Skip to main content Accessibility help
Hostname: page-component-684899dbb8-mhx7p Total loading time: 0.45 Render date: 2022-05-28T19:33:20.793Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true }

Drops and bubbles in wedges

Published online by Cambridge University Press:  06 May 2014

Etienne Reyssat*
PMMH, CNRS UMR 7636 – ESPCI – UPMC Université Paris 6 – UPD Université Paris 7, 10 rue Vauquelin, 75005 Paris, France
Email address for correspondence:


We investigate experimentally the spontaneous motion of drops and bubbles confined between two plates forming a narrow wedge. Such discoidal objects migrate under the gradient in interfacial energy induced by the non-homogeneous confinement. The resulting capillary driving force is balanced by viscous resistance. The viscous friction on a drop bridging parallel plates is estimated by measuring its sliding velocity under gravity. The viscous forces are the sum of two contributions, from the bulk of the liquid and from contact lines, the relative strength of which depends on the drop size and velocity and the physical properties of the liquid. The balance of capillarity and viscosity quantitatively explains the dynamics of spontaneous migration of a drop in a wedge. Close the tip of the wedge, bulk dissipation dominates and the migrating velocity of drops is constant and independent of drop volume. The distance between the drop and the tip of the wedge is thus linear with time $t$, $x(t) \sim t_0-t$, where $t_0$ is the time at which the drop reaches the tip of the wedge. Far away from the apex, contact lines dominate the friction, the motion is accelerated toward the tip of the wedge and velocities are higher for larger drops. In this regime, it is shown that $x(t) \sim (t_0-t)^{4/13}$. The position and time of the crossover between the two dissipation regimes are used to write a dimensionless equation of motion. Plotted in rescaled variables, all experimental trajectories collapse to the prediction of our model. In contrast to drops, gas bubbles in a liquid-filled wedge behave as non-wetting objects. They thus escape the confinement of the wedge to reduce their surface area. The physical mechanisms involved are similar for drops and bubbles, so that the forces acting have the same mathematical structures in both cases, except for the sign of the capillary driving force and a numerical factor. We thus predict and show experimentally that the trajectories of drops and bubbles obey the same equation of motion, except for a change in the sign of $t_0-t$.

© 2014 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.)


