Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-24T18:48:34.955Z Has data issue: false hasContentIssue false
  • English
  • Français

Instability of a thin viscous film flowing under an inclined substrate: the emergence and stability of rivulets

Published online by Cambridge University Press:  12 October 2020

Pier Giuseppe Ledda Open the ORCID record for Pier Giuseppe Ledda [Opens in a new window] ,
Gaétan Lerisson ,
Gioele Balestra Open the ORCID record for Gioele Balestra [Opens in a new window]  and
François Gallaire Open the ORCID record for François Gallaire [Opens in a new window]
Pier Giuseppe Ledda*
Affiliation:
Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, LausanneCH-1015, Switzerland
Gaétan Lerisson
Affiliation:
Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, LausanneCH-1015, Switzerland
Gioele Balestra
Affiliation:
iPrint Institute, University of Applied Sciences and Arts of Western Switzerland, FribourgCH-1700, Switzerland
François Gallaire
Affiliation:
Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, LausanneCH-1015, Switzerland
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the pattern formation of a thin film flowing under an inclined planar substrate. The phenomenon is studied in the context of the Rayleigh–Taylor instability using the lubrication equation. Inspired by experimental observations, we numerically study the thin film response to a streamwise-invariant sinusoidal initial condition. The numerical response shows the emergence of predominant streamwise-aligned structures, modulated along the direction perpendicular to the flow, called rivulets. Oscillations of the thickness profile along the streamwise direction do not grow significantly when the inclination is very large or the liquid layer very thin. However, for small inclinations or thick films, streamwise perturbations grow on rivulets. A secondary stability analysis of one-dimensional and steady rivulets reveals a strong stabilization mechanism for large inclinations or very thin films. The theoretical results are compared with experimental measurements of the streamwise oscillations of the rivulet profile, showing a good agreement. The emergence of rivulets is investigated by studying the impulse response. Both the experimental observation and the numerical simulation show a marked anisotropy favouring streamwise-aligned structures. A weakly nonlinear model is proposed to rationalize the levelling of all but streamwise-aligned structures.

JFM classification

Instability: Absolute/convective instability Instability: Nonlinear instability Interfacial Flows (free surface): Thin films
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 904 , 10 December 2020 , A23
Check for updates
Copyright
© The Author(s), 2020. Published by 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

