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On contact-line dynamics with mass transfer

Published online by Cambridge University Press:  10 August 2015

J. M. OLIVER ,
J. P. WHITELEY ,
M. A. SAXTON ,
D. VELLA ,
V. S. ZUBKOV  and
J. R. KING
J. M. OLIVER
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK Email: oliver@maths.ox.ac.uk, saxton@maths.ox.ac.uk, vella@maths.ox.ac.uk, zubkov@maths.ox.ac.uk
J. P. WHITELEY
Affiliation:
Department of Computer Science, University of Oxford, Parks Road, Oxford, OX1 3QD, UK Email: jonathan.Whiteley@cs.ox.ac.uk
M. A. SAXTON
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK Email: oliver@maths.ox.ac.uk, saxton@maths.ox.ac.uk, vella@maths.ox.ac.uk, zubkov@maths.ox.ac.uk
D. VELLA
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK Email: oliver@maths.ox.ac.uk, saxton@maths.ox.ac.uk, vella@maths.ox.ac.uk, zubkov@maths.ox.ac.uk
V. S. ZUBKOV
Affiliation:
Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK Email: oliver@maths.ox.ac.uk, saxton@maths.ox.ac.uk, vella@maths.ox.ac.uk, zubkov@maths.ox.ac.uk
J. R. KING
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK Email: John.King@nottingham.ac.uk
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Abstract

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We investigate the effect of mass transfer on the evolution of a thin, two-dimensional, partially wetting drop. While the effects of viscous dissipation, capillarity, slip and uniform mass transfer are taken into account, other effects, such as gravity, surface tension gradients, vapour transport and heat transport, are neglected in favour of mathematical tractability. Our focus is on a matched-asymptotic analysis in the small-slip limit, which reveals that the leading-order outer formulation and contact-line law depend delicately on both the sign and the size of the mass transfer flux. This leads, in particular, to novel generalisations of Tanner's law. We analyse the resulting evolution of the drop on the timescale of mass transfer and validate the leading-order predictions by comparison with preliminary numerical simulations. Finally, we outline the generalisation of the leading-order formulations to prescribed non-uniform rates of mass transfer and to three dimensions.

Keywords

Thin filmssurface-tension driven flowsevaporationcondensationmatched asymptotic expansions
Type
Papers
Information
European Journal of Applied Mathematics , Volume 26 , Special Anniversay Issue 5: Celebrating 75 years , October 2015 , pp. 671 - 719
Copyright
Copyright © Cambridge University Press 2015 

