Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-17T09:16:54.817Z Has data issue: false hasContentIssue false
  • English
  • Français

Film deposition and transition on a partially wetting plate in dip coating

Published online by Cambridge University Press:  22 February 2016

Peng Gao ,
Lei Li ,
James J. Feng ,
Hang Ding  and
Xi-Yun Lu
Peng Gao*
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230026, China
Lei Li
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230026, China
James J. Feng
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada Department of Chemical and Biological Engineering, University of British Columbia, Vancouver, British Columbia V6T 1Z3, Canada
Hang Ding
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230026, China
Xi-Yun Lu
Affiliation:
Department of Modern Mechanics, University of Science and Technology of China, Hefei, Anhui 230026, China
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the entrainment of liquid films on a partially wetting plate vertically withdrawn from a reservoir of viscous liquid using a combination of diffuse-interface numerical simulation and lubrication analysis. So far available theoretical investigations were commonly conducted by focusing on separate parameter regions, and a complete description of the flow regimes with increasing plate speed is still missing. By solving the full Stokes equations, we present a complete scenario of film transition in the presence of moving contact line. With increasing plate speed, we identify numerically four successive flow regimes in terms of the interfacial morphologies: (1) a stationary meniscus, (2) a speed-independent thick film connected to the liquid bath through a stationary dimple, (3) coexistence of a thick film and the classical Landau–Levich–Derjaguin (LLD) film connected by a propagating capillary shock and (4) a film with a monotonically varying thickness. The characteristics of the film profiles in different regions of the interfaces are analysed with lubrication theory as applicable, and satisfactory agreements with the numerical results are obtained. In particular, we confirm that the onset of film deposition occurs at a vanishing apparent contact angle, consistent with the predictions of lubrication theory. Numerical results suggest that the critical capillary number for the onset of film deposition is smaller than that for the onset of LLD film despite the fact that it is higher than the experimentally observed one, showing that the thick film can be realized in the two-dimensional model. We also demonstrated that the LLD film is triggered by the bifurcation of the stationary dimple, which is found to admit multiple branches of stable and unstable solutions.

JFM classification

Interfacial Flows (free surface): Capillary flows Interfacial Flows (free surface): Contact lines Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 791 , 25 March 2016 , pp. 358 - 383
Check for updates
Copyright
© 2016 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