REFERENCES

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.CrossRefGoogle Scholar
Balestra, G., Kofman, N., Brun, P.-T., Scheid, B. & Gallaire, F. 2018 a Three-dimensional Rayleigh–Taylor instability under a unidirectional curved substrate. J. Fluid Mech. 837, 19–47.CrossRefGoogle Scholar
Balestra, G., Nguyen, D. M.-P. & Gallaire, F. 2018 b Rayleigh–Taylor instability under a spherical substrate. Phys. Rev. Fluids 3 (8), 084005.CrossRefGoogle Scholar
Bers, A. 1975 Linear waves and instabilities. In Physique des Plasmas (ed. DeWitt, C & Peyraud, J.), pp. 117–215. Gordon & Breach.Google Scholar
Bertagni, M. B. & Camporeale, C. 2017 Nonlinear and subharmonic stability analysis in film-driven morphological patterns. Phys. Rev. E 96 (5), 053115.CrossRefGoogle ScholarPubMed
Briggs, R. J. 1964 Electron stream interactionwith plasmas. In Handbook of Plasma Physics (ed. Bers, A.). MIT Press.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.CrossRefGoogle Scholar
Camporeale, C. 2015 Hydrodynamically locked morphogenesis in karst and ice flutings. J. Fluid Mech. 778, 89–119.CrossRefGoogle Scholar
Chandrasekhar, S. 2013 Hydrodynamic and Hydromagnetic Stability. Courier Corporation.Google 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
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, 349–383.CrossRefGoogle Scholar
Gallaire, F. & Brun, P.-T. 2017 Fluid dynamic instabilities: theory and application to pattern forming in complex media. Phil. Trans. R. Soc. A Math. Phys. Engng Sci. 375 (2093), 20160155.CrossRefGoogle ScholarPubMed
Gaster, M. 1962 A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability. J. Fluid Mech. 14 (2), 222–224.CrossRefGoogle Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22 (1), 473–537.CrossRefGoogle Scholar
Huerre, P. & Rossi, M. 1998 Hydrodynamic Instabilities in Open Flows, pp. 81–294. Cambridge University Press.CrossRefGoogle Scholar
Kalliadasis, S., Ruyer-Quil, C., Scheid, B. & Velarde, M. G. (Ed.) 2012 Falling Liquid Films. Springer-Verlag.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, 286–293.CrossRefGoogle 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 ScholarPubMed
Lerisson, G., Ledda, P. G., Balestra, G. & Gallaire, F. 2019 Dripping down the rivulet. Phys. Rev. Fluids 4, 100504.CrossRefGoogle Scholar
Lerisson, G., Ledda, P. G., Balestra, G. & Gallaire, F. 2020 Instability of a thin viscous film flowing under an inclined substrate: steady patterns. J. Fluid Mech. 898, A6.CrossRefGoogle 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.CrossRefGoogle Scholar
Marthelot, J., Strong, E. F., Reis, P. M. & Brun, P-T. 2018 a Designing soft materials with interfacial instabilities in liquid films. Nat. Commun. 9 (1), 4477.CrossRefGoogle ScholarPubMed
Marthelot, J., Strong, E. F., Reis, P. M. & Brun, P.-T. 2018 b Solid structures generated by capillary instability in thin liquid films. Phys. Rev. Fluids 3, 100506.CrossRefGoogle Scholar
Meakin, P. & Jamtveit, B. 2010 Geological pattern formation by growth and dissolution in aqueous systems. Proc. R. Soc. A Math. Phys. Engng Sci. 466 (2115), 659–694.Google Scholar
Moisy, F., Rabaud, M. & Salsac, K. 2009 A synthetic schlieren method for the measurement of the topography of a liquid interface. Exp. Fluids 46 (6), 1021.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, 170–177.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, 189–211.CrossRefGoogle Scholar
Roman, B., Gay, C. & Clanet, C. 2020 Pendulum, 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), 705–709.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
Schmid, P. J., Henningson, D. S. & Jankowski, D. F. 2002 Stability and transition in shear flows. applied mathematical sciences, vol. 142. Appl. Mech. Rev. 55 (3), B57–B59.CrossRefGoogle Scholar
Settles, G. S. 2001 Schlieren and Shadowgraph Techniques: Visualizing Phenomena in Transparent Media. Springer.CrossRefGoogle Scholar
Short, M. B., Baygents, J. C., Beck, J. W., Stone, D. A., Toomey III, R. S. & Goldstein, R. E. 2005 Stalactite growth as a free-boundary problem: a geometric law and its platonic ideal. Phys. Rev. Lett. 94 (1), 018501.CrossRefGoogle ScholarPubMed
Taylor, G. I. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I. Proc. R. Soc. Lond. A Math. Phys. Sci. 201 (1065), 192–196.Google Scholar
Weinstein, S. J. & Ruschak, K. J. 2004 Coating flows. Annu. Rev. Fluid Mech. 36, 29–53.CrossRefGoogle Scholar
Wilson, S. D. R. 1982 The drag-out problem in film coating theory. J. Engng Math. 16 (3), 209–221.CrossRefGoogle Scholar
Yiantsios, S. G. & Higgins, B. G. 1989 Rayleigh–Taylor instability in thin viscous films. Phys. Fluids A Fluid Dyn. 1 (9), 1484–1501.CrossRefGoogle Scholar
Zaccaria, D., Bigoni, D., Noselli, G. & Misseroni, D. 2011 Structures buckling under tensile dead load. Proc. R. Soc. A Math. Phys. Engng Sci. 467 (2130), 1686–1700.Google Scholar