On contact-line dynamics with mass transfer. (English) Zbl 1386.80004
Summary: 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.
MSC:
| 80A20 | Heat and mass transfer, heat flow (MSC2010) |
| 76A20 | Thin fluid films |
| 76D45 | Capillarity (surface tension) for incompressible viscous fluids |
Keywords:
thin films; surface-tension driven flows; evaporation; condensation; matched asymptotic expansionsSoftware:
PMADReferences:
| [1] | Ajaev, V. S., Spreading of thin volatile liquid droplets on uniformly heated surfaces, J. Fluid Mech., 528, 279, (2005) · Zbl 1165.76313 · doi:10.1017/S0022112005003320 |
| [2] | Anderson, D. M.; Davis, S. H., The spreading of volatile liquid droplets on heated surfaces, Phys. Fluids, 7, 248, (1995) · Zbl 0843.76016 · doi:10.1063/1.868623 |
| [3] | Bonn, D.; Eggers, J.; Indekeu, J.; Meunier, J.; Rolley, E., Wetting and spreading, Rev. Mod. Phys., 81, 739, (2009) · doi:10.1103/RevModPhys.81.739 |
| [4] | Burelbach, J. P.; Bankoff, S. G.; Davis, S. H., Nonlinear stability of evaporating/condensing liquid films, J. Fluid Mech., 195, 463, (1988) · Zbl 0653.76035 · doi:10.1017/S0022112088002484 |
| [5] | Cachile, M.; Bénichou, O.; Poulard, C.; Cazabat, A. M., Evaporating droplets, Langmuir, 18, 8070, (2002) · doi:10.1021/la0204646 |
| [6] | Cazabat, A. M.; Guéna, G., Evaporation of macroscopic sessile droplets, Soft Matter, 6, 2591, (2010) · doi:10.1039/b924477h |
| [7] | Davis, S. H.; Hocking, L. M., Spreading and imbibition of viscous liquid on a porous base, Phys. Fluids, 12, 1646, (2000) · Zbl 1184.76126 · doi:10.1063/1.870416 |
| [8] | Deegan, R. D.; Bakajin, O.; Dupont, T. F.; Huber, G.; Nagel, S. R.; Witten, T. A., Capillary flow as the cause of ring stains from dried liquid drops, Nature, 389, 827, (1997) · doi:10.1038/39827 |
| [9] | Deegan, R. D.; Bakajin, O.; Dupont, T. F.; Huber, G.; Nagel, S. R.; Witten, T. A., Contact line deposits in an evaporating drop, Phys. Rev. E, 62, 756, (2000) · doi:10.1103/PhysRevE.62.756 |
| [10] | Dunn, G. J.; Wilson, S. K.; Duffy, B. R.; David, S.; Sefiane, K., A mathematical model for the evaporation of a thin sessile liquid droplet: Comparison between experiment and theory, Colloids Surf. A, 323, 50, (2008) · doi:10.1016/j.colsurfa.2007.09.031 |
| [11] | Dunn, G. J.; Wilson, S. K.; Duffy, B. R.; David, S.; Sefiane, K., The strong influence of substrate conductivity on droplet evaporation, J. Fluid Mech., 623, 329, (2009) · Zbl 1157.76303 · doi:10.1017/S0022112008005004 |
| [12] | Eggers, J.; Pismen, L., Nonlocal description of evaporating drops, Phys. Fluids, 22, 112101, (2010) · Zbl 1308.76079 · doi:10.1063/1.3491133 |
| [13] | Erbil, H. Y.; Mchale, G.; Newton, M. I., Drop evaporation on solid surfaces: Constant contact angle mode, Langmuir, 18, 2636, (2002) · doi:10.1021/la011470p |
| [14] | Eriksson, K.; Estep, D.; Hansbo, P.; Johnson, C., Computational Differential Equations, (1996), Cambridge University Press: Cambridge University Press, University Printing House, Cambridge, CB2 8BS, UK · Zbl 0946.65049 |
| [15] | Fried, E.; Jabbour, M., Dynamical equations for the contact line of an evaporating or condensing sessile drop, J. Fluid Mech., 703, 204, (2012) · Zbl 1248.76140 · doi:10.1017/jfm.2012.209 |
| [16] | Gelderblom, H.; Bloemen, O.; Snoeijer, J. H., Stokes flow near the contact line of an evaporating drop, J. Fluid Mech., 709, 69, (2012) · Zbl 1275.76066 · doi:10.1017/jfm.2012.321 |
| [17] | Greenspan, H. P., On the motion of a small viscous droplet that wets a surface, J. Fluid Mech., 84, 125, (1978) · Zbl 0373.76040 · doi:10.1017/S0022112078000075 |
| [18] | Hocking, L. M., A moving fluid interface on a rough surface, J. Fluid Mech., 76, 801, (1976) · Zbl 0352.76026 · doi:10.1017/S0022112076000906 |
