Hostname: page-component-76fb5796d-2lccl Total loading time: 0 Render date: 2024-04-26T10:07:01.828Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic theory of fluid entrainment in dip coating

Published online by Cambridge University Press:  16 April 2018

Jian Qin  and
Peng Gao Open the ORCID record for Peng Gao [Opens in a new window]
Jian Qin
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230026, China
Peng Gao*
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230026, China State Key Laboratory of Fire Science, University of Science and Technology of China, Hefei, Anhui 230026, China
*
†Email address for correspondence: gaopeng@ustc.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

When a contact line moves with a sufficiently large speed, liquid or gas films can be entrained on a solid depending on the direction of contact-line movement. In this work, the contact-line dynamics in the situation of a generic two-fluid system is investigated. We demonstrate that the hydrodynamics of a contact line, no matter whether advancing or receding, can formally reduce to that of a receding one with small interfacial slopes. Since the latter can be well treated under the classical lubrication approximation, this analogy allows us to derive an asymptotic solution of the interfacial profiles for arbitrary values of contact angle and viscosity ratio. For the dip-coating geometry, we obtain, with no adjustable parameters, an analytical formula for the critical speed of wetting transition, which in particular predicts the onset of both liquid and gas entrainment. Moreover, the present analysis also builds a novel connection between the Cox–Voinov law and classical lubrication theory for moving contact lines.

JFM classification

Interfacial Flows (free surface): Capillary flows Interfacial Flows (free surface): Contact lines Low-Reynolds-number flows: Lubrication theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 844 , 10 June 2018 , pp. 1026 - 1037
Copyright
© 2018 Cambridge University Press 

