Hostname: page-component-8448b6f56d-cfpbc Total loading time: 0 Render date: 2024-04-16T04:00:28.465Z Has data issue: false hasContentIssue false
  • English
  • Français

Propagation of capillary waves and ejection of small droplets in rapid droplet spreading

Published online by Cambridge University Press:  12 March 2012

H. Ding ,
E. Q. Li ,
F. H. Zhang ,
Y. Sui ,
P. D. M. Spelt  and
S. T. Thoroddsen
H. Ding
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, 230027 Hefei, China Department of Chemical Engineering, University of California at Santa Barbara, Santa Barbara, CA 93106-5080, USA
E. Q. Li
Affiliation:
Department of Mechanical Engineering, National University of Singapore, 119260 Singapore Division of Physical Sciences and Engineering & CCRC, King Abdullah University of Science and Technology, Thuwal, 23955-6900, Saudi Arabia
F. H. Zhang
Affiliation:
Department of Mechanical Engineering, National University of Singapore, 119260 Singapore Singapore-MIT Alliance, National University of Singapore, 117576 Singapore
Y. Sui
Affiliation:
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK
P. D. M. Spelt*
Affiliation:
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK Département Mécanique, Université de Lyon 1, and Laboratoire de la Mécanique des Fluides & d’Acoustique, CNRS, 69134 Ecully, France
S. T. Thoroddsen
Affiliation:
Division of Physical Sciences and Engineering & CCRC, King Abdullah University of Science and Technology, Thuwal, 23955-6900, Saudi Arabia
*
†Email address for correspondence: peter.spelt@univ-lyon1.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

A new regime of droplet ejection following the slow deposition of drops onto a near-complete wetting solid substrate is identified in experiments and direct numerical simulations; a coalescence cascade subsequent to pinch-off is also observed for the first time. Results of numerical simulations indicate that the propagation of capillary waves that lead to pinch-off is closely related to the self-similar behaviour observed in the inviscid recoil of droplets, and that motions of the crests and troughs of capillary waves along the interface do not depend on the wettability and surface tension (or Ohnesorge number). The simulations also show that a self-similar theory for universal pinch-off can be used for the time evolution of the pinching neck. However, although good agreement is also found with the double-cone shape of the pinching neck for droplet ejection in drop deposition on a pool of the same liquid, substantial deviations are observed in such a comparison for droplet ejection in rapid drop spreading (including the newly identified regime). This deviation is shown to result from interference by the solid substrate, a rapid downwards acceleration of the top of the drop surface and the rapid spreading process. The experiments also confirm non-monotonic spreading behaviour observed previously only in numerical simulations, and suggest substantial inertial effects on the relation between an apparent contact angle and the dimensionless contact-line speed.

JFM classification

Drops and Bubbles: Drops Waves/Free-surface Flows: Capillary waves Interfacial Flows (free surface): Contact lines
Type
Papers
Information
Journal of Fluid Mechanics , Volume 697 , 25 April 2012 , pp. 92 - 114
Copyright
Copyright © Cambridge University Press 2012

