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On thin evaporating drops: When is the $d^{2}$-law valid?

Published online by Cambridge University Press:  29 February 2016

M. A. Saxton ,
J. P. Whiteley ,
D. Vella  and
J. M. Oliver
M. A. Saxton
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
J. P. Whiteley
Affiliation:
Department of Computer Science, University of Oxford, Parks Road, Oxford OX1 3QD, UK
D. Vella
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
J. M. Oliver*
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
*
†Email address for correspondence: oliver@maths.ox.ac.uk
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Abstract

We study the evolution of a thin, axisymmetric, partially wetting drop as it evaporates. The effects of viscous dissipation, capillarity, slip and diffusion-dominated vapour transport are taken into account. A matched asymptotic analysis in the limit of small slip is used to derive a generalization of Tanner’s law that takes account of the effect of mass transfer. We find a criterion for when the contact-set radius close to extinction evolves as the square root of the time remaining until extinction – the famous $d^{2}$-law. However, for a sufficiently large rate of evaporation, our analysis predicts that a (slightly different) ‘$d^{13/7}$-law’ is more appropriate. Our asymptotic results are validated by comparison with numerical simulations.

JFM classification

Phase change: Condensation/evaporation Interfacial Flows (free surface): Contact lines Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 792 , 10 April 2016 , pp. 134 - 167
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© 2016 Cambridge University Press 

