluid Mech. 262, 205–221.Google Scholar
Eggers, J. & Villermaux, E. 2008 Physics of liquid jets. Rep. Prog. Phys. 71, 036601.Google Scholar
Everett, D. H. & Haynes, J. M. 1972 Model studies of capillary condensation. J. Colloid Interface Sci. 38 (1), 125–137.CrossRefGoogle 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 (1), 225–233.Google Scholar
Gauglitz, P. A. 1988 An extended evolution equation for liquid film breakup in cylindrical capillaries. Chem. Engng Sci. 43 (7), 1457–1465.Google Scholar
Goren, S. L. 1962 The instability of an annular thread of fluid. J. Fluid Mech. 12 (2), 309–319.Google Scholar
Gosh, S., Mandal, T. K., Das, G. & Das, P. K. 2009 Review of oil water core annular flow. Renew. Sustain. Energy Rev. 13, 1957–1965.Google Scholar
Govindarajan, R. & Sahu, K. C. 2014 Instabilities in viscosity-stratified flow. Annu. Rev. Fluid Mech. 46, 331–353.Google Scholar
Grotberg, J. 1994 Pulmonary flow and transport phenomena. J. Fluid Mech. 26, 529–571.Google Scholar
Grotberg, J. 2011 Respiratory fluid mechanics. Phys. Fluids 23, 021301.Google Scholar
Halpern, D., Fujioka, H. & Grotberg, J. B. 2010 The effect of viscoelasticity on the stability of a pulmonary airway liquid layer. Phys. Fluids 22, 011901.Google Scholar
Halpern, D. & Grotberg, J. B. 2003 Nonlinear saturation of the Rayleigh-instability due to oscillatory flow in a liquid-lined tube. J. Fluid Mech. 492, 251–270.Google Scholar
Hamacher, H., Fitton, B. & Kingdon, J. 1987 Fluid Sciences and Materials Science in Space. Springer.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
Heil, M. & Hazel, A. L. 2011 Fluid–structure interaction in internal physiological flows. Annu. Rev. Fluid Mech. 43, 141–162.CrossRefGoogle Scholar
Hickox, C. E. 1971 Instability due to viscosity and density stratification in axisymmetric pipe flow. Phys. Fluids 14 (2), 251–262.Google Scholar
Hu, H. H. & Joseph, D. D. 1989 Lubricated pipelining: stability of core–annular flow. Part 2. J. Fluid Mech. 205, 359–396.CrossRefGoogle Scholar
Jebson, R. S. & Chen, H. 1997 Performances of falling film evaporators on whole milk and a comparison with performance on skim milk. J. Dairy Res. 64, 57–67.Google Scholar
Jensen, O. E. 2000 Draining collars and lenses in liquid-lined vertical tubes. J. Colloid Interface Sci. 221, 38–49.Google Scholar
Johnson, M., Kamm, R. D., Ho, L. W., Shapiro, A. & Pedley, T. J. 1991 The nonlinear growth of surface-tension-driven instabilities of a thin annular film. J. Fluid Mech. 233, 141–156.Google Scholar
Joseph, D. D., Bai, R., Mata, C., Sury, K. & Grant, C. 1999 Self-lubricated transport of bitumen froth. J. Fluid Mech. 386, 127–148.Google Scholar
Joseph, D. D., Chen, K. P. & Renardy, Y. Y. 1997 Core–annular flows. Annu. Rev. Fluid Mech. 29, 65–90.Google Scholar
Kalliadasis, S., Ruyer-Quil, C., Scheid, B. & Velarde, M. G. 2012 Falling Liquid Films, Applied Mathematical Sciences, vol. 176. Springer.Google Scholar
Kapitza, P. L. 1948 Wave flow of thin layer of viscous fluid. Zh. Eksp. Teor. Fiz. 18 (1), 3–28 (in Russian).Google Scholar
Kerchman, V. 1995 Strongly nonlinear interfacial dynamics in core–annular flows. J. Fluid Mech. 290, 131–166.Google Scholar