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

Benkreira, H. & Ikin, J. B. 2010 Dynamic wetting and gas viscosity effects. Chem. Engng Sci. 65, 1790–1796.10.1016/j.ces.2009.11.019Google Scholar
Benkreira, H. & Khan, M. I. 2008 Air entrainment in dip coating under reduced air pressures. Chem. Engng Sci. 63, 448–459.10.1016/j.ces.2007.09.045Google Scholar
Blake, T. D. 2006 The physics of moving wetting lines. J. Colloid Interface Sci. 299, 1–13.10.1016/j.jcis.2006.03.051Google Scholar
Blake, T. D. & Ruschak, K. J. 1979 A maximum speed of wetting. Nature 282, 489–491.10.1038/282489a0Google Scholar
Blake, T. D. & Shikhmurzaev, Y. D. 2002 Dynamic wetting by liquids of different viscosity. J. Colloid Interface Sci. 253, 196–202.10.1006/jcis.2002.8513Google Scholar
Burley, R. & Kennedy, B. S. 1976 Experimental-study of air entrainment at a solid–liquid–gas interface. Chem. Engng Sci. 31, 901–911.10.1016/0009-2509(76)87040-6Google Scholar
Chan, T. S., Snoeijer, J. H. & Eggers, J. 2012 Theory of the forced wetting transition. Phys. Fluids 24, 072104.10.1063/1.4736531Google Scholar
Chan, T. S., Srivastava, S., Marchand, A., Andreotti, B., Biferale, L., Toschi, F. & Snoeijer, J. H. 2013 Hydrodynamics of air entrainment by moving contact lines. Phys. Fluids 25, 074105.10.1063/1.4814466Google Scholar
Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J. & Knuth, D. E. 1996 On the Lambert W function. Adv. Comput. Math. 5, 329–359.10.1007/BF02124750Google Scholar
Cox, R. G. 1986 The dynamics of the spreading of liquids on a solid surface. Part 1. Viscous flow. J. Fluid Mech. 168, 169–194.10.1017/S0022112086000332Google Scholar
Derjaguin, B. V. 1943 Thickness of liquid layer adhering to walls of vessels on their emptying. Acta Physicochim. USSR 20, 349–352.Google Scholar
Duez, C., Ybert, C., Clanet, C. & Bocquet, L. 2007 Making a splash with water repellency. Nat. Phys. 3, 180–183.10.1038/nphys545Google Scholar
Duffy, B. R. & Wilson, S. K. 1997 A third-order differential equation arising in thin-film flows and relevant to Tanner’s law. Appl. Maths Lett. 10, 63–68.10.1016/S0893-9659(97)00036-0Google Scholar
Dussan V, E. B. & Davis, S. H. 1974 On the motion of a fluid–fluid interface along a solid surface. J. Fluid Mech. 65, 71–95.10.1017/S0022112074001261Google Scholar
Eddi, A., Winkels, K. G. & Snoeijer, J. H. 2013 Short time dynamics of viscous drop spreading. Phys. Fluids 25, 013102.10.1063/1.4788693Google Scholar
Eggers, J. 2004 Hydrodynamic theory of forced dewetting. Phys. Rev. Lett. 93, 094502.10.1103/PhysRevLett.93.094502Google Scholar
Eggers, J. 2005 Existence of receding and advancing contact lines. Phys. Fluids 17, 082106.10.1063/1.2009007Google Scholar
Galvagno, M., Tseluiko, D., Lopez, H. & Thiele, U. 2014 Continuous and discontinuous dynamic unbinding transitions in drawn film flow. Phys. Rev. Lett. 112, 137803.10.1103/PhysRevLett.112.137803Google Scholar
Gao, P., Li, L., Feng, J. J., Ding, H. & Lu, X.-Y. 2016 Film deposition and transition on a partially wetting plate in dip coating. J. Fluid Mech. 791, 358–383.10.1017/jfm.2016.64Google Scholar
de Gennes, P.-G., Brochard-Wyart, F. & Quéré, D. 2004 Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls and Waves. Springer.10.1007/978-0-387-21656-0Google Scholar
Hocking, L. M. 2001 Meniscus draw-up and draining. Eur. J. Appl. Maths 12, 195–208.10.1017/S0956792501004247Google Scholar
Huh, C. & Scriven, L. E. 1971 Hydrodynamic model of steady movement of a solid/liquid/fluid contact line. J. Colloid Interface Sci. 35, 85–101.10.1016/0021-9797(71)90188-3Google Scholar
Landau, L. D. & Levich, B. V. 1942 Dragging of a liquid by a moving plate. Acta Physicochim. USSR 17, 42–54.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1984 Fluid Mechanics. Pergamon.Google Scholar
Ledesma-Aguilar, R., Hernández-Machado, A. & Pagonabarraga, I. 2013 Theory of wetting-induced fluid entrainment by advancing contact lines on dry surfaces. Phys. Rev. Lett. 110, 264502.10.1103/PhysRevLett.110.264502Google Scholar
Marchand, A., Chan, T. S., Snoeijer, J. H. & Andreotti, B. 2012 Air entrainment by contact lines of a solid plate plunged into a viscous fluid. Phys. Rev. Lett. 108, 204501.10.1103/PhysRevLett.108.204501Google Scholar
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931–980.10.1103/RevModPhys.69.931Google Scholar
Podgorski, T., Flesselles, J. M. & Limat, L. 2001 Corners, cusps, and pearls in running drops. Phys. Rev. Lett. 87, 036102.10.1103/PhysRevLett.87.036102Google Scholar
Snoeijer, J. H. 2006 Free-surface flows with large slopes: beyond lubrication theory. Phys. Fluids 18, 021701.10.1063/1.2171190Google Scholar
Snoeijer, J. H. & Andreotti, B. 2013 Moving contact lines: scales, regimes, and dynamical transitions. Annu. Rev. Fluid Mech. 45, 269–292.10.1146/annurev-fluid-011212-140734Google Scholar
Snoeijer, J. H., Delon, G., Fermigier, M. & Andreotti, B. 2006 Avoided critical behavior in dynamically forced wetting. Phys. Rev. Lett. 96, 174504.10.1103/PhysRevLett.96.174504Google Scholar
Snoeijer, J. H., Ziegler, J., Andreotti, B., Fermigier, M. & Eggers, J. 2008 Thick films of viscous fluid coating a plate withdrawn from a liquid reservoir. Phys. Rev. Lett. 100, 244502.10.1103/PhysRevLett.100.244502Google Scholar
Sprittles, J. E. 2015 Air entrainment in dynamic wetting: Knudsen effects and the influence of ambient air pressure. J. Fluid Mech. 769, 444–481.10.1017/jfm.2015.121Google Scholar
Sprittles, J. E. 2017 Kinetic effects in dynamic wetting. Phys. Rev. Lett. 118, 114502.10.1103/PhysRevLett.118.114502Google Scholar
Vandre, E., Carvalho, M. S. & Kumar, S. 2012 Delaying the onset of dynamic wetting failure through meniscus confinement. J. Fluid Mech. 707, 496–520.10.1017/jfm.2012.295Google Scholar
Vandre, E., Carvalho, M. S. & Kumar, S. 2013 On the mechanism of wetting failure during fluid displacement along a moving substrate. Phys. Fluids 25, 102103.10.1063/1.4821193Google Scholar
Voinov, O. V. 1976 Hydrodynamics of wetting. Fluid Dyn. 11, 714–721.10.1007/BF01012963Google Scholar
Weinstein, S. J. & Ruschak, K. J. 2004 Coating flows. Annu. Rev. Fluid Mech. 36, 29–53.10.1146/annurev.fluid.36.050802.122049Google Scholar
Xu, L., Zhang, W. W. & Nagel, S. R. 2005 Drop splashing on a dry smooth surface. Phys. Rev. Lett. 94, 184505.10.1103/PhysRevLett.94.184505Google Scholar