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References

Abdelrazaq, N. 2006 The solution of heat conduction equation with mixed boundary conditions. J. Math. Stat. 2, 346–350.Google Scholar
Ajaev, V. S. 2005 Spreading of thin volatile liquid droplets on uniformly heated surfaces. J. Fluid Mech. 528, 279–296.Google Scholar
Anderson, D. M. & Davis, S. H. 1995 The spreading of volatile liquid droplets on heated surfaces. Phys. Fluids 7, 248–265.Google Scholar
Bascom, W. D., Cottington, R. L. & Singleterry, C. R. 1963 Dynamic surface phenomena in the spontaneous spreading of oils on solids. Tech. Rep. DTIC Document.Google Scholar
Bonn, D., Eggers, J., Indekeu, J., Meunier, J. & Rolley, E. 2009 Wetting and spreading. Rev. Mod. Phys. 81, 739–805.Google Scholar
Brochard-Wyart, F., di Meglio, J.-M., Quére, D. & de Gennes, P.-G. 1991 Spreading of nonvolatile liquids in a continuum picture. Langmuir 7, 335–338.Google Scholar
Burelbach, J. P., Bankoff, S. G. & Davis, S. H. 1988 Nonlinear stability of evaporating/condensing liquid films. J. Fluid Mech. 195, 463–494.Google Scholar
Cachile, M., Bénichou, O., Poulard, C. & Cazabat, A.-M. 2002 Evaporating droplets. Langmuir 18, 8070–8078.Google Scholar
Cazabat, A.-M. & Guéna, G. 2010 Evaporation of macroscopic sessile droplets. Soft Matt. 6, 2591–2612.Google Scholar
Colinet, P. & Rednikov, A. 2011 On integrable singularities and apparent contact angles within a classical paradigm. Eur. Phys. J. Special Topics 197, 89–113.CrossRefGoogle Scholar
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, 827–829.Google Scholar
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, 756–765.Google Scholar
Dondlinger, M., Margerit, J. & Dauby, P. C. 2005 Weakly nonlinear study of Marangoni instabilities in an evaporating liquid layer. J. Colloid Interface Sci. 283, 522–532.Google Scholar
Dunn, G. J., Wilson, S. K., Duffy, B. R., David, S. & Sefiane, K. 2008 A mathematical model for the evaporation of a thin sessile liquid droplet: comparison between experiment and theory. Colloids Surf. A 323, 50–55.Google Scholar
Dunn, G. J., Wilson, S. K., Duffy, B. R., David, S. & Sefiane, K. 2009 The strong influence of substrate conductivity on droplet evaporation. J. Fluid Mech. 623, 329–351.Google Scholar
Erbil, H. Y. 2012 Evaporation of pure liquid sessile and spherical suspended drops: a review. Adv. Colloid Interface Sci. 170, 67–86.CrossRefGoogle ScholarPubMed
Erbil, H. Y., McHale, G. & Newton, M. I. 2002 Drop evaporation on solid surfaces: constant contact angle mode. Langmuir 18, 2636–2641.Google Scholar
Fried, E. & Jabbour, M. 2012 Dynamical equations for the contact line of an evaporating or condensing sessile drop. J. Fluid Mech. 703, 204–237.Google Scholar
Gelderblom, H., Bloemen, O. & Snoeijer, J. H. 2012 Stokes flow near the contact line of an evaporating drop. J. Fluid Mech. 709, 69–84.Google Scholar
de Gennes, P.-G. 1985 Wetting: statics and dynamics. Rev. Mod. Phys. 57, 827–863.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
Hadjittofis, A., Lister, J. R., Singh, K. & Vella, D.2015 Evaporation effects in elastocapillary aggregation. arXiv:1506.07241.Google Scholar
Haut, B. & Colinet, P. 2005 Surface-tension-driven instabilities of a pure liquid layer evaporating into an inert gas. J. Colloid Interface Sci. 285, 296–305.Google Scholar
Hervet, H. & de Gennes, P.-G. 1984 Dynamique du mouillage: films précurseurs sur solide sec. C. R. Acad. Sci. Paris 299, 499–503.Google Scholar
Hocking, L. M. 1976 A moving fluid interface on a rough surface. J. Fluid Mech. 76, 801–817.Google Scholar
Hocking, L. M. 1983 The spreading of a thin drop by gravity and capillarity. Q. J. Mech. Appl. Maths 36, 55–69.Google Scholar
Hocking, L. M. 1995 On contact angles in evaporating liquids. Phys. Fluids 7, 2950–2955.Google Scholar
Hocking, L. M. & Rivers, A. D. 1982 The spreading of a drop by capillary action. J. Fluid Mech. 121, 425–442.Google Scholar
Hu, H. & Larson, R. G. 2002 Evaporation of a sessile droplet on a substrate. J. Phys. Chem. B 106, 1334–1344.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, 85–101.CrossRefGoogle Scholar