Kouris, C. & Tsamopoulos, J. 2001 Dynamics of axisymmetric core–annular flow in a straight tube. I. The more viscous fluid in the core, bamboo waves. Phys. Fluids 13 (4), 841–858.Google Scholar
Kouris, C. & Tsamopoulos, J. 2002 Dynamics of the axisymmetric core–annular flow. II. The less viscous fluid in the core, saw tooth waves. Phys. Fluids 14 (3), 1011–1029.Google 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.Google Scholar
Mayo, L. C., McCue, S. W. & Moroney, T. J. 2013 Gravity-driven fingering simulations for a thin liquid film flowing down the outside of a vertical cylinder. Phys. Rev. E 87, 053018.Google Scholar
McKinley, G. H. & Renardy, M. 2011 Wolfgang von ohnesorge. Phys. Fluids 23, 127101.CrossRefGoogle Scholar
Mehidi, N. & Amatousse, N. 2009 Modélisation d’un écoulement coaxial en conduite circulaire de deux fluides visqueux. C. R. Méc. 337, 112–118.Google Scholar
Newhouse, L. A. & Pozrikidis, C. 1992 The capillary instability of annular layers and liquid threads. J. Fluid Mech. 242, 193–209.Google Scholar
Novbari, E. & Oron, A. 2009 Energy integral method for the nonlinear dynamics of an axisymmetric thin liquid film falling on a vertical cylinder. Phys. Fluids 21, 062107.Google Scholar
Novbari, E. & Oron, A. 2011 Analysis of time-dependent nonlinear dynamics of the axisymmetric liquid film on a vertical circular cylinder: energy integral model. Phys. Fluids 23, 012105.Google Scholar
von Ohnesorge, W. 1936 Die Bildung von Tropfen an Düsen und die Auflösung flüssiger Strahlen. Z. Angew. Math. Mech. 16, 355–358.Google Scholar
Olbricht, W. L. 1996 Pore-scale prototypes of multiphase flow in porous media. Annu. Rev. Fluid Mech. 28, 187–213.Google Scholar
d’Olce, M., Martin, J., Rakotomalala, N., Salin, D. & Talon, L. 2008 Pearl and mushroom instability patterns in two miscible fluids core annular flows. Phys. Fluids 20, 024104.Google Scholar
d’Olce, M., Martin, J., Rakotomalala, N., Salin, D. & Talon, L. 2009 Convective/absolute instability in miscible core–annular flow. Part 1: experiments. J. Fluid Mech. 618, 305–322.Google Scholar
Papageorgiou, D. T., Maldarelli, C. & Rumschitzki, D. S. 1990 Nonlinear interfacial stability of core–annular film flows. Phys. Fluids 2, 340–352.Google Scholar
Piroird, K., Clanet, C. & Quéré, D. 2011 Detergency in a tube. Soft Matt. 7, 7498–7503.Google Scholar
Popinet, S. 2009 An accurate adaptive solver for surface-tension-driven interfacial flows. J. Comput. Phys. 228, 5838–5866.Google Scholar
Pozrikidis, C. 1999 Capillary instability and breakup of a viscous thread. J. Engng Maths 36, 255–275.Google Scholar
Preziosi, L., Chen, K. P. & Joseph, D. D. 1989 Lubricated pipelining – stability of core annular flow. J. Fluid Mech. 201, 323–356.Google Scholar
Pumir, A., Manneville, P. & Pomeau, Y. 1983 On solitary waves running down an inclined plane. J. Fluid Mech. 135, 27–50.Google Scholar
Quéré, D. 1999 Fluid coating on a fibre. Annu. Rev. Fluid Mech. 31, 347–384.CrossRefGoogle Scholar
Rayleigh, L. 1892 On the instability of cylindrical fluid surfaces. Phil. Mag. 34 (207), 177–180.Google Scholar
Ribe, N. 2002 A general theory for the dynamics of thin viscous sheets. J. Fluid Mech. 457, 255–283.Google Scholar
Ribe, N. 2012 Liquid rope coiling. Annu. Rev. Fluid Mech. 44, 249–266.Google Scholar
Ruyer-Quil, C. & Kalliadasis, S. 2012 Wavy regimes of film flow down a fibre. Phys. Rev. E 85, 046302.Google Scholar
Ruyer-Quil, C., Kofman, N. & Chasseur, D. 2014 Dynamics of falling liquid films. Eur. Phys. J. E 37 (30), 1–17.Google Scholar
Ruyer-Quil, C. & Manneville, P. 2000 Improved modeling of flows down inclined planes. Eur. Phys. J. B 15 (2), 357–369.CrossRefGoogle Scholar
Ruyer-Quil, C. & Manneville, P. 2002 Further accuracy and convergence results on the modeling of flows down inclined planes by weighted-residual approximations. Phys. Fluids 14 (1), 170–183.Google Scholar
Ruyer-Quil, C., Trevelyan, P., Giorgiutti-Dauphine, F., Duprat, C. & Kalliadasis, S. 2008 Modelling film flows down a fibre. J. Fluid Mech. 603, 431–462.Google Scholar
Selvam, B., Talon, L., Lesshafft, L. & Meiburg, E. 2009 Convective/absolute instability in miscible core–annular flow. Part 2. Numerical simulations and nonlinear global modes. J. Fluid Mech. 618, 328–348.Google Scholar
Shkadov, V. Ya. 1967 Wave flow regimes of a thin layer of viscous fluid subject to gravity. Fluid Dyn. 2 (1), 29–34.Google Scholar
Sierou, A. & Lister, J. R. 2003 Self-similar solutions for viscous capillary pinch-off. J. Fluid Mech. 497, 381–403.Google Scholar
Slim, A. C., Balmforth, N. J., Craster, R. V. & Miller, J. C. 2009 Surface wrinkling of a channelized flow. Proc. R. Soc. Lond. A 465, 123–142.Google Scholar
Slim, A. C., Teichman, J. & Mahadevan, L. 2012 Buckling of a thin-layer Couette flow. J. Fluid Mech. 694, 5–28.Google Scholar
Suleiman, M. & Munson, B. R. 1981 Viscous buckling of thin fluid layers. Phys. Fluids 24, 1–5.Google Scholar
Tai, C.-F., Bian, S., Halpern, D., Zheng, Y., Filoche, M. & Grotberg, J. B. 2011 Numerical study of flow fields in an airway closure model. J. Fluid Mech. 677, 483–502.Google Scholar
Thiele, U., Todorova, D. V. & Lopez, H. 2013 Gradient dynamics description for films of mixtures and suspensions: dewetting triggered by coupled film height and concentration fluctuations. Phys. Rev. Lett. 111, 117801.Google Scholar
Timmermans, M.-L. E. & Lister, J. R. 2002 The effect of surfactant on the stability of a liquid thread. J. Fluid Mech. 459, 289–306.Google Scholar
Trifonov, Y. Y. 1992 Steady-state traveling waves on the surface of a viscous liquid film falling down on vertical wires and tubes. AIChE J. 38 (6), 821–834.Google Scholar
Trifonov, Y. Y. 2010a Counter-current gas–liquid wavy film flow between the vertical plates analyzed using the Navier–Stokes equations. AIChE J. 56 (8), 1975–1987.Google Scholar
Trifonov, Y. Y. 2010b 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
Wang, Q. 2013 Capillary instability of a viscous liquid thread in a cylindrical tube. Phys. Fluids 25, 014104.Google Scholar
Wray, A. W. 2013 Electrostatically controlled large-amplitude, non-axisymetric waves in thin film flows down a cylinder. J. Fluid Mech. 736, R2.Google Scholar
Yarin, A. L. & Tchadarov, B. M. 1996 Onset of folding in plane liquid films. J. Fluid Mech. 307, 85–99.Google Scholar
Yiantsios, S. G. & Higgins, B. G. 1989 Rayleigh–Taylor instability in thin viscous films. Phys. Fluids A 1, 1484–1501.Google Scholar
Yih, C.-S. 1967 Instability due to viscosity stratification. J. Fluid Mech. 27 (2), 337–352.Google Scholar