Ahmadlouydarab, M. & Feng, J. J. 2014 Motion and coalescence of sessile drops driven by substrate wetting gradient and external flow. J. Fluid Mech. 746, 214–235.CrossRefGoogle Scholar
Benilov, E. S., Benilov, M. S. & O’Brien, S. B. G. 2009 Existence and stability of regularized shock solutions, with applications to rimming flows. J. Engng Maths 63, 197–212.CrossRefGoogle Scholar
Benilov, E. S., Chapman, S. J., Mcleod, J. B., Ockendon, J. R. & Zubkov, V. S. 2010 On liquid films on an inclined plate. J. Fluid Mech. 663, 53–69.CrossRefGoogle Scholar
Bertozzi, A. L., Munch, A., Fanton, X. & Cazabat, A. M. 1998 Contact line stability and ‘undercompressive shocks’ in driven thin film flow. Phys. Rev. Lett. 81, 5169–5172.Google Scholar
Blake, T. D. 2006 The physics of moving wetting lines. J. Colloid Interface Sci. 299, 1–13.Google Scholar
Blake, T. D. & Ruschak, K. J. 1979 Maximum speed of wetting. Nature 282, 489–491.CrossRefGoogle Scholar
Bonn, D., Eggers, J., Indekeu, J., Meunier, J. & Rolley, E. 2009 Wetting and spreading. Rev. Mod. Phys. 81, 739–805.CrossRefGoogle Scholar
Chan, T. S., Snoeijer, J. H. & Eggers, J. 2012 Theory of the forced wetting transition. Phys. Fluids 24, 072104.Google 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.Google 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.Google Scholar
Delon, G., Fermigier, M., Snoeijer, J. H. & Andreotti, B. 2008 Relaxation of a dewetting contact line. Part 2. Experiments. J. Fluid Mech. 604, 55–75.Google 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
Derjaguin, B. V. & Levi, S. M. 1964 Film Coating Theory. Focal.Google Scholar
Ding, H. & Spelt, P. D. M. 2007 Inertial effects in droplet spreading: a comparison between diffuse-interface and level-set simulations. J. Fluid Mech. 576, 287–296.CrossRefGoogle Scholar
Ding, H. & Spelt, P. D. M. 2008 Onset of motion of a three-dimensional droplet on a wall in shear flow at moderate Reynolds numbers. J. Fluid Mech. 599, 341–362.Google Scholar
Dussan, E. B. & Davis, S. H. 1974 Motion of a fluid–fluid interface along a solid-surface. J. Fluid Mech. 65, 71–95.Google Scholar
Eggers, J. 2004 Hydrodynamic theory of forced dewetting. Phys. Rev. Lett. 93, 094502.Google Scholar
Eggers, J. 2005 Existence of receding and advancing contact lines. Phys. Fluids 17, 082106.Google Scholar
Evans, P. L. & Münch, A. 2006 Interaction of advancing fronts and meniscus profiles formed by surface-tension-gradient-driven liquid films. SIAM J. Appl. Maths 66, 1610–1631.Google 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.CrossRefGoogle ScholarPubMed
Gao, P. & Feng, J. J. 2011a A numerical investigation of the propulsion of water walkers. J. Fluid Mech. 668, 363–383.Google Scholar
Gao, P. & Feng, J. J. 2011b Spreading and breakup of a compound drop on a partially wetting substrate. J. Fluid Mech. 682, 415–433.Google Scholar
Gao, P., Li, L. & Lu, X.-Y 2015 Dewetting films with inclined contact lines. Phys. Rev. E 91, 023008.Google Scholar
Gao, P. & Lu, X.-Y 2013 On the wetting dynamics in a Couette flow. J. Fluid Mech. 724, R1.Google Scholar
de Gennes, P.-G., Brochard-Wyart, F. & Quéré, D. 2004 Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls and Waves. Springer.Google Scholar
Hocking, L. M. 2001 Meniscus draw-up and draining. Eur. J. Appl. Maths 12, 195–208.CrossRefGoogle 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.Google Scholar
Jacqmin, D. 2000 Contact-line dynamics of a diffuse fluid interface. J. Fluid Mech. 402, 57–88.Google Scholar
Jacqmin, D. 2004 Onset of wetting failure in liquid–liquid systems. J. Fluid Mech. 517, 209–228.Google 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
Le Grand, N., Daerr, A. & Limat, L. 2005 Shape and motion of drops sliding down an inclined plane. J. Fluid Mech. 541, 293–315.CrossRefGoogle Scholar
Maleki, M., Reyssat, E., Quéré, D. & Golestanian, R. 2007 On the Landau–Levich transition. Langmuir 23, 10116–10122.CrossRefGoogle ScholarPubMed
