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Three-dimensional Rayleigh–Taylor instability under a unidirectional curved substrate

Published online by Cambridge University Press:  19 December 2017

Gioele Balestra ,
Nicolas Kofman ,
P.-T. Brun ,
Benoit Scheid  and
François Gallaire Open the ORCID record for François Gallaire [Opens in a new window]
Gioele Balestra*
Affiliation:
Laboratory of Fluid Mechanics and Instabilities, EPFL, CH 1015 Lausanne, Switzerland
Nicolas Kofman
Affiliation:
Laboratory of Fluid Mechanics and Instabilities, EPFL, CH 1015 Lausanne, Switzerland
P.-T. Brun
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ 08544, USA
Benoit Scheid
Affiliation:
TIPs Laboratory, Université Libre de Bruxelles, C.P. 165/67, Avenue Franklin Roosevelt 50, 1050 Bruxelles, Belgium
François Gallaire
Affiliation:
Laboratory of Fluid Mechanics and Instabilities, EPFL, CH 1015 Lausanne, Switzerland
*
†Email address for correspondence: [email protected]
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Abstract

We investigate the Rayleigh–Taylor instability of a thin liquid film coated on the inside of a cylinder whose axis is orthogonal to gravity. We are interested in the effects of geometry on the instability, and contrast our results with the classical case of a thin film coated under a flat substrate. In our problem, gravity is the destabilizing force at the origin of the instability, but also yields the progressive drainage and stretching of the coating along the cylinder’s wall. We find that this flow stabilizes the film, which is asymptotically stable to infinitesimal perturbations. However, the short-time algebraic growth that these perturbations can achieve promotes the formation of different patterns, whose nature depends on the Bond number that prescribes the relative magnitude of gravity and capillary forces. Our experiments indicate that a transverse instability arises and persists over time for moderate Bond numbers. The liquid accumulates in equally spaced rivulets whose dominant wavelength corresponds to the most amplified mode of the classical Rayleigh–Taylor instability. The formation of rivulets allows for a faster drainage of the liquid from top to bottom when compared to a uniform drainage. For higher Bond numbers, a two-dimensional stretched lattice of droplets is found to form on the top part of the cylinder. Rivulets and the lattice of droplets are inherently three-dimensional phenomena and therefore require a careful three-dimensional analysis. We found that the transition between the two types of pattern may be rationalized by a linear optimal transient growth analysis and nonlinear numerical simulations.

JFM classification

Instability: Instability Nonlinear Dynamical Systems: Pattern formation Interfacial Flows (free surface): Thin films
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 837 , 25 February 2018 , pp. 19 - 47
Copyright
© 2017 Cambridge University Press 

