Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-28T18:49:36.002Z Has data issue: false hasContentIssue false
  • English
  • Français

Drops and bubbles in wedges

Published online by Cambridge University Press:  06 May 2014

Etienne Reyssat
Etienne Reyssat*
Affiliation:
PMMH, CNRS UMR 7636 – ESPCI – UPMC Université Paris 6 – UPD Université Paris 7, 10 rue Vauquelin, 75005 Paris, France
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

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 $t_0-t$.

JFM classification

Interfacial Flows (free surface): Capillary flows Drops and Bubbles: Drops and Bubbles Low-Reynolds-number flows: Hele-Shaw flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 748 , 10 June 2014 , pp. 641 - 662
Copyright
© 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.)

References

Bico, J. & Quéré, D. 2001 Falling slugs. J. Colloid Interface Sci. 243, 262–264.Google Scholar
Bonn, D., Eggers, J., Indekeu, J., Meunier, J. & Rolley, E. 2009 Wetting and spreading. Rev. Mod. Phys. 81 (2), 739–805.Google 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), 107–109.Google Scholar
Brochard, F. 1989 Motion of droplets on solid surfaces induced by chemical or thermal gradients. Langmuir 5 (3), 432–438.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, 283–303.CrossRefGoogle Scholar
Cantat, I. 2013 Liquid meniscus friction on a wet plate: bubbles, lamellae, and foams. Phys. Fluids 25 (3), 031303.Google Scholar
Chaudhury, M. K. & Whitesides, G. M. 1992 How to make water run uphill. Science 256 (5063), 1539–1541.Google Scholar
Dangla, R., Kayi, S. C. & Baroud, C. N. 2013 Droplet microfluidics driven by gradients of confinement. Proc. Natl Acad. Sci. USA 110 (3), 853–858.Google Scholar
Daniel, S., Sircar, S., Gliem, J. & Chaudhury, M. K. 2004 Ratcheting motion of liquid drops on gradient surfaces. Langmuir 20 (10), 4085–4092.CrossRefGoogle ScholarPubMed
de Gennes, P.-G. 1985 Wetting: statics and dynamics. Rev. Mod. Phys. 57 (3), 827–863.Google Scholar
de Gennes, P.-G., Brochard-Wyart, F. & Quéré, D. 2004 Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls and Waves. Springer.Google Scholar
Domingues dos Santos, F. & Ondarçuhu, T. 1995 Free-running droplets. Phys. Rev. Lett. 75 (16), 2972–2975.Google 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, 153–168.Google 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, 309–321.Google Scholar
Eri, A. & Okumura, K. 2011 Viscous drag friction acting on a fluid drop confined in between two plates. Soft Matt. 7 (12), 5648–5653.Google Scholar
Genzer, J. & Bhat, R. R. 2008 Surface-bound soft matter gradients. Langmuir 24, 2294–2317.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), 879–884.Google Scholar
Greenspan, H. P. 1978 On the motion of a small viscous droplet that wets a surface. J. Fluid Mech. 84, 125–143.Google 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), 395–396.Google 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), 85–101.Google Scholar
Jacqmin, D. 2000 Contact-line dynamics of a diffuse fluid interface. J. Fluid Mech. 402, 57–88.Google Scholar
Landau, L. & Levich, V. G. 1942 Dragging of a liquid by a moving plate. Acta Physicochim. USSR 17, 42–54.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), 144–152.CrossRefGoogle ScholarPubMed
Lorenceau, E. & Quéré, D. 2004 Drops on a conical wire. J. Fluid Mech. 510, 29–45.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), 1023–1028.Google Scholar
Maruvada, S. R. K. & Park, C. W. 1996 Retarded motion of bubbles in Hele-Shaw cells. Phys. Fluids 8, 3229–3233.Google Scholar
Maxworthy, T. 1986 Bubble formation, motion and interaction in a Hele-Shaw cell. J. Fluid Mech. 173, 95–114.Google Scholar
Metz, T., Paust, N., Zengerle, R. & Koltay, P. 2009 Capillary driven movement of gas bubbles in tapered structures. Microfluid Nanofluid. 9 (2–3), 341–355.Google 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), 2613–2618.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), 931–934.Google Scholar
Renvoisé, P., Bush, J. W. M., Prakash, M. & Quéré, D. 2009 Drop propulsion in tapered tubes. Eur. Phys. Lett. 86 (6), 64003.Google Scholar
Reyssat, M., Pardo, F. & Quéré, D. 2009 Drops onto gradients of texture. Eur. Phys. Lett. 87, 36003.Google 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.Google Scholar
Selva, B., Cantat, I. & Jullien, M.-C. 2011 Temperature-induced migration of a bubble in a soft microcavity. Phys. Fluids 23 (5), 052002.Google Scholar
Seppecher, P. 1996 Moving contact lines in the Cahn–Hilliard theory. Intl J. Engng Sci. 34 (9), 977–992.Google Scholar
Siekmann, J., Eck, W. & Johann, W. 1974 Experimentelle Untersuchungen ûber das Verhalten von Gasblasen in einem Null- $g$ -Simulator. Z. Flugwiss. 22 (3), 83–92.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, 12541–12544.Google Scholar
Tanner, L. H. 1979 The spreading of silicone oil drops on horizontal surfaces. J. Phys. D: Appl. Phys. 12, 1473–1484.Google Scholar
Thompson, P. A. & Troian, S. M. 1997 A general boundary condition for liquid flow at solid surfaces. Nature 389 (September), 360–362.Google 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), 5127–5134.Google Scholar
Wayner, P. C. Jr 1993 Spreading of a liquid film with a finite contact angle by the evaporation/condensation process. Langmuir 9 (14), 294–299.Google Scholar
Weislogel, M. M. 1997 Steady spontaneous capillary flow in partially coated tubes. AIChE J. 43 (3), 645–654.Google 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, 271–305.Google Scholar
Weislogel, M. M. & Lichter, S. 1996 A spreading drop in an interior corner: theory and experiment. Microgravity Sci. Technol. 9, 175–184.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), 640–643.Google Scholar