Maleki, M., Reyssat, M., Restagno, F., Quéré, D. & Clanet, C. 2011 Landau–Levich menisci. J. Colloid Interface Sci. 354, 359–363.Google 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.Google Scholar
Münch, A. & Evans, P. L. 2005 Marangoni-driven liquid films rising out of a meniscus onto a nearly-horizontal substrate. Physica D 209, 164–177.Google Scholar
Oron, A., Davis, S. H. & Bankoff, G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931–980.Google Scholar
Petrov, J. G. & Sedev, R. V. 1985 On the existence of a maximum speed of wetting. Colloids Surf. 13, 313–322.Google Scholar
Podgorski, T., Flesselles, J. M. & Limat, L. 2001 Corners, cusps, and pearls in running drops. Phys. Rev. Lett. 87, 036102.Google ScholarPubMed
Qian, T., Wang, X.-P. & Sheng, P. 2006 A variational approach to moving contact line hydrodynamics. J. Fluid Mech. 564, 333–360.Google Scholar
Quéré, D. 1991 On the minimal velocity of forced spreading in partial wetting. C. R. Acad. Sci. Paris II 313, 313–318.Google Scholar
Rio, E., Daerr, A., Andreotti, B. & Limat, L. 2005 Boundary conditions in the vicinity of a dynamic contact line: experimental investigation of viscous drops sliding down an inclined plane. Phys. Rev. Lett. 94, 024503.CrossRefGoogle ScholarPubMed
Ruschak, K. J. 1985 Coating flows. Annu. Rev. Fluid Mech. 17, 65–89.Google Scholar
Sedev, R. V. & Petrov, J. G. 1991 The critical condition for transition from steady wetting to film entrainment. Colloids Surf. 53, 147–156.Google Scholar
Snoeijer, J. H. 2006 Free-surface flows with large slopes: beyond lubrication theory. Phys. Fluids 18, 021701.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
Snoeijer, J. H., Andreotti, B., Delon, G. & Fermigier, M. 2007 Relaxation of a dewetting contact line. Part 1. A full-scale hydrodynamic calculation. J. Fluid Mech. 579, 63–83.CrossRefGoogle Scholar
Snoeijer, J. H., Delon, G., Fermigier, M. & Andreotti, B. 2006 Avoided critical behavior in dynamically forced wetting. Phys. Rev. Lett. 96, 174504.Google 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.CrossRefGoogle ScholarPubMed
Spiers, R. P., Subbaraman, C. V. & Wilkinson, W. L. 1974 Free coating of a Newtonian liquid onto a vertical surface. Chem. Engng Sci. 29, 389–396.CrossRefGoogle Scholar
Srivastava, S., Perlekar, P., Biferale, L., Sbragaglia, M., Boonkkamp, J. H. M., ten Thije & Toschi, F. 2013 A study of fluid interfaces and moving contact lines using the lattice Boltzmann method. Commun. Comput. Phys. 13, 725–740.Google Scholar
Sui, Y., Ding, H. & Spelt, P. D. M. 2014 Numerical simulations of flows with moving contact lines. Annu. Rev. Fluid Mech. 46, 97–119.Google Scholar
Tseluiko, D., Galvagno, M. & Thiele, U. 2014 Collapsed heteroclinic snaking near a heteroclinic chain in dragged meniscus problems. Eur. Phys. J. E 37, 33.Google 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.Google Scholar
Voinov, O. V. 1976 Hydrodynamics of wetting. Fluid Dyn. 11, 714–721.Google Scholar
Weinstein, S. J. & Ruschak, K. J. 2004 Coating flows. Annu. Rev. Fluid Mech. 36, 29–53.Google Scholar
White, D. A. & Tallmadge, J. A. 1965 Theory of drag out of liquids on flat plates. Chem. Engng Sci. 20, 33–37.Google Scholar
Wilson, S. D. R. 1982 The drag-out problem in film coating theory. J. Engng Maths 16, 209–221.Google Scholar
Yue, P. T., Feng, J. J., Liu, C. & Shen, J. 2004 A diffuse-interface method for simulating two-phase flows of complex fluids. J. Fluid Mech. 515, 293–317.CrossRefGoogle Scholar
Yue, P. T., Zhou, C. F. & Feng, J. J. 2010 Sharp-interface limit of the Cahn–Hilliard model for moving contact lines. J. Fluid Mech. 645, 279–294.Google Scholar
Yue, P. T., Zhou, C. F., Feng, J. J., Ollivier-Gooch, C. F. & Hu, H. H. 2006 Phase-field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing. J. Comput. Phys. 219, 47–67.Google Scholar
Zhou, C. F., Yue, P. T. & Feng, J. J. 2010 3D phase-field simulations of interfacial dynamics in Newtonian and viscoelastic fluids. J. Comput. Phys. 229, 498–511.Google Scholar
Ziegler, J., Snoeijer, J. H. & Eggers, J. 2009 Film transitions of receding contact lines. Eur. Phys. J. Special Topics 166, 177–180.Google Scholar