Skip to main content Accessibility help
×
Home

Dynamic drying transition via free-surface cusps

  • Catherine Kamal (a1), James E. Sprittles (a2), Jacco H. Snoeijer (a3) and Jens Eggers (a1)

Abstract

We study air entrainment by a solid plate plunging into a viscous liquid, theoretically and numerically. At dimensionless speeds $Ca=U\unicode[STIX]{x1D702}/\unicode[STIX]{x1D6FE}$ of order unity, a near-cusp forms due to the presence of a moving contact line. The radius of curvature of the cusp’s tip scales with the slip length multiplied by an exponential of $-Ca$ . The pressure from the air flow drawn inside the cusp leads to a bifurcation, at which air is entrained, i.e. there is ‘wetting failure’. We develop an analytical theory of the threshold to air entrainment, which predicts the critical capillary number to depend logarithmically on the viscosity ratio, with corrections coming from the slip in the gas phase.

  • View HTML
    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      Dynamic drying transition via free-surface cusps
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      Dynamic drying transition via free-surface cusps
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      Dynamic drying transition via free-surface cusps
      Available formats
      ×

Copyright

This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

Corresponding author

Email address for correspondence: Jens.Eggers@bristol.ac.uk

Footnotes

Hide All

Present address: School of Engineering and Materials Science, Queen Mary, University of London, Mile End Road, London E1 4NS, UK