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

1. Biance, A.-L., Clanet, C. & Queré, D. 2004 First steps in the spreading of a liquid droplet. Phys. Rev. E 69, 016301.CrossRefGoogle ScholarPubMed
2. Billingham, J. 1999 Surface-tension-driven flow in fat fluid wedges and cones. J. Fluid Mech. 397, 45–71.CrossRefGoogle Scholar
3. Blanchette, F. & Bigioni, T. P. 2006 Partial coalescence of drops at liquid interfaces. Nature Phys. 2, 254–257.CrossRefGoogle Scholar
4. Blanchette, F. & Bigioni, T. P. 2009 Dynamics of drop coalescence at fluid interfaces. J. Fluid Mech. 620, 333–352.CrossRefGoogle Scholar
5. Cox, R. G. 1986 The dynamics of the spreading of liquids on a solid surface. Part 1. Viscous flow. J. Fluid Mech. 168, 169.CrossRefGoogle Scholar
6. Day, R. F., Hinch, E. J. & Lister, J. R. 1998 Self-similar capillary pinch off of an inviscid fluid. Phys. Rev. Lett. 80, 704–707.Google Scholar
7. Ding, H. & Spelt, P. D. M. 2007a Inertial effects in droplet spreading: a comparison between diffuse interface and level-set simulations. J. Fluid Mech. 576, 287–296.CrossRefGoogle Scholar
8. Ding, H. & Spelt, P. D. M. 2007b Wetting condition in diffuse interface simulation of contact line motion. Phys. Rev. E 75, 046708.CrossRefGoogle ScholarPubMed
9. Ding, H., Spelt, P. D. M. & Shu, C. 2007 Diffuse interface model for incompressible two-phase flows with large density ratios. J. Comput. Phys. 226, 2078–2095.CrossRefGoogle Scholar
10. Ding, H. & Spelt, P. D. M. 2008 Onset of motion of a 3D droplet on a wall in shear flow at moderate Reynolds numbers. J. Fluid Mech. 599, 341–362.CrossRefGoogle Scholar
11. Ding, H., Gilani, M. N. H. & Spelt, P. D. M. 2010 Sliding, pinch off and detachment of a droplet on a wall in shear flow. J. Fluid Mech. 644, 217–244.CrossRefGoogle Scholar
12. Doshi, P., Cohen, I., Zhang, W. W., Siegel, M., Howell, P., Basaran, O. A. & Nagel, S. R. 2003 Persistence of memory in drop breakup: the breakdown of universality. Science 302, 1185–1188.CrossRefGoogle ScholarPubMed
13. Eggers, J. 1997 Nonlinear dynamics and breakup of free-surface flows. Rev. Mod. Phys. 69, 865–929.Google Scholar
14. 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
15. de Gennes, P. G. 1985 Wetting: Statistics and dynamics. Rev. Mod. Phys. 57, 827–863.CrossRefGoogle Scholar
16. Gilet, T., Mulleners, K., Lecomte, J. P., Vandewalle, N. & Dorbolo, S. 2007 Critical parameters for the partial coalescence of a droplet. Phys. Rev. E 75, 036303.CrossRefGoogle ScholarPubMed
17. Goriely, A. & McMillen, T. 2002 Shape of a cracking whip. Phys. Rev. Lett. 88, 244301.CrossRefGoogle ScholarPubMed
18. Hocking, L. M. 1983 The spreading of a thin drop by gravity and capillarity. Q. J. Mech. Appl. Maths 36, 55–69.Google Scholar
19. Jacqmin, D. 1999 Calculation of two-phase Navier–Stokes flows using phase-field modelling. J. Comput. Phys. 155, 96–127.CrossRefGoogle Scholar
20. Jacqmin, D. 2000 Contact-line dynamics of a diffuse fluid interface. J. Fluid Mech. 402, 57.CrossRefGoogle Scholar
21. Jang, E. Y., Seo, D. K., Kim, T., Kang, T. J. & Kim, Y. H. 2010 Electrical resistance variation of carbon-nanotube networks due to surface modification of glass substrate. Phys. Status Solidi A 1.Google Scholar
22. Keller, J. B. & Miksis, M. J. 1983 Surface tension driven flows. SIAM J. Appl. Maths 43, 268–277.CrossRefGoogle Scholar
23. King, A. C. 1991 Moving contact lines in slender fuid wedges. Q. J. Mech. Appl. Maths 44, 173–192.Google Scholar
24. Lawrie, J. B. 1990 Surface-tension-driven flow in a wedge. Q. J. Mech. Appl. Maths 43, 251–273.CrossRefGoogle Scholar
25. Lister, J. R. & Stone, H. A. 1999 Capillary breakup of a viscous thread surrounded by another viscous fluid. Phys. Fluids 10, 2758–2764.CrossRefGoogle Scholar
26. MacNeice, P., Olson, K. M., Mobarry, C., deFainchtein, R. & Packer, C. 2000 PARAMESH: a parallel adaptive mesh refinement community toolkit. Comput. Phys. Commun. 126, 330–354.CrossRefGoogle Scholar
27. Manasseh, R., Riboux, G. & Risso, F. 2008 Sound generation on bubble coalescence following detachment. Intl J. Multiphase Flow 34, 938–949.CrossRefGoogle Scholar
28. Marsh, J. A., Garoff, S. & Dussan, V. E. B. 1993 Dynamic contact angles and hydrodynamics near a moving contact line. Phys. Rev. Lett. 70, 2778–2781.Google Scholar
29. Rayleigh, L. 1879 On the capillary phenomena of jets. Proc. R. Soc. 29, 71–97.Google Scholar
30. Renardy, Y., Popinet, S., Duchemin, L., Renardy, M., Zaleski, S., Josserand, C., Drumright-Clarke, M. A., Richard, D., Clanet, C. & Quéré, 2003 Pyramidal and toroidal water drops after impact on a solid surface. J. Fluid Mech. 484, 69–83.CrossRefGoogle Scholar
31. 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, 024503.Google Scholar
32. Rioboo, R., Adao, M. H., Voué, M. & De Coninck, J. 2006 Experimental evidence of liquid drop breakup in complete wetting experiments. J. Mater. Sci. 41, 5068–5080.Google Scholar
33. Roux, D. C. D. & Cooper-White, J. J. 2004 Dynamics of water spreading on a glass surface. J. Colloid Interface Sci. 277, 424–436.Google Scholar
34. Sierou, A. & Lister, J. R. 2004 Self-similar recoil of inviscid drops. Phys. Fluids 16, 1379–1394.Google Scholar
35. Shaw, S. J. & Spelt, P. D. M. 2010 Shock emission from collapsing gas bubbles. J. Fluid Mech. 646, 363–373.Google Scholar
36. Spelt, P. D. M. 2005 A level-set approach for simulations of flows with multiple moving contact lines with hysteresis. J. Comput. Phys. 207, 389–404.CrossRefGoogle Scholar
37. Tanner, L. H. 1979 The spreading of silicone oil drops on horizontal surfaces. J. Phys. D 12, 1473–1483.Google Scholar
38. Thoroddsen, S. T., Etoh, T. G. & Takehara, K. 2003 Air entrapment under an impacting drop. J. Fluid Mech. 478, 125–134.CrossRefGoogle Scholar
39. Thoroddsen, S. T., Etoh, T. G. & Takehara, K. 2007a Microjetting from wave focusing on oscillating drops. Phys. Fluids 19, 152101.CrossRefGoogle Scholar
40. Thoroddsen, S. T., Qian, B., Etoh, T. G. & Takehara, K. 2007b The initial coalescence of miscible drops. Phys. Fluids 19, 072110.CrossRefGoogle Scholar
41. Thoroddsen, S. T. & Takehara, K. 2000 The coalescence cascade of a drop. Phys. Fluids 12, 1265–1267.Google Scholar
42. Wheeler, D., Warren, J. A. & Boettinger, W. J. 2010 Modelling the early stages of reactive wetting. Phys. Rev. E 82, 051601.CrossRefGoogle ScholarPubMed
43. Yarin, A. L. 2006 Droplet impact dynamics: splashing, spreading, receding, bouncing. Annu. Rev. Fluid Mech. 38, 159–192.CrossRefGoogle Scholar
44. Yiantsios, S. G. & Davis, R. H. 1990 On the buoyancy-driven motion of a drop towards a rigid or a deformable surface. J. Fluid Mech. 217, 547–573.CrossRefGoogle Scholar
45. Zhang, F. H. & Thoroddsen, S. T. 2008 Satellite generation during bubble coalescence. Phys. Fluids 20, 022104.CrossRefGoogle Scholar
46. Zhang, F. H., Li, E. Q. & Thoroddsen, S. T. 2009 Satellite formation during coalescence of unequal size drops. Phys. Rev. Lett. 102, 104502.Google Scholar

