Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T19:54:57.051Z Has data issue: false hasContentIssue false
  • English
  • Français

The dynamics of three-dimensional liquid bridges with pinned and moving contact lines

Published online by Cambridge University Press:  02 August 2012

Shawn Dodds ,
Marcio S. Carvalho  and
Satish Kumar
Shawn Dodds
Affiliation:
Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455, USA
Marcio S. Carvalho*
Affiliation:
Department of Mechanical Engineering, Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, RJ 22453-900, Brazil
Satish Kumar*
Affiliation:
Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455, USA
*
†Email addresses for correspondence: kumar030@umn.edu, msc@puc-rio.br
†Email addresses for correspondence: kumar030@umn.edu, msc@puc-rio.br
Get access Rights & Permissions [Opens in a new window]

Abstract

Liquid bridges with moving contact lines are relevant in a variety of natural and industrial settings, ranging from printing processes to the feeding of birds. While it is often assumed that the liquid bridge is two-dimensional in nature, there are many applications where either the stretching motion or the presence of a feature on a bounding surface lead to three-dimensional effects. To investigate this we solve Stokes equations using the finite-element method for the stretching of a three-dimensional liquid bridge between two flat surfaces, one stationary and one moving. We first consider an initially cylindrical liquid bridge that is stretched using either a combination of extension and shear or extension and rotation, while keeping the contact lines pinned in place. We find that whereas a shearing motion does not alter the distribution of liquid between the two plates, rotation leads to an increase in the amount of liquid resting on the stationary plate as breakup is approached. This suggests that a relative rotation of one surface can be used to improve liquid transfer to the other surface. We then consider the extension of non-cylindrical bridges with moving contact lines. We find that dynamic wetting, characterized through a contact line friction parameter, plays a key role in preventing the contact line from deviating significantly from its original shape as breakup is approached. By adjusting the friction on both plates it is possible to drastically improve the amount of liquid transferred to one surface while maintaining the fidelity of the liquid pattern.

