Skip to main content
×
×
Home

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

  • Gioele Balestra (a1), Nicolas Kofman (a1), P.-T. Brun (a2) (a3), Benoit Scheid (a4) and François Gallaire (a1)...
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.

Copyright
Corresponding author
Email address for correspondence: gioele.balestra@epfl.ch
References
Hide All
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), 15291546.
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, 350373.
Alexeev, A. & Oron, A. 2007 Suppression of the Rayleigh–Taylor instability of thin liquid films by the Marangoni effect. Phys. Fluids 19 (8), 082101.
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), 31593161.
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.
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, 107129.
Boos, W. & Thess, A. 1999 Cascade of structures in long-wavelength Marangoni instability. Phys. Fluids 11 (6), 14841494.
Brun, P.-T., Damiano, A., Rieu, P., Balestra, G. & Gallaire, F. 2015 Rayleigh–Taylor instability under an inclined plane. Phys. Fluids 27 (8), 084107.
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), 15991605.
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), 12031206.
Chandrasekhar, S. 1981 Hydrodynamic and Hydromagnetic Stability. Dover.
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.
Duclaux, V., Clanet, C. & Quéré, D. 2006 The effects of gravity on the capillary instability in tubes. J. Fluid Mech. 556, 217226.
Eggers, J. & Villermaux, E. 2008 Physics of liquid jets. Rep. Prog. Phys. 71 (3), 036601.
Fauve, S. 2005 Pattern forming instabilities. In Hydrodynamics and Nonlinear Instabilities (ed. Godrèche, C. & Manneville, P.), chap. 4, pp. 387491. Cambridge University Press.
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, 349383.
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.
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, 363384.
Hosoi, A. E. & Mahadevan, L. 1999 Axial instability of a free-surface front in a partially filled horizontal rotating cylinder. Phys. Fluids 11 (1), 97106.
Indeikina, A., Veretennikov, I. & Chang, H.-C. 1997 Drop fall-off from pendent rivulets. J. Fluid Mech. 338, 173201.
Jensen, O. E. 1997 The thin liquid lining of a weakly curved cylindrical tube. J. Fluid Mech. 331, 373403.
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), 874881.
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.
Kofman, N., Ruyer-Quil, C. & Mergui, S. 2016 Selection of solitary waves in vertically falling liquid films. Intl J. Multiphase Flow 84, 7585.
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.
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.
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), 563568.
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), 166182.
Lin, T.-S., Kondic, L. & Filippov, A. 2012 Thin films flowing down inverted substrates: three-dimensional flow. Phys. Fluids 24 (2), 022105.
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, 311343.
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, 239264.
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.
Maurel, A., Pagneux, V. & Wesfreid, J. E. 1995 Mean-flow correction as non-linear saturation mechanism. Europhys. Lett. 32 (3), 217.
Melo, F. 1993 Localized states in a film-dragging experiment. Phys. Rev. E 48 (4), 2704.
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69 (3), 931.
Oron, A. & Rosenau, P. 1989 Nonlinear evolution and breaking of interfacial Rayleigh–Taylor waves. Phys. Fluids A 1 (7), 11551165.
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.
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), 170177.
Reisfeld, B. & Bankoff, S. G. 1992 Non-isothermal flow of a liquid film on a horizontal cylinder. J. Fluid Mech. 236, 167196.
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, 189211.
Rohlfs, W., Pischke, P. & Scheid, B. 2017 Hydrodynamic waves in films flowing under an inclined plane. Phys. Rev. Fluids 2 (4), 044003.
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.
Scheid, B., Kofman, N. & Rohlfs, W. 2016 Critical inclination for absolute/convective instability transition in inverted falling films. Phys. Fluids 28 (4), 044107.
Seiden, G. & Thomas, P. J. 2011 Complexity, segregation, and pattern formation in rotating-drum flows. Rev. Mod. Phys. 83, 13231365.
Sharp, D. H. 1984 An overview of Rayleigh–Taylor instability. Physica D 12 (1–3), 318.
Takagi, D. & Huppert, H. E. 2010 Flow and instability of thin films on a cylinder and sphere. J. Fluid Mech. 647, 221.
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), 192196.
Thoroddsen, S. T. & Mahadevan, L. 1997 Experimental study of coating flows in partially-filled horizontally rotating cylinder. Exp. Fluids 23, 113.
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.
Troian, S. M., Herbolzheimer, E., Safran, S. A. & Joanny, J. F. 1989 Fingering instabilities of driven spreading films. Europhys. Lett. 10 (1), 25.
Van Saarloos, W. 2003 Front propagation into unstable states. Phys. Rep. 386 (2), 29222.
Weidner, D. E. 2012 The effect of surfactant convection and diffusion on the evolution of an axisymmetric pendant droplet. Phys. Fluids 24 (6), 062104.
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), 243258.
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.
Witelski, T. P. & Bowen, M. 2003 ADI schemes for higher-order nonlinear diffusion equations. App. Num. Math. 45 (2), 331351.
Yiantsios, S. G. & Higgins, B. G. 1989 Rayleigh–Taylor instability in thin viscous films. Phys. Fluids A 1 (9), 14841501.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

JFM classification

Type Description Title
VIDEO
Movies

Balestra et al. supplementary movie 1
Dynamics of the dripping droplets

 Video (41.2 MB)
41.2 MB
VIDEO
Movies

Balestra et al. supplementary movie 2
Dynamics of the rivulets

 Video (28.9 MB)
28.9 MB
VIDEO
Movies

Balestra et al. supplementary movie 3
Dynamics of the mixed regime

 Video (28.6 MB)
28.6 MB
VIDEO
Movies

Balestra et al. supplementary movie 4
Dynamics of the dripping rivulets

 Video (44.1 MB)
44.1 MB
VIDEO
Movies

Balestra et al. supplementary movie 5
Dynamics of rivulets in the cylinder experiment

 Video (18.9 MB)
18.9 MB
VIDEO
Movies

Balestra et al. supplementary movie 6
Independent repetition of movie 5 for the same experimental conditions

 Video (17.8 MB)
17.8 MB

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed