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Stokes flow near the contact line of an evaporating drop

  • Hanneke Gelderblom (a1), Oscar Bloemen (a1) and Jacco H. Snoeijer (a1)

The evaporation of sessile drops in quiescent air is usually governed by vapour diffusion. For contact angles below , the evaporative flux from the droplet tends to diverge in the vicinity of the contact line. Therefore, the description of the flow inside an evaporating drop has remained a challenge. Here, we focus on the asymptotic behaviour near the pinned contact line, by analytically solving the Stokes equations in a wedge geometry of arbitrary contact angle. The flow field is described by similarity solutions, with exponents that match the singular boundary condition due to evaporation. We demonstrate that there are three contributions to the flow in a wedge: the evaporative flux, the downward motion of the liquid–air interface and the eigenmode solution which fulfils the homogeneous boundary conditions. Below a critical contact angle of , the evaporative flux solution will dominate, while above this angle the eigenmode solution dominates. We demonstrate that for small contact angles, the velocity field is very accurately described by the lubrication approximation. For larger contact angles, the flow separates into regions where the flow is reversing towards the drop centre.

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1. Anderson D. & Davis S. 1995 The spreading of volatile liquid droplets on heated surfaces. Phys. Fluids 7 (2), 248265.
2. Berteloot G., Pham C. T., Daerr A., Lequeux F. & Limat L. 2008 Evaporation-induced flow near a contact line: consequences on coating and contact angle. Europhys. Lett. 83, 14003.
3. Bigioni T. P., Lin X. M., Nguyen T. T., Corwin E. I., Witten T. A. & Jaeger H. M. 2006 Kinetically driven self assembly of highly ordered nanoparticle monolayers. Nature Mater. 5 (4), 265270.
4. Bodiguel H. & Leng J. 2010 Imaging the drying of a colloidal suspension. Soft Matt. 6 (21), 54515460.
5. Brutin D., Sobac B., Loquet B. & Sampol J. 2011 Pattern formation in drying drops of blood. J. Fluid Mech. 667, 8595.
6. Burelbach J. P., Bankoff S. G. & Davis S. H. 1988 Nonlinear stability of evaporating condensing liquid films. J. Fluid Mech. 195, 463494.
7. Cazabat A. M. & Guéna G. 2010 Evaporation of macroscopic sessile droplets. Soft Matt. 6 (12), 25912612.
8. Colinet P. & Rednikov A. 2011 On integrable singularities and apparent contact angles within a classical paradigm. Eur. Phys. J. Spec. Top. 197 (1), 89113.
9. Dean W. R. & Montagnon P. E. 1949 On the steady motion of viscous liquid in a corner. Proc. Camb. Phil. Soc. 45, 389394.
10. Deegan R. D., Bakajin O., Dupont T. F., Huber G., Nagel S. R. & Witten T. A. 1997 Capillary flow as the cause of ring stains from dried liquid drops. Nature 389 (6653), 827828.
11. Deegan R. D., Bakajin O., Dupont T. F., Huber G., Nagel S. R. & Witten T. A. 2000 Contact line deposits in an evaporating drop. Phys. Rev. E 62 (1), 756765.
12. Dufresne E. R., Corwin E. I., Greenblatt N. A., Ashmore J., Wang D. Y., Dinsmore A. D., Cheng J. X., Xie X. S., Hutchinson J. W. & Weitz D. A. 2003 Flow and fracture in drying nanoparticle suspensions. Phys. Rev. Lett. 91 (22), 224501.
13. Eggers J. & Pismen L. M. 2010 Non-local description of evaporating drops. Phys. Fluids 22 (11), 112101.
14. Eral H. B., Augustine D. M., Duits M. H. G. & Mugele F. 2011 Suppressing the coffee stain effect: how to control colloidal self-assembly in evaporating drops using electrowetting. Soft Matt. 7, 15.
15. Fischer B. J. 2002 Particle convection in an evaporating colloidal droplet. Langmuir 18, 6067.
16. Gelderblom H., Marín A. G., Nair H., van Housselt A., Lefferts L., Snoeijer J. H. & Lohse D. 2011 How water droplets evaporate on a superhydrophobic substrate. Phys. Rev. E 83 (2), 026306.
17. Guéna G., Poulard C. & Cazabat A. M. 2007 The leading edge of evaporating droplets. J. Colloid Interface Sci. 312 (1), 164171.
18. Haut B. & Colinet P. 2005 Surface-tension-driven instabilities of a pure liquid layer evaporating into an inert gas. J. Colloid Interface Sci. 285 (1), 296305.
19. Hu H. & Larson R. G. 2002 Evaporation of a sessile droplet on a substrate. J. Phys. Chem. B 106 (6), 13341344.
20. Hu H. & Larson R. G. 2005 Analysis of the microfluidic flow in an evaporating sessile droplet. Langmuir 21 (9), 39633971.
21. Hu H. & Larson R. G. 2006 Marangoni effect reverses coffee-ring depositions. J. Phys. Chem. B 110 (14), 70907094.
22. 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.
23. Marín A. G., Gelderblom H., Lohse D. & Snoeijer J. H. 2011 Order-to-disorder transition in ring-shaped colloidal stains. Phys. Rev. Lett. 107, 085502.
24. Masoud H. & Felske J. D. 2009 Analytical solution for Stokes flow inside an evaporating drop: spherical and cylindrical cap shapes. Phys. Fluids 21, 042102.
25. Michell J. H. 1899 On the direct determination of stress in an elastic solid, with application to the theory of plates. Proc. Lond. Math. Soc. 100 (31), 100124.
26. Moffatt H. K. 1964 Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18, 118.
27. Moffatt H. K. & Duffy B. R. 1980 Local similarity solutions and their limitations. J. Fluid Mech. 96, 299313.
28. Murisic N. & Kondic L. 2011 On evaporation of sessile drops with moving contact lines. J. Fluid Mech. 679, 219246.
29. Petsi A. J. & Burganos V. N. 2008 Stokes flow inside an evaporating liquid line for any contact angle. Phys. Rev. E 78, 036324.
30. Pham C. T., Berteloot G., Lequeux F. & Limat L. 2010 Dynamics of complete wetting liquid under evaporation. Europhys. Lett. 92 (5), 54005.
31. Popov Y. O. 2005 Evaporative deposition patterns: spatial dimensions of the deposit. Phys. Rev. E 71 (3), 036313.
32. Poulard C., Guena G., Cazabat A. M., Boudaoud A. & Ben Amar M. 2005 Rescaling the dynamics of evaporating drops. Langmuir 21 (18), 82268233.
33. Ristenpart W. D., Kim P. G., Domingues C., Wan J. & Stone H. A. 2007 Influence of substrate conductivity on circulation reversal in evaporating drops. Phys. Rev. Lett. 99 (23), 234502.
34. Semenov S., Starov V. M., Velarde M. G. & Rubio R. G. 2011 Droplets evaporation: problems and solutions. Eur. Phys. J. Spec. Top. 197 (1), 265278.
35. Sobac B. & Brutin D. 2011 Triple-line behaviour and wettability controlled by nanocoated substrates: influence on sessile drop evaporation. Langmuir 27 (24), 1499915007.
36. Velikov K. P. 2002 Layer-by-layer growth of binary colloidal crystals. Science 296 (5565), 106109.
37. Yunker P. J., Still T., Lohr M. A. & Yodh A. G. 2011 Suppression of the coffee-ring effect by shape-dependent capillary interactions. Nature 476 (7360), 308311.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
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