Footnotes

References

Hide All
Benilov, E. S. & Vynnycky, M. 2013 Contact lines with a contact angle. J. Fluid Mech. 718, 481506.
Benkreira, H. & Ikin, J. B. 2010 Dynamic wetting and gas viscosity effects. Chem. Engng Sci. 65, 17901796.
Benkreira, H. & Khan, M. I. 2008 Air entrainment in dip coating under reduced air pressures. Chem. Engng Sci. 63, 448459.
Benney, D. J. & Timson, W. J. 1980 The rolling motion of a viscous fluid on and off a rigid surface. Stud. Appl. Maths 63, 9398.
Blake, T. D. & Ruschak, K. J. 1979 A maximum speed of wetting. Nature 282, 489491.
Blake, T. D. & Shikhmurzaev, Y. D. 2002 Dynamic wetting by liquids of different viscosity. J. Colloid Interface Sci. 253, 196202.
Bonn, D., Eggers, J., Indekeu, J., Meunier, J. & Rolley, E. 2009 Wetting and spreading. Rev. Mod. Phys. 81, 739805.
Burley, R. & Kennedy, B. S. 1976 An experimental study of air entrainment at a solid/liquid/gas interface. Chem. Engng Sci. 31, 901911.
Chan, T. S., Eggers, J., Kamal, C. & Snoeijer, J. H.2018 Cox–Voinov theory with slip (unpublished).
Chan, T. S., Snoeijer, J. H. & Eggers, J. 2012 Theory of the forced wetting transition. Phys. Fluids 24, 072104.
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.
Cox, R. G. 1986 The dynamics of the spreading of liquids on a solid surface. Part 1. Viscous flow. J. Fluid Mech. 168, 169194.
Duez, C., Ybert, C., Clanet, C. & Bocquet, L. 2007 Making a splash with water repellency. Nat. Phys. 3, 180183.
Eddi, A., Winkels, K. G. & Snoeijer, J. H. 2013 Short time dynamics of viscous drop spreading. Phys. Fluids 25, 013102.
Eggers, J. 2001 Air entrainment through free-surface cusps. Phys. Rev. Lett. 86, 42904293.
Eggers, J. 2004 Hydrodynamic theory of forced dewetting. Phys. Rev. Lett. 93, 094502.
Eggers, J. 2005 Existence of receding and advancing contact lines. Phys. Fluids 17, 082106.
Eggers, J. & Fontelos, M. A. 2013 Cusps in interfacial problems. Panoramas et Synthèses 38, 6990.
Eggers, J. & Fontelos, M. A. 2015 Singularities: Formation, Structure, and Propagation. Cambridge University Press.
Eggers, J. & Suramlishvili, N. 2017 Singularity theory of plane curves and its applications. Eur. J. Mech. B 65, 107131.
Hocking, L. M. 1977 A moving fluid interface. Part 2. The removal of the force singularity by a slip flow. J. Fluid Mech. 79, 209229.
Huh, C. & Scriven, L. E. 1971 Hydrodynamic model of steady movement of a solid/liquid/fluid contact line. J. Colloid Interface Sci. 35, 85101.
Jacqmin, D. 2002 Very, very fast wetting. J. Fluid Mech. 455, 347358.
Jeong, J.-T. & Moffatt, H. K. 1992 Free-surface cusps associated with flow at low Reynolds number. J. Fluid Mech. 241, 122.
Kiger, K. T. & Duncan, J. H. 2012 Air-entrainment mechanisms in plunging jets and breaking waves. Annu. Rev. Fluid Mech. 44, 563596.
Kistler, S. 1993 Hydrodynamics of wetting. In Wettability (ed. Berg, J. C.), p. 311. Marcel Dekker.
Latka, A., Boelens, A. M. P., Nagel, S. R. & de Pablo, J. J. 2018 Drop splashing is independent of substrate wetting. Phys. Fluids 30, 022105.
Lauga, E., Brenner, M. P. & Stone, H. A. 2008 Microfluidics: the no-slip boundary condition. In Springer Handbook of Experimental Fluid Mechanics (ed. Tropea, C., Foss, J. F. & Yarin, A.), p. 1219. Springer.
Lauga, E. & Squires, T. M. 2005 Brownian motion near a partial-slip boundary: a local probe of the no-slip condition. Phys. Fluids 17, 103102.
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.
Ledesma-Aguilar, R., Nistal, R., Hernández-Machado, A. & Pagonabarraga, I. 2011 Controlled drop emission by wetting properties in driven liquid filaments. Nat. Mater. 10, 367371.
Li, J. 2016 Macroscopic model for head-on binary droplet collisions in a gaseous medium. Phys. Rev. Lett. 117, 214502.
Liu, C.-Y, Vandre, E., Carvalho, M. S. & Kumar, S. 2016 Dynamic wetting failure in surfactant solutions. J. Fluid Mech. 789, 285309.
Lorenceau, É., Restagno, F. & Quéré, D. 2003 Fracture of a viscous liquid. Phys. Rev. Lett. 90, 184501.
Mahadevan, L. & Pomeau, Y. 1999 Rolling droplets. Phys. Fluids 11, 24492453.
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.
Moffatt, H. K. 1964 Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18, 118.
Moffatt, H. K. 1993 Fluid mechanics, topology, cusp singularities and related matters. In Science et Perspective 1er Séminaire International de la Fédération de Mécanique de Grenoble, France, 19–21 May 1992.
Navier, C. L. 1827 Sur les lois du mouvement des fluides. Mem. Acad. R. Sci. France 6, 389440.
Ngan, C. G. & Dussan V., E. B. 1984 The moving contact line with a 180° advancing contact angle. Phys. Fluids 27, 27852787.
Quéré, D. 1999 Fluid coating on a fiber. Annu. Rev. Fluid Mech. 31, 347384.
Snoeijer, J., Le Grand, N., Limat, L., Stone, H. A. & Eggers, J. 2007a Cornered drop and rivulets. Phys. Fluids 19, 042104.
Snoeijer, J. H. 2006 Free-surface flows with large slopes: beyond lubrication theory. Phys. Fluids 18, 021701.
Snoeijer, J. H. & Andreotti, B. 2013 Moving contact lines: scales, regimes, and dynamical transitions. Annu. Rev. Fluid Mech. 45, 269292.
Snoeijer, J. H., Andreotti, B., Delon, G. & Fermigier, M. 2007b Relaxation of a dewetting contact line. Part 1: A full-scale hydrodynamic calculation. J. Fluid Mech. 579, 6383.
Sprittles, J. E. 2015 Air entrainment in dynamic wetting: Knudsen effects and the influence of ambient air pressure. J. Fluid Mech. 769, 444481.
Sprittles, J. E. 2017 Kinetic effects in dynamic wetting. Phys. Rev. Lett. 118, 114502.
Sprittles, J. E. & Shikhmurzaev, Y. D. 2012 Finite element framework for describing dynamic wetting phenomena. Intl J. Numer. Meth. Fluids 68, 12571298.
Sun, C.-T. 2011 Fracture Mechanics. Elsevier.
Vandre, E., Carvalho, M. S. & Kumar, S. 2012 Delaying the onset of dynamic wetting failure through meniscus confinement. J. Fluid Mech. 707, 496520.
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.
Vandre, E., Carvalho, M. S. & Kumar, S. 2014 Characteristics of air entrainment during dynamic wetting failure along a planar substrate. J. Fluid Mech. 747, 119140.
Voinov, O. V. 1976 Hydrodynamics of wetting. Fluid Dyn. 11, 714721.
Weinstein, S. J. & Ruschak, K. J. 2004 Coating flows. Annu. Rev. Fluid Mech. 36, 2953.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed