Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-17T18:22:27.842Z Has data issue: false hasContentIssue false
  • English
  • Français

Surface waves along liquid cylinders. Part 1. Stabilising effect of gravity on the Plateau–Rayleigh instability

Published online by Cambridge University Press:  18 March 2020

Chi-Tuong Pham Open the ORCID record for Chi-Tuong Pham [Opens in a new window] ,
Stéphane Perrard Open the ORCID record for Stéphane Perrard [Opens in a new window]  and
Gabriel Le Doudic Open the ORCID record for Gabriel Le Doudic [Opens in a new window]
Chi-Tuong Pham*
Affiliation:
Université Paris-Saclay, CNRS, LIMSI, rue John von Neumann, 91400Orsay, France
Stéphane Perrard
Affiliation:
Laboratoire de Physique de l’École normale supérieure, ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris, 24 rue Lhomond, 75005Paris, France
Gabriel Le Doudic
Affiliation:
Matière et Systèmes Complexes, Université de Paris, CNRS, 10 rue Alice Domon et Léonie Duquet, 75013Paris, France
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the shape and the geometrical properties of sessile drops with translational invariance (namely ‘liquid cylinders’) deposited upon a flat superhydrophobic substrate. We account for the flattening effects of gravity on the shape of the drop using a pendulum rotation motion analogy. In the framework of the inviscid Saint-Venant equations, we show that liquid cylinders are always unstable because of the Plateau–Rayleigh instability. However, a cylindrical drop deposited upon a superhydrophobic non-flat channel (here, wedge-shaped channels) is stabilised beyond a critical cross-sectional area. The critical threshold of the Plateau–Rayleigh instability is analytically computed for various profiles of the channel. The stability analysis is performed in terms of an effective propagation speed of varicose waves. Experiments are performed in order to test these analytical results. We measure the critical drop size at which breakup occurs, together with the decreasing effective propagation speed of varicose waves as the threshold is approached. Our theoretical predictions are in excellent agreement with the experimental measurements.

JFM classification

Drops and Bubbles: Breakup/coalescence Interfacial Flows (free surface): Capillary flows Waves/Free-surface Flows: Surface gravity waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 891 , 25 May 2020 , A8
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