JFM classification

Mathematical Foundations: Computational methods Interfacial Flows (free surface): Contact lines Interfacial Flows (free surface): Liquid bridges
Type
Papers
Information
Journal of Fluid Mechanics , Volume 707 , 25 September 2012 , pp. 521 - 540
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. Ahmed, D. H., Sung, H. J. & Kim, D.-S. 2011 Simulation of non-Newtonian ink transfer between two separating plates for gravure-offset printing. Intl J. Heat Fluid Flow 32, 298–307.CrossRefGoogle Scholar
2. Ambravaneswaran, B. & Basaran, O. A. 1999 Effects of insoluble surfactants on the nonlinear deformation and breakup of stretching liquid bridges. Phys. Fluids 11, 997–1015.CrossRefGoogle Scholar
3. Amestoy, P. R., Duff, I. S., Koster, J. & L’Excellent, J. Y. 2001 A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. A 23, 15–41.CrossRefGoogle Scholar
4. Baer, T. A., Cairncross, R. A., Schunk, P. R., Rao, R. R. & Sackinger, P. A. 2000 A finite element method for free surface flows of incompressible fluids in three dimensions. Part II. Dynamic wetting lines. Intl J. Numer. Meth. Fluids 33, 405–427.3.0.CO;2-4>CrossRefGoogle Scholar
5. Balu, B., Berry, A. D., Hess, D. W. & Breedveld, V. 2009 Patterning of superhydrophobic paper to control the mobility of micro-liter drops for two-dimensional lab-on-paper applications. Lab on a Chip 9, 3066–3075.CrossRefGoogle ScholarPubMed
6. Blake, T. D. 2006 The physics of moving wetting lines. J. Colloid Interface Sci. 299, 1–13.Google ScholarPubMed
7. Bonito, A., Picasso, M. & Laso, M. 2006 Numerical simulation of 3D viscoelastic flows with free surfaces. J. Comput. Phys. 215, 691–716.CrossRefGoogle Scholar
8. Bonn, D., Eggers, J., Indekeu, J., Meunier, J. & Rolley, E. 2009 Wetting and spreading. Rev. Mod. Phys. 81, 739–805.CrossRefGoogle Scholar
9. Cairncross, R. A., Schunk, P. R., Baer, T. A., Rao, R. R. & Sackinger, P. A. 2000 A finite element method for free surface flows of incompressible fluids in three dimensions. Part I. Boundary fitted mesh motion. Intl J. Numer. Meth. Fluids 33, 375–403.3.0.CO;2-O>CrossRefGoogle Scholar
10. Christodoulou, K. N., Kistler, S. F. & Schunk, P. R. 1997 Advances in computational methods for free-surface flows. In Liquid Film Coating (ed. Kistler, S. F. & Schweizer, P. M. ), pp. 297–366. Chapman Hall.CrossRefGoogle Scholar
11. Chuang, H.-K., Lee, C.-C. & Liu, T.-J. 2008 An experimental study on the pickout of scaled up gravure cells. Intern. Polymer Processing xxiii, 216–222.Google Scholar
12. Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 1131–1198.Google Scholar
13. Darhuber, A. A., Troian, S. M. & Wagner, S. 2001 Physical mechanisms governing pattern fidelity in microscale offset printing. J. Appl. Phys. 90, 3602–3609.Google Scholar
14. Dimitrakopoulos, P. & Higdon, J. J. L. 2003 On the displacement of fluid bridges from solid surfaces in viscous pressure-driven flows. Phys. Fluids 15, 3255–3258.Google Scholar
15. Dodds, S., Carvalho, M. & Kumar, S. 2009 Stretching and slipping of liquid bridges near plates and cavities. Phys. Fluids 21, 092103.CrossRefGoogle Scholar
16. Dodds, S., Carvalho, M. S. & Kumar, S. 2011a Stretching liquid bridges with bubbles: the effect of air bubbles on liquid transfer. Langmuir 27, 1556–1559.CrossRefGoogle ScholarPubMed
17. Dodds, S., Carvalho, M. S. & Kumar, S. 2011b Stretching liquid bridges with moving contact lines: the role of inertia. Phys. Fluids 23, 092101.CrossRefGoogle Scholar
18. Eggers, J. 1997 Nonlinear dynamics and breakup of free-surface flows. Rev. Mod. Phys. 69, 865–929.CrossRefGoogle Scholar
19. Eggers, J. & Villermaux, E. 2008 Physics of liquid jets. Rep. Prog. Phys. 71, 036601.CrossRefGoogle Scholar
20. Gaudet, S., McKinley, G. H. & Stone, H. A. 1996 Extensional deformation of Newtonian liquid bridges. Phys. Fluids 8, 2568–2579.Google Scholar
21. Ghadiri, F., Ahmed, D. H., Sung, H. J. & Shirani, E. 2011 Non-Newtonian ink transfer in gravure-offset printing. Intl J. Heat Fluid Flow 32, 308–317.CrossRefGoogle Scholar