Iwamoto, C. & Tanaka, S.-I. 2002 Atomic morphology and chemical reactions of the reactive wetting front. Acta Mater. 50, 749–755.Google Scholar
King, J. R. 2001 Thin-film flows and high-order degenerate parabolic equations. In IUTAM Symposium on Free Surface Flows, pp. 7–18. Springer.Google Scholar
King, J. R. & Bowen, M. 2001 Moving boundary problems and non-uniqueness for the thin film equation. Eur. J. Appl. Maths 12, 321–356.Google Scholar
Lacey, A. A. 1982 The motion with slip of a thin viscous droplet over a solid surface. Stud. Appl. Maths 67, 217–230.CrossRefGoogle Scholar
Ledesma-Aguilar, R., Vella, D. & Yeomans, J. M. 2014 Lattice-Boltzmann simulations of droplet evaporation. Soft Matt. 10, 8267–8275.Google Scholar
Lide, D. R. 2004 CRC Handbook of Chemistry and Physics, 85th edn. CRC Press.Google Scholar
Machrafi, H., Sadoun, N., Rednikov, A., Dehaeck, S., Dauby, P. C. & Colinet, P. 2013 Evaporation rates and Bénard-Marangoni supercriticality levels for liquid layers under an inert gas flow. Microgravity Sci. Technol. 25, 251–265.Google Scholar
Margerit, J., Dondlinger, M. & Dauby, P. C. 2005 Improved 1.5-sided model for the weakly nonlinear study of Bénard–Marangoni instabilities in an evaporating liquid layer. J. Colloid Interface Sci. 290, 220–230.Google Scholar
Murisic, N. & Kondic, L. 2008 Modeling evaporation of sessile drops with moving contact lines. Phys. Rev. E 78, 065301.Google Scholar
Murisic, N. & Kondic, L. 2011 On evaporation of sessile drops with moving contact lines. J. Fluid Mech. 679, 219–246.Google Scholar
Myers, T. G. 1998 Thin films with high surface tension. SIAM Rev. 40, 441–462.Google Scholar
Oliver, J. M., Whiteley, J. P., Saxton, M. A., Vella, D., Zubkov, V. S. & King, J. R. 2015 On contact-line dynamics with mass transfer. Eur. J. Appl. Maths 26, 671–719.Google Scholar
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931–980.Google Scholar
Plawsky, J. L., Ojha, M., Chatterjee, A. & Wayner, P. C. Jr. 2008 Review of the effects of surface topography, surface chemistry, and fluid physics on evaporation at the contact line. Chem. Engng Commun. 196, 658–696.CrossRefGoogle Scholar
Popov, Y. O. 2005 Evaporative deposition patterns: spatial dimensions of the deposit. Phys. Rev. E 71, 036313.Google Scholar
Poulard, C., Bénichou, O. & Cazabat, A.-M. 2003 Freely receding evaporating droplets. Langmuir 19, 8828–8834.Google Scholar
Poulard, C., Guéna, G., Cazabat, A.-M., Boudaoud, A. & Ben Amar, M. 2005 Rescaling the dynamics of evaporating drops. Langmuir 21, 8226–8233.CrossRefGoogle ScholarPubMed
Sáenz, P. J., Sefiane, K., Kim, J., Matar, O. K. & Valluri, P. 2015 Evaporation of sessile drops: a three-dimensional approach. J. Fluid Mech. 772, 705–739.CrossRefGoogle Scholar
Savva, N., Rednikov, A. & Colinet, P.2014 Asymptotic analysis of evaporating droplets. http://bura.brunel.ac.uk/handle/2438/9469.Google Scholar
Sefiane, K., David, S. & Shanahan, M. E. R. 2008 Wetting and evaporation of binary mixture drops. J. Phys. Chem. B 112, 11317–11323.Google Scholar
Semenov, S., Trybala, A., Rubio, R. G., Kovalchuk, N., Starov, V. & Velarde, M. G. 2014 Simultaneous spreading and evaporation: recent developments. Adv. Colloid Interface Sci. 206, 382–398.Google Scholar
Shahidzadeh-Bonn, N., Rafaï, S., Azouni, A. & Bonn, D. 2006 Evaporating droplets. J. Fluid Mech. 549, 307–313.Google Scholar
Snoeijer, J. H. & Andreotti, B. 2013 Moving contact lines: scales, regimes, and dynamical transitions. Annu. Rev. Fluid Mech. 45, 269–292.Google Scholar
Sodtke, C., Ajaev, V. S. & Stephan, P. 2007 Evaporation of thin liquid droplets on heated surfaces. Heat Mass Transfer 43, 649–657.Google Scholar
Stauber, J. M., Wilson, S. K., Duffy, B. R. & Sefiane, K. 2014 On the lifetimes of evaporating droplets. J. Fluid Mech. 744, R2.Google Scholar
Sultan, E., Boudaoud, A. & Ben Amar, M. 2005 Evaporation of a thin film: diffusion of the vapour and Marangoni instabilities. J. Fluid Mech. 543, 183–202.Google Scholar
Voinov, O. V. 1976 Hydrodynamics of wetting. Fluid Dyn. 11, 714–721.Google Scholar
Weber, H. 1873 Ueber die Besselschen functionen und ihre Anwendung auf die Theorie der elektrischen Ströme. J. Reine Angew. Math. 75, 75–105.Google Scholar