Ding et al. supplementary movie

Movie 1. Typical sequence of a second-stage pinchoff for a drop of water and glycerin mixture. Oh=0.009, We=0.026 and a static angle of 13 degrees.

Download Ding et al. supplementary movie(Video)
Video 3.1 MB

Ding et al. supplementary movie

Movie 1. Typical sequence of a second-stage pinchoff for a drop of water and glycerin mixture. Oh=0.009, We=0.026 and a static angle of 13 degrees.

Download Ding et al. supplementary movie(Video)
Video 426.7 KB

Ding et al. supplementary movie

Movie 2. Six-stage coalescence cascade after 1st-stage pinchoff for Oh=0.006, We=0.033 and a static angle of 12+/- 2 degrees.

Download Ding et al. supplementary movie(Video)
Video 45.5 MB

Ding et al. supplementary movie

Movie 2. Six-stage coalescence cascade after 1st-stage pinchoff for Oh=0.006, We=0.033 and a static angle of 12+/- 2 degrees.

Download Ding et al. supplementary movie(Video)
Video 1.1 MB

Ding et al. supplementary movie

Movie 3. Propagation of the capillary wave from the contact line (right end), in terms of the radial coordinate at the drop surface as a function of the polar angle, as defined in figure 4. Oh=0.008, We=0.016, contact angle is 30 degrees.

Download Ding et al. supplementary movie(Video)
Video 17.2 MB

Ding et al. supplementary movie

Movie 3. Propagation of the capillary wave from the contact line (right end), in terms of the radial coordinate at the drop surface as a function of the polar angle, as defined in figure 4. Oh=0.008, We=0.016, contact angle is 30 degrees.

Download Ding et al. supplementary movie(Video)
Video 1.5 MB