22. Gresho, P. M., Lee, R. L. & Sani, R. L. 1980 On the time-dependent solution of the incompressible Navier–Stokes equations in two and three dimensions. In Recent Advances in Numerical Methods in Fluids (ed. Taylor, C. & Morgan, K. ), pp. 27–79. Pineridge.Google Scholar
23. Gupta, C., Mensing, G. A., Shannon, M. A. & Kenis, P. J. A. 2007 Double transfer printing of small volumes of liquids. Langmuir 23, 2906–2914.CrossRefGoogle ScholarPubMed
24. Hagberg, J., Pudas, M., Leppävuori, S., Elsey, K. & Logan, A. 2001 Gravure offset printing development for fine line thick film circuits. Microelectron. Int. 18, 32–35.CrossRefGoogle Scholar
25. Hanumanthu, R. 1996 Patterned roll coating. PhD thesis, University of Minnesota.Google Scholar
26. Hoda, N. & Kumar, S. 2008 Boundary integral simulations of liquid emptying from a model gravure cell. Phys. Fluids 20, 092106.CrossRefGoogle Scholar
27. Huang, W.-X., Lee, S.-H., Sung, H. J., Lee, T.-M. & Kim, D.-S. 2008 Simulation of liquid transfer between separating walls for modelling micro-gravure-offset printing. Intl J. Heat Fluid Flow 29, 1436–1446.Google Scholar
28. Huh, C. & Scriven, L. E. 1971 Hydrodynamic model of a steady movement of a solid/liquid/fluid contact line. J. Colloid Interface Sci. 35, 85–101.CrossRefGoogle Scholar
29. Jung, S., Reis, P. M., James, J., Clanet, C. & Bush, J. W. M. 2009 Capillary origami in nature. Phys. Fluids 21, 091110.CrossRefGoogle Scholar
30. Kang, H. W., Sung, H. J., Lee, T.-M., Kim, D.-S. & Kim, C.-J. 2009 Liquid transfer between two separating plates for micro-gravure-offset printing. J. Micromech. Microengng 19, 015025.CrossRefGoogle Scholar
31. Kapur, N. 2003 A parametric study of direct gravure coating. Chem. Engng Sci. 58, 2875–2882.CrossRefGoogle Scholar
32. Kistler, S. F. & Scriven, L. E. 1984 Coating flow theory by finite element and asymptotic analysis of the Navier–Stokes system. Intl J. Numer. Meth. Fluids 4, 207–229.Google Scholar
33. Ku, H. C., Hirsh, R. S. & Taylor, T. D. 1987 A pseudospectral method for solution of the three-dimensional incompressible Navier–Stokes equations. J. Comput. Phys. 70, 439–462.CrossRefGoogle Scholar
34. Lenz, R. D. & Kumar, S. 2007 Competitive displacement of thin liquid films on chemically patterned substrates. J. Fluid Mech. 571, 33–57.CrossRefGoogle Scholar
35. Lockwood, A. J., Anantheshwara, K., Bobji, M. S. & Inkson, B. J. 2011 Friction-formed liquid drops. Nanotechnology 22, 105703.CrossRefGoogle Scholar
36. McKinley, G. H. & Sridhar, T. 2002 Filament-stretching rheometry of complex fluids. Annu. Rev. Fluid Mech. 34, 375–415.CrossRefGoogle Scholar
37. Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931–980.CrossRefGoogle Scholar
38. Powell, C. A., Savage, M. D. & Guthrie, J. T. 2002 Computational simulation of the printing of Newtonian liquid from a trapezoidal cavity. Intl J. Numer. Meth. Heat Fluid Flow 12, 338–355.CrossRefGoogle Scholar
39. Prakash, M., Quéré, D. & Bush, J. W. M. 2008 Surface tension transport of prey by feeding shorebirds: the capillary ratchet. Science 320, 931–934.CrossRefGoogle ScholarPubMed
40. Puah, L. S., Sedev, R., Fornasiero, D., Ralston, J. & Blake, T. 2010 Influence of surface charge on wetting kinetics. Langmuir 26, 17218–17224.CrossRefGoogle ScholarPubMed
41. Pudas, M., Hagberg, J. & Leppavuori, S. 2004 Printing parameters and ink components affecting ultra-fine-line gravure-offset printing for electronics applications. J. Eur. Ceram. Soc. 24, 2943–2950.CrossRefGoogle Scholar
42. Pudas, M., Halonen, N., Granat, P. & Vähäkangas, J. 2005 Gravure printing of conductive particulare polymer inks on flexible substrates. Prog. Org. Coat. 54, 310–316.CrossRefGoogle Scholar
43. Py, C., Reverdy, P., Doppler, L., Bico, J., oman, B. & Baroud, C. N. 2007 Capillary origami: spontaneous wrapping of a droplet with an elastic sheet. Phys. Rev. Lett. 98, 156103.CrossRefGoogle ScholarPubMed