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References

Abdelall, F. F., Abdel-Khalik, S. I., Sadowski, D. L., Shin, S. & Yoda, M. 2006 On the Rayleigh–Taylor instability for confined liquid films with injection through the bounding surfaces. Intl J. Heat Mass Transfer 49 (7), 1529–1546.Google Scholar
Alekseenko, S. V., Aktershev, S. P., Bobylev, A. V., Kharlamov, S. M. & Markovich, D. M. 2015 Nonlinear forced waves in a vertical rivulet flow. J. Fluid Mech. 770, 350–373.Google Scholar
Alexeev, A. & Oron, A. 2007 Suppression of the Rayleigh–Taylor instability of thin liquid films by the Marangoni effect. Phys. Fluids 19 (8), 082101.Google Scholar
Babchin, A. J., Frenkel, A. L., Levich, B. G. & Sivashinsky, G. I. 1983 Nonlinear saturation of Rayleigh–Taylor instability in thin films. Phys. Fluids 26 (11), 3159–3161.Google Scholar
Balestra, G., Brun, P.-T. & Gallaire, F. 2016 Rayleigh–Taylor instability under curved substrates: an optimal transient growth analysis. Phys. Rev. Fluids 1 (8), 083902.Google Scholar
Benilov, E. S. & Lapin, V. N. 2013 Inertial instability of flows on the inside or outside of a rotating horizontal cylinder. J. Fluid Mech. 736, 107–129.Google Scholar
Boos, W. & Thess, A. 1999 Cascade of structures in long-wavelength Marangoni instability. Phys. Fluids 11 (6), 1484–1494.Google Scholar
Brun, P.-T., Damiano, A., Rieu, P., Balestra, G. & Gallaire, F. 2015 Rayleigh–Taylor instability under an inclined plane. Phys. Fluids 27 (8), 084107.Google Scholar
de Bruyn, J. R. 1997 Crossover between surface tension and gravity-driven instabilities of a thin fluid layer on a horizontal cylinder. Phys. Fluids 9 (6), 1599–1605.Google Scholar
Burgess, J. M., Juel, A., McCormick, W. D., Swift, J. B. & Swinney, H. L. 2001 Suppression of dripping from a ceiling. Phys. Rev. Lett. 86 (7), 1203–1206.Google Scholar
Chandrasekhar, S. 1981 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
Cimpeanu, R., Papageorgiou, D. T. & Petropoulos, P. G. 2014 On the control and suppression of the Rayleigh–Taylor instability using electric fields. Phys. Fluids 26 (2), 022105.Google Scholar
Duclaux, V., Clanet, C. & Quéré, D. 2006 The effects of gravity on the capillary instability in tubes. J. Fluid Mech. 556, 217–226.Google Scholar
Eggers, J. & Villermaux, E. 2008 Physics of liquid jets. Rep. Prog. Phys. 71 (3), 036601.Google Scholar
Fauve, S. 2005 Pattern forming instabilities. In Hydrodynamics and Nonlinear Instabilities (ed. Godrèche, C. & Manneville, P.), chap. 4, pp. 387–491. Cambridge University Press.Google Scholar
Fermigier, M., Limat, L., Wesfreid, J. E., Boudinet, P. & Quilliet, C. 1992 Two-dimensional patterns in Rayleigh–Taylor instability of a thin layer. J. Fluid Mech. 236, 349–383.Google Scholar
Gallaire, F. & Brun, P.-T. 2017 Fluid dynamic instabilities: theory and application to pattern forming in complex media. Phil. Trans. R. Soc. Lond. A 375 (2093), 20160155.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
Hosoi, A. E. & Mahadevan, L. 1999 Axial instability of a free-surface front in a partially filled horizontal rotating cylinder. Phys. Fluids 11 (1), 97–106.Google Scholar
Indeikina, A., Veretennikov, I. & Chang, H.-C. 1997 Drop fall-off from pendent rivulets. J. Fluid Mech. 338, 173–201.Google Scholar
Jensen, O. E. 1997 The thin liquid lining of a weakly curved cylindrical tube. J. Fluid Mech. 331, 373–403.Google Scholar
Kaita, R., Berzak, L., Boyle, D., Gray, T., Granstedt, E., Hammett, G., Jacobson, C. M., Jones, A., Kozub, T., Kugel, H. & Others 2010 Experiments with liquid metal walls: status of the lithium tokamak experiment. Fusion Engng Des. 85 (6), 874–881.Google Scholar
King, A. A., Cummings, L. J., Naire, S. & Jensen, O. E. 2007 Liquid film dynamics in horizontal and tilted tubes: dry spots and sliding drops. Phys. Fluids 19 (4), 042102.Google Scholar
Kofman, N., Ruyer-Quil, C. & Mergui, S. 2016 Selection of solitary waves in vertically falling liquid films. Intl J. Multiphase Flow 84, 75–85.Google Scholar
Lapuerta, V., Mancebo, F. J. & Vega, J. M. 2001 Control of Rayleigh–Taylor instability by vertical vibration in large aspect ratio containers. Phys. Rev. E 64 (1), 016318.Google Scholar
Lee, A., Brun, P.-T., Marthelot, J., Balestra, G., Gallaire, F. & Reis, P. M. 2016 Fabrication of slender elastic shells by the coating of curved surfaces. Nat. Commun. 7, 11155.Google Scholar
Limat, L. 1993 Instabilité d’un liquide suspendu sous un surplomb solide: influence de l’épaisseur de la couche. C. R. Acad. Sci. Paris 317 (5), 563–568.Google Scholar
Limat, L., Jenffer, P., Dagens, B., Touron, E., Fermigier, M. & Wesfreid, J. E. 1992 Gravitational instabilities of thin liquid layers: dynamics of pattern selection. Physica D 61 (1), 166–182.Google Scholar
Lin, T.-S., Kondic, L. & Filippov, A. 2012 Thin films flowing down inverted substrates: three-dimensional flow. Phys. Fluids 24 (2), 022105.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