References

[1]Ajaev, V. S. (2005) Spreading of thin volatile liquid droplets on uniformly heated surfaces. J. Fluid Mech. 528, 279.CrossRefGoogle Scholar
[2]Anderson, D. M. & Davis, S. H. (1995) The spreading of volatile liquid droplets on heated surfaces. Phys. Fluids 7, 248.CrossRefGoogle Scholar
[3]Bonn, D., Eggers, J., Indekeu, J., Meunier, J. & Rolley, E. (2009) Wetting and spreading. Rev. Mod. Phys. 81, 739.CrossRefGoogle Scholar
[4]Burelbach, J. P., Bankoff, S. G. & Davis, S. H. (1988) Nonlinear stability of evaporating/condensing liquid films. J. Fluid Mech. 195, 463.CrossRefGoogle Scholar
[5]Cachile, M., Bénichou, O., Poulard, C. & Cazabat, A. M. (2002) Evaporating droplets. Langmuir 18, 8070.CrossRefGoogle Scholar
[6]Cazabat, A. M. & Guéna, G. (2010) Evaporation of macroscopic sessile droplets. Soft Matter 6, 2591.CrossRefGoogle Scholar
[7]Davis, S. H. & Hocking, L. M. (2000) Spreading and imbibition of viscous liquid on a porous base. Phys. Fluids 12 (7), 1646.CrossRefGoogle Scholar
[8]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.CrossRefGoogle Scholar
[9]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.CrossRefGoogle Scholar
[10]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.CrossRefGoogle Scholar
[11]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.CrossRefGoogle Scholar
[12]Eggers, J. & Pismen, L. (2010) Nonlocal description of evaporating drops. Phys. Fluids 22 (11), 112101.CrossRefGoogle Scholar
[13]Erbil, H. Y., McHale, G. & Newton, M. I. (2002) Drop evaporation on solid surfaces: Constant contact angle mode. Langmuir 18, 2636.CrossRefGoogle Scholar
[14]Eriksson, K., Estep, D., Hansbo, P. & Johnson, C. (1996) Computational Differential Equations, Cambridge University Press, University Printing House, Cambridge, CB2 8BS, UK.Google Scholar
[15]Fried, E. & Jabbour, M. (2012) Dynamical equations for the contact line of an evaporating or condensing sessile drop. J. Fluid Mech. 703, 204.CrossRefGoogle Scholar
[16]Gelderblom, H., Bloemen, O. & Snoeijer, J. H. (2012) Stokes flow near the contact line of an evaporating drop. J. Fluid Mech. 709, 69.CrossRefGoogle Scholar
[17]Greenspan, H. P. (1978) On the motion of a small viscous droplet that wets a surface. J. Fluid Mech. 84, 125.CrossRefGoogle Scholar
[18]Hocking, L. M. (1976) A moving fluid interface on a rough surface. J. Fluid Mech. 76 (4), 801.CrossRefGoogle Scholar
[19]Hocking, L. M. (1983) The spreading of a thin drop by gravity and capillarity. Q. J. Mech. Appl. Math. 36, 55.CrossRefGoogle Scholar
[20]Hocking, L. M. (1995) On contact angles in evaporating liquids. Phys. Fluids 7, 2950.CrossRefGoogle Scholar
[21]Hocking, L. M. & Rivers, A. D. (1982) The spreading of a drop by capillary action. J. Fluid Mech. 121, 425–442.CrossRefGoogle Scholar
[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), 85.CrossRefGoogle Scholar
[23]King, J. R. (2001) Thin-film flows and high-order degenerate parabolic equations. In: King, A. C. & Shikhmurzaev, Y. D. (editors), IUTAM Symposium on Free Surface Flows, Kluwer, Dordrecht, pp. 7–18.CrossRefGoogle Scholar
[24]King, J. R. & Bowen, M. (2001) Moving boundary problems and non-uniqueness for the thin film equation. Eur. J. Appl. Math. 12, 321.CrossRefGoogle Scholar
[25]King, J. R. & Oliver, J. M. (2005) Thin-film modelling of poroviscous free surface flows. Eur. J. Appl. Math. 15, 519.CrossRefGoogle Scholar
[26]Lacey, A. A. (1982) The motion with slip of a thin viscous droplet over a solid surface. Stud. Appl. Math. 67, 217.CrossRefGoogle Scholar
[27]Murisic, N. & Kondic, L. (2008) Modeling evaporation of sessile drops with moving contact lines. Phys. Rev. E 78, 065301.CrossRefGoogle ScholarPubMed
[28]Murisic, N. & Kondic, L. (2011) On evaporation of sessile drops with moving contact lines. J. Fluid Mech. 679, 219.CrossRefGoogle Scholar
[29]Myers, T. G. (1998) Thin films with high surface tension. SIAM Rev. 40, 441.CrossRefGoogle Scholar
[30]Oron, A., Davis, S. H. & Bankoff, S. G. (1997) Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931.CrossRefGoogle Scholar
[31]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. Eng. Comm. 196, 658.CrossRefGoogle Scholar
[32]Poulard, C., Benichou, O. & Cazabat, A. M. (2003) Freely receding evaporating droplets. Langmuir 19, 8828.CrossRefGoogle Scholar
[33]Poulard, C., Guéna, G., Cazabat, A. M., Boudaoud, A. & Ben Amar, M. (2005) Rescaling the dynamics of evaporating drops. Langmuir 21, 8226.CrossRefGoogle ScholarPubMed
[34]Schwartz, L. W. & Eley, R. R. (1998) Simulation of droplet motion on low-energy and heterogeneous surfaces. J. Colloid Interface Sci. 202, 173.CrossRefGoogle Scholar
[35]Sefiane, K. & Ward, C. A. (2007) Recent advances on thermocapillary flows and interfacial conditions during the evaporation of liquids. Adv. Colloid Interface Sci. 134, 201.CrossRefGoogle ScholarPubMed
[36]Shahidzadeh-Bonn, N., Rafaï, S., Azouni, A. & Bonn, D. (2006) Evaporating droplets. J. Fluid Mech. 549, 307.CrossRefGoogle Scholar
[37]Shampine, L. F. (2007) Accurate numerical derivatives in MATLAB. ACM Trans. Math. Softw. 33 (4).CrossRefGoogle Scholar
[38]Sodtke, C., Ajaev, V. S. & Stephan, P. (2007) Evaporation of thin liquid droplets on heated surfaces. Heat Mass Transfer 43, 649.CrossRefGoogle Scholar
[39]Tanner, L. H. (1979) The spreading of silicone oil drops on horizontal surfaces. J. Phys. D: Appl. Phys. 12, 1473.CrossRefGoogle Scholar
[40]Voinov, O. V. (1976) Hydrodynamics of wetting. Fluid Dyn. 11, 714.CrossRefGoogle Scholar
[41]Ward, J. P. & King, J. R. (2012) Thin-film modelling of biofilm growth and quorum sensing. J. Eng. Math. 73, 71.CrossRefGoogle Scholar
[42]Zhornitskaya, L. & Bertozzi, A. L. (2000) Positivity-preserving numerical schemes for lubrication-type equations. SIAM J. Numer. 37 (2), 523.CrossRefGoogle Scholar