44. Qian, B. & Breuer, K. S. 2011 The motion, stability and breakup of a stretching liquid bridge with a receding contact line. J. Fluid Mech. 666, 554–572.CrossRefGoogle Scholar
45. Qian, B., Loureiro, M., Gagnon, D. A., Tripathi, A. & Breuer, K. S. 2009 Micron-scale droplet deposition on a hydrophobic surface using a retreating syringe. Phys. Rev. Lett. 102, 164502.CrossRefGoogle ScholarPubMed
46. Ranabothu, S. R., Karnezis, C. & Dai, L. L. 2005 Dynamic wetting: hydrodynamic or molecular-kinetic?. J. Colloid Interface Sci. 288, 213–221.CrossRefGoogle ScholarPubMed
47. Rasmussen, H. K. & Hassager, O. 1999 Three-dimensional simulations of viscoelastic instability in polymeric filaments. J. Non-Newtonian Fluid Mech. 82, 189–202.CrossRefGoogle Scholar
48. Reis, P. M., Jung, S., Aristoff, J. M. & Stocker, R. 2010 How cats lap: water uptake by Felis catus . Science 330, 1231–1234.CrossRefGoogle ScholarPubMed
49. Saha, A. A. & Mitra, S. K. 2009 Effect of dynamic contact angle in a volume of fluid (VOF) model for a microfluidic capillary flow. J. Colloid Interface Sci. 339, 461–480.CrossRefGoogle Scholar
50. Scardovelli, R. & Zaleski, S. 1999 Direct numerical simulation of free-surface and interfacial flow. Annu. Rev. Fluid Mech. 31, 567–603.CrossRefGoogle Scholar
51. Schwartz, L. W. 2002 Numerical modelling of liquid withdrawl from gravure cavities in coating operations: the effect of cell pattern. J. Engng Maths 42, 243–253.CrossRefGoogle Scholar
52. Schwartz, L. W., Moussalli, P., Campbell, P. & Eley, R. R. 1998 Numerical modelling of liquid withdrawl from gravure cavities in coating operations. Trans. Inst. Chem. Engrs. 76, 22–28.CrossRefGoogle Scholar
53. Sethian, J. A. & Smereka, P. 2003 Level set methods for fluid interfaces. Annu. Rev. Fluid Mech. 35, 341–372.CrossRefGoogle Scholar
54. Sprittles, J. E. & Shikhmurzaev, Y. D. 2011 Finite element framework for describing dynamic wetting phenomena. Intl J. Numer. Meth. Fluids 68, 1257–1298.Google Scholar
55. Stone, H. A. 2010 Interfaces: in fluid mechanics and across disciplines. J. Fluid Mech. 645, 1–25.CrossRefGoogle Scholar
56. Villaneuva, W., Sjödahl, J., Stjernström, M., Roeraade, J. & Amberg, G. 2007 Microdroplet deposition under a liquid medium. Langmuir 23, 1171–1177.CrossRefGoogle Scholar
57. Vogel, M. J. & Steen, P. H. 2010 Capillarity-based switchable adhesion. Proc. Natl Acad. Sci. 107, 3377–3381.CrossRefGoogle ScholarPubMed
58. Walkley, M. A., Gaskell, P. H., Jimack, P. K., Kelmanson, M. A. & Summers, J. L. 2005a Finite element simulation of three-dimensional free-surface flow problems. J. Sci. Comput. 24, 147–162.CrossRefGoogle Scholar
59. Walkley, M. A., Gaskell, P. H., Jimack, P. K., Kelmanson, M. A. & Summers, J. L. 2005b Finite element simulation of three-dimensional free-surface flow problems with dynamic contact lines. Intl J. Numer. Meth. Fluids 47, 1353–1359.CrossRefGoogle Scholar
60. Weinstein, S. J. & Ruschak, K. J. 2004 Coating flows. Annu. Rev. Fluid Mech. 36, 29–53.CrossRefGoogle Scholar
61. Xie, X., Musson, L. C. & Pasquali, M. 2007 An isochoric domain deformation method for computing steady free surface flows with conserved volumes. J. Comput. Phys. 226, 398–413.CrossRefGoogle Scholar
62. Yao, M. & McKinley, G. H. 1998 Numerical simulation of extensional deformations of viscoelastic liquid bridges in filament stretching devices. J. Non-Newtonian Fluid Mech. 74, 47–54.CrossRefGoogle Scholar
63. Yildirim, O. E. & Basaran, O. A. 2001 Deformation and breakup of stretching bridges of Newtonian and shear-thinning liquids: comparison of one- and two-dimensional models. Chem. Engng Sci. 56, 211–233.CrossRefGoogle Scholar
64. Zhang, X., Padgett, R. S. & Basaran, O. A. 1996 Nonlinear deformation and breakup of stretching liquid bridges. J. Fluid Mech. 329, 207–245.CrossRefGoogle Scholar
65. Zhou, C., Yue, P., Feng, J. J., Ollivier-Gooch, C. F. & Hu, H. H. 2010 3D phase-field simulations of interfacial dynamics in Newtonian and viscoelastic fluids. J. Comput. Phys. 229, 498–511.CrossRefGoogle Scholar