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Instability of a thin viscous film flowing under an inclined substrate: steady patterns

Published online by Cambridge University Press:  30 June 2020

Gaétan Lerisson
Affiliation:
Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, Lausanne, CH-1015, Switzerland
Pier Giuseppe Ledda
Affiliation:
Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, Lausanne, CH-1015, Switzerland
Gioele Balestra
Affiliation:
iPrint Institute, University of Applied Sciences and Arts of Western Switzerland, Fribourg, CH-1700, Switzerland
François Gallaire*
Affiliation:
Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, Lausanne, CH-1015, Switzerland
*
Email address for correspondence: [email protected]

Abstract

The flow of a thin film coating the underside of an inclined substrate is studied. We measure experimentally spatial growth rates and compare them to the linear stability analysis of a flat film modelled by the lubrication equation. When forced by a stationary localized perturbation, a front develops that we predict with the group velocity of the unstable wave packet. We compare our experimental measurements with numerical solutions of the nonlinear lubrication equation with complete curvature. Streamwise structures dominate and saturate after some distance. We recover their profile with a one-dimensional lubrication equation suitably modified to ensure an invariant profile along the streamwise direction and compare them with the solution of a purely two-dimensional pendent drop, showing overall a very good agreement. Finally, those different profiles agree also with a two-dimensional simulation of the Stokes equations.