Amini, G. & Dolatabadi, A. 2011 Capillary instability of elliptic liquid jets. Phys. Fluids 23, 084109.CrossRefGoogle Scholar
Amini, G., Lv, L., Dolatabadi, A. & Ihme, M. 2014 Instability of elliptic liquid jets: temporal linear stability theory and experimental analysis. Phys. Fluids 26, 114105.CrossRefGoogle Scholar
Arkhipenko, V. I., Barkov, Y. D., Bashtovoi, V. G. & Krakov, M. S. 1980 Investigation into the stability of a stationary cylindrical column of magnetizable liquid. Fluid Dyn. 15, 477–481.Google Scholar
Birnir, B., Mertens, K., Putkaradze, V. & Vorobieff, P. 2008 Morphology of a stream flowing down an inclined plane. Part 2. Meandering. J. Fluid Mech. 603, 401–411.CrossRefGoogle Scholar
Bostwick, J. B. & Steen, P. H. 2010 Stability of constrained cylindrical interfaces and the torus lift of Plateau–Rayleigh. J. Fluid Mech. 647, 201–219.CrossRefGoogle Scholar
Bostwick, J. B. & Steen, P. H. 2018 Static rivulet instabilities: varicose and sinuous modes. J. Fluid Mech. 837, 819–838.CrossRefGoogle Scholar
Couvreur, S.2013 Instabilités de filets liquides sur plan incliné. PhD thesis, Université Paris Diderot.Google Scholar
Daerr, A., Eggers, J., Limat, L. & Valade, N. 2011 General mechanism for the meandering instability of rivulets of Newtonian fluids. Phys. Rev. Lett. 106, 184501.CrossRefGoogle ScholarPubMed
Davis, S. H. 1980 Moving contact lines and rivulet instabilities. Part 1. The static rivulet. J. Fluid Mech. 98, 225–242.CrossRefGoogle Scholar
Decoene, A., Bonaventura, L., Miglio, E. & Saleri, F. 2009 Asymptotic derivation of the section-averaged shallow water equations. Math. Models Meth. Appl. Sci. 19, 387–417.CrossRefGoogle Scholar
Diez, J. A., González, A. G. & Kondic, L. 2009 On the breakup of fluid rivulets. Phys. Fluids 21 (8), 082105.CrossRefGoogle Scholar
Duclaux, V., Clanet, C. & Quéré, D. 2006 The effects of gravity on the capillary instability in tubes. J. Fluid Mech. 556, 217–226.CrossRefGoogle Scholar
Goren, S. L. 1962 The instability of an annular thread of fluid. J. Fluid Mech. 12, 309–319.CrossRefGoogle Scholar
Gupta, R., Vaikuntanathan, V. & Siakumar, S. 2016 Superhydrophobic qualities of an aluminum surface coated with hydrophobic solution NeverWet. Colloids Surf. A 500, 45–53.CrossRefGoogle Scholar
Gutmark, E. J. & Grinstein, F. F. 1999 Flow control with noncircular jets. Annu. Rev. Fluid Mech. 31, 239–272.CrossRefGoogle Scholar
Ku, T. C., Ramsey, J. H. & Clinton, W. C. 1968 Calculation of liquid droplet profiles from closed-form solution of Young–Laplace equation. IBM J. Res. Dev. 12, 441–447.CrossRefGoogle Scholar
Lamb, H. 1928 Statics, including Hydrostatics and the Elements of the Theory of Elasticity, 3rd edn. Cambridge University Press.Google Scholar
Landau, L. D. & Lifschitz, E. M. 1987 Fluid Mechanics, 2nd edn. Pergamon.Google Scholar
Langbein, D. 1990 The shape and stability of liquid menisci at solid edges. J. Fluid Mech. 213, 251–265.CrossRefGoogle Scholar
McCuan, J.2017 The stability of cylindrical pendant drop. In Memoirs of the American Mathematical Society, 250 (1189), doi:10.1090/memo/1189.CrossRefGoogle Scholar
Mertens, K., Putkaradze, V. & Vorobieff, P. 2005 Morphology of a stream flowing down an inclined plane. Part 1. Braiding. J. Fluid Mech. 531, 49–58.CrossRefGoogle Scholar
Michael, D. H. & Williams, P. G. 1977 The equilibrium and stability of sessile drops. Proc. R. Soc. Lond. A 354, 127–136.Google Scholar
Mora, S., Phou, T., Fromental, J.-M., Pismen, L. M. & Pomeau, Y. 2010 Capillary driven instability of a soft solid. Phys. Rev. Lett. 105, 214301.CrossRefGoogle ScholarPubMed
Morris, P. J. 1988 Instability of elliptic jets. AIAA J. 26, 172–178.CrossRefGoogle Scholar
Myers, T. G., Liang, H. X. & Wetton, B. 2004 The stability and flow of a rivulet driven by interfacial shear and gravity. Intl J. Nonlinear Mech. 39, 1239–1249.CrossRefGoogle Scholar