Lister, J. R., Rallison, J. M. & Rees, S. J. 2010 The nonlinear dynamics of pendent drops on a thin film coating the underside of a ceiling. J. Fluid Mech. 647, 239–264.Google Scholar
Mantič-Lugo, V., Arratia, C. & Gallaire, F. 2014 Self-consistent mean flow description of the nonlinear saturation of the vortex shedding in the cylinder wake. Phys. Rev. Lett. 113 (8), 084501.Google Scholar
Maurel, A., Pagneux, V. & Wesfreid, J. E. 1995 Mean-flow correction as non-linear saturation mechanism. Europhys. Lett. 32 (3), 217.Google Scholar
Melo, F. 1993 Localized states in a film-dragging experiment. Phys. Rev. E 48 (4), 2704.Google Scholar
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69 (3), 931.Google Scholar
Oron, A. & Rosenau, P. 1989 Nonlinear evolution and breaking of interfacial Rayleigh–Taylor waves. Phys. Fluids A 1 (7), 1155–1165.Google Scholar
Pougatch, K. & Frigaard, I. 2011 Thin film flow on the inside surface of a horizontally rotating cylinder: steady state solutions and their stability. Phys. Fluids 23 (2), 022102.Google Scholar
Rayleigh, Lord 1882 Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. Lond. Math. Soc. s1–14 (1), 170–177.Google Scholar
Reisfeld, B. & Bankoff, S. G. 1992 Non-isothermal flow of a liquid film on a horizontal cylinder. J. Fluid Mech. 236, 167–196.Google Scholar
Rietz, M., Scheid, B., Gallaire, F., Kofman, N., Kneer, R. & Rohlfs, W. 2017 Dynamics of falling films on the outside of a vertical rotating cylinder: waves, rivulets and dripping transitions. J. Fluid Mech. 832, 189–211.Google Scholar
Rohlfs, W., Pischke, P. & Scheid, B. 2017 Hydrodynamic waves in films flowing under an inclined plane. Phys. Rev. Fluids 2 (4), 044003.Google Scholar
Scheid, B., Kalliadasis, S., Ruyer-Quil, C. & Colinet, P. 2008 Interaction of three-dimensional hydrodynamic and thermocapillary instabilities in film flows. Phys. Rev. E 78 (6), 066311.Google Scholar
Scheid, B., Kofman, N. & Rohlfs, W. 2016 Critical inclination for absolute/convective instability transition in inverted falling films. Phys. Fluids 28 (4), 044107.Google Scholar
Seiden, G. & Thomas, P. J. 2011 Complexity, segregation, and pattern formation in rotating-drum flows. Rev. Mod. Phys. 83, 1323–1365.Google Scholar
Sharp, D. H. 1984 An overview of Rayleigh–Taylor instability. Physica D 12 (1–3), 3–18.Google Scholar
Takagi, D. & Huppert, H. E. 2010 Flow and instability of thin films on a cylinder and sphere. J. Fluid Mech. 647, 221.Google Scholar
Taylor, G. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I. Proc. R. Soc. Lond. A 201 (1065), 192–196.Google Scholar
Thoroddsen, S. T. & Mahadevan, L. 1997 Experimental study of coating flows in partially-filled horizontally rotating cylinder. Exp. Fluids 23, 1–13.Google Scholar
Trinh, P. H., Kim, H., Hammoud, N., Howell, P. D., Chapman, S. J. & Stone, H. A. 2014 Curvature suppresses the Rayleigh–Taylor instability. Phys. Fluids 26 (5), 051704.Google Scholar
Troian, S. M., Herbolzheimer, E., Safran, S. A. & Joanny, J. F. 1989 Fingering instabilities of driven spreading films. Europhys. Lett. 10 (1), 25.Google Scholar
Van Saarloos, W. 2003 Front propagation into unstable states. Phys. Rep. 386 (2), 29–222.Google Scholar
Weidner, D. E. 2012 The effect of surfactant convection and diffusion on the evolution of an axisymmetric pendant droplet. Phys. Fluids 24 (6), 062104.Google Scholar
Weidner, D. E., Schwartz, L. W. & Eres, M. H. 1997 Simulation of coating layer evolution and drop formation on horizontal cylinders. J. Coll. Int. Sci. 187 (1), 243–258.Google Scholar
Weidner, D. E., Schwartz, L. W. & Eres, M. H. 2007 Suppression and reversal of drop formation in a model paint film. Chem. Prod. Process Model. 2 (3), 19.Google Scholar
Witelski, T. P. & Bowen, M. 2003 ADI schemes for higher-order nonlinear diffusion equations. App. Num. Math. 45 (2), 331–351.Google Scholar
Yiantsios, S. G. & Higgins, B. G. 1989 Rayleigh–Taylor instability in thin viscous films. Phys. Fluids A 1 (9), 1484–1501.Google Scholar

Balestra et al. supplementary movie 1

Dynamics of the dripping droplets

Download Balestra et al. supplementary movie 1(Video)
Video 41.2 MB

Balestra et al. supplementary movie 2

Dynamics of the rivulets

Download Balestra et al. supplementary movie 2(Video)
Video 28.9 MB

Balestra et al. supplementary movie 3

Dynamics of the mixed regime

Download Balestra et al. supplementary movie 3(Video)
Video 28.6 MB

Balestra et al. supplementary movie 4

Dynamics of the dripping rivulets

Download Balestra et al. supplementary movie 4(Video)
Video 44.1 MB

Balestra et al. supplementary movie 5

Dynamics of rivulets in the cylinder experiment

Download Balestra et al. supplementary movie 5(Video)
Video 18.9 MB

Balestra et al. supplementary movie 6

Independent repetition of movie 5 for the same experimental conditions

Download Balestra et al. supplementary movie 6(Video)
Video 17.8 MB