Type
JFM Papers
Copyright
© The Author(s), 2020. Published by 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 (7–8), 15291546.CrossRefGoogle 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.CrossRefGoogle 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), 31593161.CrossRefGoogle Scholar
Balestra, G., Kofman, N., Brun, P.-T., Scheid, B. & Gallaire, F. 2018a Three-dimensional Rayleigh–Taylor instability under a unidirectional curved substrate. J. Fluid Mech. 837, 1947.CrossRefGoogle Scholar
Balestra, G., Nguyen, D. M.-P. & Gallaire, F. 2018b Rayleigh–Taylor instability under a spherical substrate. Phys. Rev. Fluids 3 (8), 084005.CrossRefGoogle Scholar
Barannyk, L. L., Papageorgiou, D. T. & Petropoulos, P. G. 2012 Suppression of Rayleigh–Taylor instability using electric fields. Maths Comput. Simul. 82 (6), 10081016.CrossRefGoogle Scholar
Barlow, N. S., Helenbrook, B. T. & Weinstein, S. J. 2015 Algorithm for spatio-temporal analysis of the signalling problem. IMA J. Appl. Maths 82 (1), 132.Google Scholar
Bestehorn, M. & Merkt, D. 2006 Regular surface patterns on Rayleigh–Taylor unstable evaporating films heated from below. Phys. Rev. Lett. 97 (12), 127802.CrossRefGoogle ScholarPubMed
Boos, W. & Thess, A. 1999 Cascade of structures in long-wavelength Marangoni instability. Phys. Fluids 11 (6), 14841494.CrossRefGoogle 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.CrossRefGoogle Scholar
Brunet, P., Flesselles, J. M. & Limat, L. 2007 Dynamics of a circular array of liquid columns. Eur. Phys. J. B 55 (3), 297322.CrossRefGoogle 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.CrossRefGoogle ScholarPubMed
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon Press, Oxford University Press.Google Scholar
Chang, H. 1994 Wave evolution on a falling film. Annu. Rev. Fluid Mech. 26 (1), 103136.CrossRefGoogle Scholar
Charogiannis, A., Denner, F., van Wachem, B. G. M., Kalliadasis, S., Scheid, B. & Markides, C. N. 2018 Experimental investigations of liquid falling films flowing under an inclined planar substrate. Phys. Rev. Fluids 3 (11), 114002.CrossRefGoogle 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.CrossRefGoogle Scholar
Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81 (3), 1131.CrossRefGoogle Scholar
Dietze, G. F., Picardo, J. R. & Narayanan, R. 2018 Sliding instability of draining fluid films. J. Fluid Mech. 857, 111141.CrossRefGoogle Scholar
Duprat, C., Ruyer-Quil, C., Kalliadasis, S. & Giorgiutti-Dauphiné, F. 2007 Absolute and convective instabilities of a viscous film flowing down a vertical fiber. Phys. Rev. Lett. 98 (24), 244502.CrossRefGoogle Scholar
Duprat, C. & Stone, H. A. 2015 Fluid–Structure Interactions in Low-Reynolds-Number Flows. Royal Society of Chemistry.CrossRefGoogle 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, 349383.CrossRefGoogle Scholar
Gaster, M. 1962 A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability. J. Fluid Mech. 14 (2), 222224.CrossRefGoogle Scholar
Glasner, K. B. 2007 The dynamics of pendant droplets on a one-dimensional surface. Phys. Fluids 19 (10), 102104.CrossRefGoogle Scholar
Högberg, M. & Henningson, D. 1998 Secondary instability of cross-flow vortices in Falkner–Skan–Cooke boundary layers. J. Fluid Mech. 368, 339357.CrossRefGoogle Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22 (1), 473537.CrossRefGoogle Scholar
Kalliadasis, S., Ruyer-Quil, C., Scheid, B. & Velarde, M. G. 2011 Falling Liquid Films, vol. 176. Springer Science and Business Media.Google Scholar
Kapitza, P. L. 1965 43 – wave flow of thin layers of a viscous fluid. In Collected Papers of P. L. Kapitza (ed. Haar, D. T.), pp. 662709. Pergamon.Google Scholar
Kheshgi, H. S., Kistler, S. F. & Scriven, L. E. 1992 Rising and falling film flows: viewed from a first-order approximation. Chem. Engng Sci. 47 (3), 683694.CrossRefGoogle Scholar
King, K. R., Weinstein, S. J., Zaretzky, P. M., Cromer, M. & Barlow, N. S. 2016 Stability of algebraically unstable dispersive flows. Phys. Rev. Fluids 1, 073604.CrossRefGoogle Scholar
Kofman, N., Rohlfs, W., Gallaire, F., Scheid, B. & Ruyer-Quil, C. 2018 Prediction of two-dimensional dripping onset of a liquid film under an inclined plane. Intl J. Multiphase Flow 104, 286293.CrossRefGoogle Scholar
Krechetnikov, R. 2010 On application of lubrication approximations to nonunidirectional coating flows with clean and surfactant interfaces. Phys. Fluids 22 (9), 092102.CrossRefGoogle 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 ScholarPubMed
Lerisson, G., Ledda, P. G., Balestra, G. & Gallaire, F. 2019 Dripping down the rivulet. Phys. Rev. Fluids 4 (10), 100504.CrossRefGoogle 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: Nonlinear Phenomena 61 (1–4), 166182.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, 239264.CrossRefGoogle Scholar
Marthelot, J., Strong, E. F., Reis, P. M. & Brun, P.-T. 2018 Designing soft materials with interfacial instabilities in liquid films. Nat. Commun. 9 (1), 4477.CrossRefGoogle ScholarPubMed
Maxwell, J. C. 1875 Capillary action. In Encyclopaedia Britannica, 9th edn. Reprinted in 2011: The Scientific Papers of James Clerk Maxwell (Cambridge Library Collection - Physical Sciences, pp. 541–591). Cambridge University Press.Google Scholar
Monkewitz, P. A. & Sohn, K. 1988 Absolute instability in hot jets. AIAA J. 26 (8), 911916.CrossRefGoogle Scholar
Pirat, C., Mathis, C., Maissa, P. & Gil, L. 2004 Structures of a continuously fed two-dimensional viscous film under a destabilizing gravitational force. Phys. Rev. Lett. 92 (10), 104501.CrossRefGoogle 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), 170177.CrossRefGoogle 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, 189211.CrossRefGoogle Scholar
Roman, B., Gay, C. & Clanet, C.2020 Pendulums, drops and rods: a physical analogy, arXiv:2006.02742.Google Scholar
Ruschak, K. J. 1978 Flow of a falling film into a pool. AIChE J. 24 (4), 705709.CrossRefGoogle Scholar
Scheid, B. 2013 Rivulet structures in falling liquid films. In Without Bounds: A Scientific Canvas of Nonlinearity and Complex Dynamics, pp. 435441. Springer.CrossRefGoogle Scholar
Scheid, B., Kofman, N. & Rohlfs, W. 2016 Critical inclination for absolute/convective instability transition in inverted falling films. Phys. Fluids 28 (4), 044107.CrossRefGoogle Scholar
Sterman-Cohen, E., Bestehorn, M. & Oron, A. 2017 Rayleigh–Taylor instability in thin liquid films subjected to harmonic vibration. Phys. Fluids 29 (5), 052105.Google Scholar
Taylor, G. I. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I. Proc. R. Soc. Lond. A 201 (1065), 192196.Google Scholar
Van Saarloos, W. 2003 Front propagation into unstable states. Phys. Rep. 386 (2–6), 29222.CrossRefGoogle 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).Google Scholar
Weinstein, S. J. & Ruschak, K. J. 2004 Coating flows. Annu. Rev. Fluid Mech. 36, 2953.CrossRefGoogle Scholar
Wilson, S. D. R. 1982 The drag-out problem in film coating theory. J. Engng Maths 16 (3), 209221.CrossRefGoogle Scholar
Yiantsios, S. G. & Higgins, B. G. 1989 Rayleigh–Taylor instability in thin viscous films. Phys. Fluids A 1 (9), 14841501.CrossRefGoogle Scholar
Yoshikawa, H. N., Mathis, C., Satoh, S. & Tasaka, Y. 2019 Inwardly rotating spirals in a nonoscillatory medium. Phys. Rev. Lett. 122 (1), 014502.CrossRefGoogle Scholar
Zaccaria, D., Bigoni, D., Noselli, G. & Misseroni, D. 2011 Structures buckling under tensile dead load. Proc. R. Soc. Lond. A 467 (2130), 16861700.CrossRefGoogle Scholar