Nakagawa, T. 1992 Rivulet meanders on a smooth hydrophobic surface. Intl J. Multiphase Flow 18, 455–463.CrossRefGoogle Scholar
Nakagawa, T. & Scott, J. C. 1984 Stream meanders on a smooth hydrophobic surface. J. Fluid Mech. 149, 88–99.CrossRefGoogle Scholar
Perrard, S., Deike, L., Duchêne, C. & Pham, C.-T. 2015 Capillary solitons on a levitated medium. Phys. Rev. E 92, 011002(R).Google ScholarPubMed
Plateau, J. 1849 Recherches expérimentales et théoriques sur les figures d’une masse liquide sans pesanteur. Mémoires de l’Académie royale des sciences, des lettres et des beaux arts de Belgique 23, 1–50.Google Scholar
Plateau, J. 1873 Statique expérimentale et théorique des liquides soumis aux seules forces moléculaires. Gauthier-Villars.Google Scholar
Quinn, W. R. 1992 Streamwise evolution of a square jet cross section. AIAA J. 30, 2852–2857.CrossRefGoogle Scholar
Rayleigh, Lord 1878 On the instability of jets. Proc. Lond. Math. Soc. 10, 4–13.CrossRefGoogle Scholar
Rayleigh, Lord 1879 On the capillary phenomena of jets. Proc. R. Soc. Lond. 29, 71–97.Google Scholar
Rayleigh, Lord 1892a On the instability of a cylinder of viscous liquid under capillary force. Philos. Mag. 34, 145–154.CrossRefGoogle Scholar
Rayleigh, Lord 1892b On the instability of cylindrical fluid surfaces. Philos. Mag. 34, 177–180.CrossRefGoogle Scholar
Roman, B., Gay, C. & Clanet, C.2003 Pendulum, drops and rods: an analogy. Available at:https://www.researchgate.net/publication/237480399_Pendulum_Drops_and_Rods_a_physical_analogy.Google Scholar
Roy, V. & Schwartz, L. W. 1999 On the stability of liquid ridges. J. Fluid Mech. 291, 293–318.CrossRefGoogle Scholar
Saint-Venant, A. J. C. B. de 1871 Théorie du mouvement non permanent des eaux, avec application aux crues des rivières et a l’introduction de marées dans leurs lits. C. R. Acad. Sci. 73, 147–154 and 237–240.Google Scholar
Savart, F. 1833 Mémoire sur la constitution des veines liquides lancées par des orifices circulaires en mince paroi. Ann. Chim. Phys. 53, 337–386.Google Scholar
Schmuki, P. & Laso, M. 1990 On the stability of rivulet flow. J. Fluid Mech. 215, 125–143.CrossRefGoogle Scholar
Sekimoto, K., Oguma, R. & Kawasaki, K. 1987 Morphological stability analysis of partial wetting. Ann. Phys. 176, 359–392.CrossRefGoogle Scholar
Speth, R. L. & Lauga, E. 2009 Capillary instability on a hydrophilic stripe. New J. Phys. 11, 075024.Google Scholar
Stone, H. A. & Leal, L. G. 1989a The influence of initial deformation on drop breakup in subcritical time-dependent flows at low Reynolds numbers. J. Fluid Mech. 206, 223–263.CrossRefGoogle Scholar
Stone, H. A. & Leal, L. G. 1989b Relaxation and breakup of an initially extended drop in an otherwise quiescent fluid. J. Fluid Mech. 198, 399–427.CrossRefGoogle Scholar
Tam, C. K. W. & Thies, A. T. 1992 Instability of rectangular jets. J. Fluid Mech. 248, 425–448.CrossRefGoogle Scholar
Tomotika, S. 1935 On the instability of a cylindrical thread of a viscous liquid surrounded by another viscous fluid. Proc. R. Soc. Lond. A 150, 322–337.Google Scholar
Yang, L. & Homsy, G. M. 2007 Capillary instabilities of liquid films inside a wedge. Phys. Fluids 19, 044101.CrossRefGoogle Scholar
Supplementary material: File

Pham et al. supplementary movie 1

Below a critical volume (close to 15 mL), a 49 centimeter long drop, pinned at both ends, breaks up after a slow pinch-off dynamics. Pinch-off occurs about the center of the drop and both halves end up retracting, owing to surface tension (real time).

Download Pham et al. supplementary movie 1(File)
File 1.2 MB
Supplementary material: File

Pham et al. supplementary movie 2

Close-up of the pinch-off region (real time) and remaining satellite drop after the break-up.

Download Pham et al. supplementary movie 2(File)
File 1.3 MB