Skip to main content Accessibility help

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

  • Chi-Tuong Pham (a1), Stéphane Perrard (a2) and Gabriel Le Doudic (a3)


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.


Corresponding author

Email address for correspondence:


Hide All
Amini, G. & Dolatabadi, A. 2011 Capillary instability of elliptic liquid jets. Phys. Fluids 23, 084109.
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.
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, 477481.
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, 401411.
Bostwick, J. B. & Steen, P. H. 2010 Stability of constrained cylindrical interfaces and the torus lift of Plateau–Rayleigh. J. Fluid Mech. 647, 201219.
Bostwick, J. B. & Steen, P. H. 2018 Static rivulet instabilities: varicose and sinuous modes. J. Fluid Mech. 837, 819838.
Couvreur, S.2013 Instabilités de filets liquides sur plan incliné. PhD thesis, Université Paris Diderot.
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.
Davis, S. H. 1980 Moving contact lines and rivulet instabilities. Part 1. The static rivulet. J. Fluid Mech. 98, 225242.
Decoene, A., Bonaventura, L., Miglio, E. & Saleri, F. 2009 Asymptotic derivation of the section-averaged shallow water equations. Math. Models Meth. Appl. Sci. 19, 387417.
Diez, J. A., González, A. G. & Kondic, L. 2009 On the breakup of fluid rivulets. Phys. Fluids 21 (8), 082105.
Duclaux, V., Clanet, C. & Quéré, D. 2006 The effects of gravity on the capillary instability in tubes. J. Fluid Mech. 556, 217226.
Goren, S. L. 1962 The instability of an annular thread of fluid. J. Fluid Mech. 12, 309319.
Gupta, R., Vaikuntanathan, V. & Siakumar, S. 2016 Superhydrophobic qualities of an aluminum surface coated with hydrophobic solution NeverWet. Colloids Surf. A 500, 4553.
Gutmark, E. J. & Grinstein, F. F. 1999 Flow control with noncircular jets. Annu. Rev. Fluid Mech. 31, 239272.
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, 441447.
Lamb, H. 1928 Statics, including Hydrostatics and the Elements of the Theory of Elasticity, 3rd edn. Cambridge University Press.
Landau, L. D. & Lifschitz, E. M. 1987 Fluid Mechanics, 2nd edn. Pergamon.
Langbein, D. 1990 The shape and stability of liquid menisci at solid edges. J. Fluid Mech. 213, 251265.
McCuan, J.2017 The stability of cylindrical pendant drop. In Memoirs of the American Mathematical Society, 250 (1189), doi:10.1090/memo/1189.
Mertens, K., Putkaradze, V. & Vorobieff, P. 2005 Morphology of a stream flowing down an inclined plane. Part 1. Braiding. J. Fluid Mech. 531, 4958.
Michael, D. H. & Williams, P. G. 1977 The equilibrium and stability of sessile drops. Proc. R. Soc. Lond. A 354, 127136.
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.
Morris, P. J. 1988 Instability of elliptic jets. AIAA J. 26, 172178.
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, 12391249.
Nakagawa, T. 1992 Rivulet meanders on a smooth hydrophobic surface. Intl J. Multiphase Flow 18, 455463.
Nakagawa, T. & Scott, J. C. 1984 Stream meanders on a smooth hydrophobic surface. J. Fluid Mech. 149, 8899.
Perrard, S., Deike, L., Duchêne, C. & Pham, C.-T. 2015 Capillary solitons on a levitated medium. Phys. Rev. E 92, 011002(R).
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, 150.
Plateau, J. 1873 Statique expérimentale et théorique des liquides soumis aux seules forces moléculaires. Gauthier-Villars.
Quinn, W. R. 1992 Streamwise evolution of a square jet cross section. AIAA J. 30, 28522857.
Rayleigh, Lord 1878 On the instability of jets. Proc. Lond. Math. Soc. 10, 413.
Rayleigh, Lord 1879 On the capillary phenomena of jets. Proc. R. Soc. Lond. 29, 7197.
Rayleigh, Lord 1892a On the instability of a cylinder of viscous liquid under capillary force. Philos. Mag. 34, 145154.
Rayleigh, Lord 1892b On the instability of cylindrical fluid surfaces. Philos. Mag. 34, 177180.
Roman, B., Gay, C. & Clanet, C.2003 Pendulum, drops and rods: an analogy. Available at:
Roy, V. & Schwartz, L. W. 1999 On the stability of liquid ridges. J. Fluid Mech. 291, 293318.
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, 147154 and 237–240.
Savart, F. 1833 Mémoire sur la constitution des veines liquides lancées par des orifices circulaires en mince paroi. Ann. Chim. Phys. 53, 337386.
Schmuki, P. & Laso, M. 1990 On the stability of rivulet flow. J. Fluid Mech. 215, 125143.
Sekimoto, K., Oguma, R. & Kawasaki, K. 1987 Morphological stability analysis of partial wetting. Ann. Phys. 176, 359392.
Speth, R. L. & Lauga, E. 2009 Capillary instability on a hydrophilic stripe. New J. Phys. 11, 075024.
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, 223263.
Stone, H. A. & Leal, L. G. 1989b Relaxation and breakup of an initially extended drop in an otherwise quiescent fluid. J. Fluid Mech. 198, 399427.
Tam, C. K. W. & Thies, A. T. 1992 Instability of rectangular jets. J. Fluid Mech. 248, 425448.
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, 322337.
Yang, L. & Homsy, G. M. 2007 Capillary instabilities of liquid films inside a wedge. Phys. Fluids 19, 044101.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Type Description Title

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).

 Unknown (1.2 MB)
1.2 MB

Pham et al. supplementary movie 2
Close-up of the pinch-off region (real time) and remaining satellite drop after the break-up.

 Unknown (1.3 MB)
1.3 MB

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

  • Chi-Tuong Pham (a1), Stéphane Perrard (a2) and Gabriel Le Doudic (a3)


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.