Bico, J. & Quéré, D. 2001 Falling slugs. J. Colloid Interface Sci. 243, 262264.CrossRefGoogle Scholar
Bonn, D., Eggers, J., Indekeu, J., Meunier, J. & Rolley, E. 2009 Wetting and spreading. Rev. Mod. Phys. 81 (2), 739805.CrossRefGoogle Scholar
Bouasse, H. 1924 Capillarité, Phénomènes Superficiels. Delagrave.Google Scholar
Boudaoud, A. 2007 Non-Newtonian thin films with normal stresses: dynamics and spreading. Eur. Phys. J. E 22 (2), 107109.CrossRefGoogle ScholarPubMed
Brochard, F. 1989 Motion of droplets on solid surfaces induced by chemical or thermal gradients. Langmuir 5 (3), 432438.CrossRefGoogle Scholar
Bush, J. W. M. 1997 The anomalous wake accompanying bubbles rising in a thin gap: a mechanically forced Marangoni flow. J. Fluid Mech. 352, 283303.CrossRefGoogle Scholar
Cantat, I. 2013 Liquid meniscus friction on a wet plate: bubbles, lamellae, and foams. Phys. Fluids 25 (3), 031303.CrossRefGoogle Scholar
Chaudhury, M. K. & Whitesides, G. M. 1992 How to make water run uphill. Science 256 (5063), 15391541.CrossRefGoogle ScholarPubMed
Dangla, R., Kayi, S. C. & Baroud, C. N. 2013 Droplet microfluidics driven by gradients of confinement. Proc. Natl Acad. Sci. USA 110 (3), 853858.CrossRefGoogle ScholarPubMed
Daniel, S., Sircar, S., Gliem, J. & Chaudhury, M. K. 2004 Ratcheting motion of liquid drops on gradient surfaces. Langmuir 20 (10), 40854092.CrossRefGoogle ScholarPubMed
de Gennes, P.-G. 1985 Wetting: statics and dynamics. Rev. Mod. Phys. 57 (3), 827863.CrossRefGoogle Scholar
de Gennes, P.-G., Brochard-Wyart, F. & Quéré, D. 2004 Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls and Waves. Springer.CrossRefGoogle Scholar
Domingues dos Santos, F. & Ondarçuhu, T. 1995 Free-running droplets. Phys. Rev. Lett. 75 (16), 29722975.CrossRefGoogle Scholar
Eck, W. & Siekmann, J. 1978 On bubble motion in a Hele-Shaw cell, a possibility to study two-phase flows under reduced gravity. Ing.-Arch. 47, 153168.CrossRefGoogle Scholar
Eggers, J. & Stone, H. A. 2004 Characteristic lengths at moving contact lines for a perfectly wetting fluid: the influence of speed on the dynamic contact angle. J. Fluid Mech. 505, 309321.CrossRefGoogle Scholar
Eri, A. & Okumura, K. 2011 Viscous drag friction acting on a fluid drop confined in between two plates. Soft Matt. 7 (12), 56485653.CrossRefGoogle Scholar
Genzer, J. & Bhat, R. R. 2008 Surface-bound soft matter gradients. Langmuir 24, 22942317.CrossRefGoogle ScholarPubMed
Gorodtsov, V. A. 1989 Spreading of a film of nonlinearly viscous liquid over a horizontal smooth solid surface. J. Engng Phys. 57 (2), 879884.CrossRefGoogle Scholar
Greenspan, H. P. 1978 On the motion of a small viscous droplet that wets a surface. J. Fluid Mech. 84, 125143.CrossRefGoogle Scholar
Hauksbee, F. 1710 An account of an experiment touching the direction of a drop of oil of oranges, between two glass planes, towards any side of them that is nearest press’d together. By Mr. Fr. Hauksbee, F. R. S. Phil. Trans. R. Soc. Lond. 27 (325–336), 395396.CrossRefGoogle Scholar
Huh, C. & Scriven, L. E. 1971 Hydrodynamic model of steady movement of a solid/liquid/fluid contact line. J. Colloid Interface Sci. 35 (1), 85101.CrossRefGoogle Scholar
Jacqmin, D. 2000 Contact-line dynamics of a diffuse fluid interface. J. Fluid Mech. 402, 5788.CrossRefGoogle Scholar
Landau, L. & Levich, V. G. 1942 Dragging of a liquid by a moving plate. Acta Physicochim. USSR 17, 4254.Google Scholar
Lo, C.-M., Wang, H.-Bei., Dembo, M. & Wang, Y.-L. 2000 Cell movement is guided by the rigidity of the substrate. Biophys. J. 79 (1), 144152.CrossRefGoogle Scholar
Lorenceau, E. & Quéré, D. 2004 Drops on a conical wire. J. Fluid Mech. 510, 2945.CrossRefGoogle Scholar
Malvadkar, N. A., Hancock, M. J., Sekeroglu, K., Dressick, W. J. & Demirel, M. C. 2010 An engineered anisotropic nanofilm with unidirectional wetting properties. Nat. Mater. 9 (October), 10231028.CrossRefGoogle ScholarPubMed