| [19] | Hocking, L. M., The spreading of a thin drop by gravity and capillarity, Q. J. Mech. Appl. Math., 36, 55, (1983) · Zbl 0507.76100 · doi:10.1093/qjmam/36.1.55 |
| [20] | Hocking, L. M., On contact angles in evaporating liquids, Phys. Fluids, 7, 2950, (1995) · Zbl 1026.76562 · doi:10.1063/1.868672 |
| [21] | Hocking, L. M.; Rivers, A. D., The spreading of a drop by capillary action, J. Fluid Mech., 121, 425-442, (1982) · Zbl 0492.76101 · doi:10.1017/S0022112082001979 |
| [22] | Huh, C.; Scriven, L. E., Hydrodynamic model of steady movement of a solid/liquid/fluid contact line, J. Colloid Interface Sci., 35, 85, (1971) · doi:10.1016/0021-9797(71)90188-3 |
| [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. · Zbl 1077.76007 |
| [24] | King, J. R.; Bowen, M., Moving boundary problems and non-uniqueness for the thin film equation, Eur. J. Appl. Math., 12, 321, (2001) · Zbl 0985.35113 |
| [25] | King, J. R.; Oliver, J. M., Thin-film modelling of poroviscous free surface flows, Eur. J. Appl. Math., 15, 519, (2005) · Zbl 1112.76021 |
| [26] | Lacey, A. A., The motion with slip of a thin viscous droplet over a solid surface, Stud. Appl. Math., 67, 217, (1982) · Zbl 0505.76112 |
| [27] | Murisic, N.; Kondic, L., Modeling evaporation of sessile drops with moving contact lines, Phys. Rev. E, 78, 065301, (2008) · Zbl 1241.76383 · doi:10.1103/PhysRevE.78.065301 |
| [28] | Murisic, N.; Kondic, L., On evaporation of sessile drops with moving contact lines, J. Fluid Mech., 679, 219, (2011) · Zbl 1241.76383 · doi:10.1017/jfm.2011.133 |
| [29] | Myers, T. G., Thin films with high surface tension, SIAM Rev., 40, 441, (1998) · Zbl 0908.35057 · doi:10.1137/S003614459529284X |
| [30] | Oron, A.; Davis, S. H.; Bankoff, S. G., Long-scale evolution of thin liquid films, Rev. Mod. Phys., 69, 931, (1997) · doi:10.1103/RevModPhys.69.931 |
| [31] | Plawsky, J. L.; Ojha, M.; Chatterjee, A.; Wayner, P. C. Jr, Review of the effects of surface topography, surface chemistry, and fluid physics on evaporation at the contact line, Chem. Eng. Comm., 196, 658, (2008) · doi:10.1080/00986440802569679 |
| [32] | Poulard, C.; Benichou, O.; Cazabat, A. M., Freely receding evaporating droplets, Langmuir, 19, 8828, (2003) · doi:10.1021/la030162j |
| [33] | Poulard, C.; Guéna, G.; Cazabat, A. M.; Boudaoud, A.; Ben Amar, M., Rescaling the dynamics of evaporating drops, Langmuir, 21, 8226, (2005) · Zbl 1161.76464 · doi:10.1021/la050406v |
| [34] | Schwartz, L. W.; Eley, R. R., Simulation of droplet motion on low-energy and heterogeneous surfaces, J. Colloid Interface Sci., 202, 173, (1998) · doi:10.1006/jcis.1998.5448 |
| [35] | Sefiane, K.; Ward, C. A., Recent advances on thermocapillary flows and interfacial conditions during the evaporation of liquids, Adv. Colloid Interface Sci., 134, 201, (2007) |
| [36] | Shahidzadeh-Bonn, N.; Rafaï, S.; Azouni, A.; Bonn, D., Evaporating droplets, J. Fluid Mech., 549, 307, (2006) · doi:10.1017/S0022112005008190 |
| [37] | Shampine, L. F., Accurate numerical derivatives in MATLAB, ACM Trans. Math. Softw., 33, (2007) · Zbl 1365.65057 |
| [38] | Sodtke, C.; Ajaev, V. S.; Stephan, P., Evaporation of thin liquid droplets on heated surfaces, Heat Mass Transfer, 43, 649, (2007) · doi:10.1007/s00231-006-0126-6 |
| [39] | Tanner, L. H., The spreading of silicone oil drops on horizontal surfaces, J. Phys. D: Appl. Phys., 12, 1473, (1979) · doi:10.1088/0022-3727/12/9/009 |
| [40] | Voinov, O. V., Hydrodynamics of wetting, Fluid Dyn., 11, 714, (1976) |
| [41] | Ward, J. P.; King, J. R., Thin-film modelling of biofilm growth and quorum sensing, J. Eng. Math., 73, 71, (2012) · Zbl 1398.76017 · doi:10.1007/s10665-011-9490-4 |
| [42] | Zhornitskaya, L.; Bertozzi, A. L., Positivity-preserving numerical schemes for lubrication-type equations, SIAM J. Numer., 37, 523, (2000) · Zbl 0961.76060 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