Maruvada, S. R. K. & Park, C. W. 1996 Retarded motion of bubbles in Hele-Shaw cells. Phys. Fluids 8, 32293233.CrossRefGoogle Scholar
Maxworthy, T. 1986 Bubble formation, motion and interaction in a Hele-Shaw cell. J. Fluid Mech. 173, 95114.CrossRefGoogle Scholar
Metz, T., Paust, N., Zengerle, R. & Koltay, P. 2009 Capillary driven movement of gas bubbles in tapered structures. Microfluid Nanofluid. 9 (2–3), 341355.CrossRefGoogle Scholar
Pallares, G., Grimaldi, A., George, M., Ponson, L. & Ciccotti, M. 2011 Quantitative analysis of crack closure driven by Laplace pressure in silica glass. J. Am. Ceram. Soc. 94 (8), 26132618.CrossRefGoogle Scholar
Prakash, M., Quéré, D. & Bush, J. W. M. 2008 Surface tension transport of prey by feeding shorebirds: the capillary ratchet. Science 320 (5878), 931934.CrossRefGoogle ScholarPubMed
Renvoisé, P., Bush, J. W. M., Prakash, M. & Quéré, D. 2009 Drop propulsion in tapered tubes. Eur. Phys. Lett. 86 (6), 64003.CrossRefGoogle Scholar
Reyssat, M., Pardo, F. & Quéré, D. 2009 Drops onto gradients of texture. Eur. Phys. Lett. 87, 36003.CrossRefGoogle Scholar
Rio, E., Daerr, A., Andreotti, B. & Limat, L. 2005 Boundary conditions in the vicinity of a dynamic contact line: experimental investigation of viscous drops sliding down an inclined plane. Phys. Rev. Lett. 94 (January), 024503.CrossRefGoogle ScholarPubMed
Selva, B., Cantat, I. & Jullien, M.-C. 2011 Temperature-induced migration of a bubble in a soft microcavity. Phys. Fluids 23 (5), 052002.CrossRefGoogle Scholar
Seppecher, P. 1996 Moving contact lines in the Cahn–Hilliard theory. Intl J. Engng Sci. 34 (9), 977992.CrossRefGoogle Scholar
Siekmann, J., Eck, W. & Johann, W. 1974 Experimentelle Untersuchungen ûber das Verhalten von Gasblasen in einem Null- $g$ -Simulator. Z. Flugwiss. 22 (3), 8392.Google Scholar
Style, R. W., Che, Y., Park, S. Ji., Weon, B. Mook., Je, J. Ho., Hyland, C., German, G. K., Rooks, M., Wilen, L. A., Wettlaufer, J. S. & Dufresne, E. R. 2013 Patterning droplets with durotaxis. Proc. Natl Acad. Sci. USA 110, 1254112544.CrossRefGoogle ScholarPubMed
Tanner, L. H. 1979 The spreading of silicone oil drops on horizontal surfaces. J. Phys. D: Appl. Phys. 12, 14731484.CrossRefGoogle Scholar
Thompson, P. A. & Troian, S. M. 1997 A general boundary condition for liquid flow at solid surfaces. Nature 389 (September), 360362.CrossRefGoogle Scholar
Verneuil, E., Cordero, M. L., Gallaire, F. & Baroud, C. N. 2009 Laser-induced force on a microfluidic drop: origin and magnitude. Langmuir 25 (9), 51275134.CrossRefGoogle ScholarPubMed
Wayner, P. C. Jr 1993 Spreading of a liquid film with a finite contact angle by the evaporation/condensation process. Langmuir 9 (14), 294299.CrossRefGoogle Scholar
Weislogel, M. M. 1997 Steady spontaneous capillary flow in partially coated tubes. AIChE J. 43 (3), 645654.CrossRefGoogle Scholar
Weislogel, M. M., Baker, J. A. & Jenson, R. M. 2011 Quasi-steady capillarity-driven flows in slender containers with interior edges. J. Fluid Mech. 685, 271305.CrossRefGoogle Scholar
Weislogel, M. M. & Lichter, S. 1996 A spreading drop in an interior corner: theory and experiment. Microgravity Sci. Technol. 9, 175184.Google Scholar
Zheng, Y., Bai, H., Huang, Z., Tian, X., Nie, F.-Q., Zhao, Y., Zhai, J. & Jiang, L. 2010 Directional water collection on wetted spider silk. Nature 463 (7281), 640643.CrossRefGoogle Scholar
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the or variations. ‘’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Drops and bubbles in wedges
Available formats

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Drops and bubbles in wedges
Available formats

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Drops and bubbles in wedges
Available formats

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *