Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-20T01:00:19.483Z Has data issue: false hasContentIssue false
  • English
  • Français

Healing capillary films

Published online by Cambridge University Press:  16 January 2018

Zhong Zheng Open the ORCID record for Zhong Zheng [Opens in a new window] ,
Marco A. Fontelos ,
Sangwoo Shin ,
Michael C. Dallaston ,
Dmitri Tseluiko ,
Serafim Kalliadasis  and
Howard A. Stone
Zhong Zheng*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0EZ, UK
Marco A. Fontelos*
Affiliation:
Instituto de Ciencias Matemáticas, C/ Nicolás Cabrera, Madrid, 28049, Spain
Sangwoo Shin
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA Department of Mechanical Engineering, University of Hawaii at Manoa, Honolulu, HI 96822, USA
Michael C. Dallaston
Affiliation:
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK
Dmitri Tseluiko
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK
Serafim Kalliadasis
Affiliation:
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK
Howard A. Stone*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
*
†Email addresses for correspondence: [email protected], [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Consider the dynamics of a healing film driven by surface tension, that is, the inward spreading process of a liquid film to fill a hole. The film is modelled using the lubrication (or thin-film) approximation, which results in a fourth-order nonlinear partial differential equation. We obtain a self-similar solution describing the early-time relaxation of an initial step-function condition and a family of self-similar solutions governing the finite-time healing. The similarity exponent of this family of solutions is not determined purely from scaling arguments; instead, the scaling exponent is a function of the finite thickness of the prewetting film, which we determine numerically. Thus, the solutions that govern the finite-time healing are self-similar solutions of the second kind. Laboratory experiments and time-dependent computations of the partial differential equation are also performed. We compare the self-similar profiles and exponents, obtained by matching the estimated prewetting film thickness, with both measurements in experiments and time-dependent computations near the healing time, and we observe good agreement in each case.

JFM classification

Interfacial Flows (free surface): Capillary flows Interfacial Flows (free surface): Contact lines Interfacial Flows (free surface): Thin films
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 838 , 10 March 2018 , pp. 404 - 434
Check for updates
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.)

Footnotes

‡

Z. Zheng and M. A. Fontelos contributed equally to this work.

§

Present address: School of Computing, Electronics and Mathematics, and Flow Measurement and Fluid Mechanics Research Centre, Coventry University, Coventry CV1 5FB, UK.

References

Backholm, M., Benzaquen, M., Salez, T., Raphael, E. & Dalnoki-Veress, K. 2014 Capillary leveling of a cylindrical hole in a viscous film. Soft Matt. 10, 2550–2558.CrossRefGoogle Scholar
Bankoff, S. G., Johnson, M. F. G., Miksis, M. J., Schluter, R. A. & López, P. G. 2003 Dynamics of a dry spot. J. Fluid Mech. 486, 239–259.CrossRefGoogle Scholar
Barenblatt, G. I.1979 Similarity, Self-Similarity, and Intermediate Asymptotics. Consultants Bureau.CrossRefGoogle Scholar
Bender, C. M. & Orszag, S. 1978 Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill.Google Scholar
Bertozzi, A. L., Brenner, M. P., Dupont, T. F. & Kadanoff, L. P. 1994 Trends and perspectives in applied mathematics. In Applied Mathematical Sciences, pp. 155–208. Springer.Google Scholar
Bischofberger, I., Ramachandran, R. & Nagel, S. R. 2014 Fingering versus stability in the limit of zero interfacial tension. Nat. Commun. 5, 5265.Google Scholar
Blossey, R. 2003 Self-cleaning surfaces: virtual realities. Nat. Mater. 2, 301–306.Google Scholar
Bonn, D., Eggers, J., Indekeu, J., Meunier, J. & Rolley, E. 2009 Wetting and spreading. Rev. Mod. Phys. 81, 739–805.Google Scholar
Bostwick, J. B., Dijksman, J. A. & Shearer, M. 2017 Wetting dynamics of a collapsing fluid hole. Phys. Rev. Fluids 2, 014006.Google Scholar
Brenner, M. P., Lister, J. R. & Stone, H. A. 1996 Pinching threads, singularities and the number 0.0304. Phys. Fluids 8, 2827–2836.Google Scholar
Bretherton, F. P. 1961 The motion of long bubbles in a tube. J. Fluid Mech. 10, 166–188.Google Scholar
Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 1131–1198.Google Scholar
Dallaston, M. C., Tseluiko, D., Zheng, Z., Fontelos, M. A. & Kalliadasis, S. 2017 Self-similar finite-time singularity formation in degenerate parabolic equations arising in thin-film flows. Nonlinearity 30, 2647–2666.Google Scholar
Diez, J. A., Gratton, R. & Gratton, J. 1992 Self-similar solution of the second kind for a convergent viscous gravity current. Phys. Fluids A 6, 1148–1155.Google Scholar
Dijksman, J. A., Mukhopadhyay, S., Gaebler, C., Witelski, T. P. & Behringer, R. P. 2015 Obtaining self-similar scalings in focusing flows. Phys. Rev. E 92, 043016.Google Scholar
Doedel, E. J., Champneys, R., Dercole, F., Fairgrieve, T. F., Kuznetsov, Yu. A., Oldeman, B., Paffenroth, R. C., Sandstede, B., Wang, X. J. & Zhang, C. H.2007 Auto 07p: continuation and bifurcation software for ordinary differential equations. Montreal Concordia University; http://indy.cs.concordia.ca/auto/.Google Scholar
Duffy, B. R. & Wilson, S. K. 1996 A third-order differential equation arising in thin-film flows and relevant to Tanners law. Appl. Maths Lett. 10, 63–68.Google Scholar
Eggers, J. 1993 Universal pinching of 3D axisymmetric free surface flow. Phys. Rev. Lett. 71, 3458–3460.CrossRefGoogle ScholarPubMed
Eggers, J. 2004 Toward a description of contact line motion at higher capillary numbers. Phys. Fluids 16, 3491–3494.Google Scholar
Eggers, J. & Fontelos, M. A. 2009 The role of self-similarity in singularities of partial differential equations. Nonlinearity 22, R1–R44.CrossRefGoogle Scholar
Eggers, J. & Fontelos, M. A. 2015 Singularities: Formation, Structure, and Propagation. Cambridge University Press.CrossRefGoogle Scholar
Feng, J., Roche, M., Vigolo, D., Arnaudov, L. N., Stoyanov, S. D., Tsutsumanova, G. G. & Stone, H. A. 2014 Nanoemulsions obtained via bubble-bursting at a compound interface. Nat. Phys. 10, 606–612.CrossRefGoogle Scholar
de Gennes, P. G., Hua, X. & Levinson, P. 1990 Dynamics of wetting: local contact angle. J. Fluid Mech. 212, 55–63.Google Scholar
Gratton, J. & Minotti, F. 1990 Self-similar viscous gravity currents: phase plane formalism. J. Fluid Mech. 210, 155–182.CrossRefGoogle Scholar
Herminghaus, S., Brinkmann, M. & Seemann, R. 2008 Wetting and dewetting of complex surface geometries. Annu. Rev. Mater. Sci. 38, 101–121.Google Scholar
Hocking, L. M. 1983 The spreading of a thin drop by gravity and capillarity. Q. J. Mech. Appl. Maths 36 (Feb), 55–69.Google Scholar
Huppert, H. E. & Woods, A. W. 1995 Gravity driven flows in porous layers. J. Fluid Mech. 292, 55–69.Google Scholar
Jensen, O. E. 1994 Self-similar, surfactant-driven flows. Phys. Fluids 6, 1084–1094.Google Scholar
Kalliadasis, S., Bielarz, C. & Homsy, G. M. 2000 Steady free-surface thin film flows over topography. Phys. Fluids 12, 1889–1898.Google Scholar
Kataoka, D. E. & Troian, S. M. 1997 A theoretical study of instabilities at the advancing front of thermally driven coating films. J. Colloid Interface Sci. 192, 350–362.CrossRefGoogle ScholarPubMed
Landau, L. & Levich, B. 1942 Dragging of a liquid by a moving plate. Acta Physicochim. URSS 17, 42–54.Google Scholar
Levy, R. & Shearer, M. 2004 Comparison of two dynamic contact line models for driven thin liquid films. Eur. J. Appl. Maths 15, 625–642.Google Scholar
Lister, J. R. & Kerr, R. C. 1989 The propagation of two-dimensional and axisymmetric viscous gravity currents at a fluid interface. J. Fluid Mech. 203, 215–249.Google Scholar
Lister, J. R. & Stone, H. A. 1998 Capillary breakup of a viscous thread surrounded by another viscous fluid. Phys. Fluids 11, 2758–2764.CrossRefGoogle Scholar
López, P. G., Miksis, M. J. & Bankoff, S. G. 2001 Stability and evolution of a dry spot. Phys. Fluids 13, 1601–1614.Google Scholar
McGraw, J. D., Salez, T., Baumchen, O., Raphael, E. & Dalnoki-Veress, K. 2012 Self-similarity and energy dissipation in stepped polymer films. Phys. Rev. Lett. 109, 128303.Google Scholar
Moriarty, J. A. & Schwartz, L. W. 1993 Dynamic considerations in the closing and opening of holes in thin liquid films. J. Colloid Interface Sci. 161, 335–342.CrossRefGoogle Scholar
Myers, T. G. 1998 Thin films with high surface tension. SIAM Rev. 40, 441–462.CrossRefGoogle Scholar
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931–980.CrossRefGoogle Scholar
Padday, J. F. 1971 The profile of axially symmetric menisci. Phil. Trans. R. Soc. Lond. A 269, 265–293.Google Scholar
Papageorgiou, D. T. 1995 On the breakup of viscous liquid threads. Phys. Fluids 7, 1529–1544.Google Scholar
Savva, N. & Kalliadasis, S. 2009 Two-dimensional droplet spreading over topographical substrates. Phys. Fluids 21, 092102.Google Scholar
Savva, N. & Kalliadasis, S. 2011 Dynamics of moving contact lines: a comparison between slip and precursor film models. Europhys. Lett. 94, 64004.Google Scholar
Sharma, A. & Ruckenstein, E. 1990 Energetic criteria for the breakup of liquid films on nonwetting solid surfaces. J. Colloid Interface Sci. 137, 433–445.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
Stone, H. A. & Duprat, C.2016 Low-Reynolds-number flows. In RSC Soft Matter Series, chap. 2, pp. 25–77. Royal Society of Chemistry (RSC).Google Scholar
Tanner, L. H. 1979 The spreading of silicone oil drops on horizontal surfaces. J. Phys. D: Appl. Phys. 12, 1473–1485.Google Scholar
Tseluiko, D., Baxter, J. & Thiele, U. 2013 A homotopy continuation approach for analysing finite-time singularities in thin liquid films. IMA J. Appl. Maths 78, 762–776.CrossRefGoogle Scholar
Voinov, O. V. 1976 Hydrodynamics of wetting. Fluid Dyn. 11, 714–721.Google Scholar
Witelski, T. P. & Bernoff, A. J. 1999 Stability of self-similar solutions for van der Waals driven thin film rupture. Phys. Fluids 11, 2443–2445.Google Scholar
Yatsyshin, P., Parry, A. O. & Kalliadasis, S. 2016 Complete prewetting. J. Phys.: Condens. Matter 28, 275001.Google Scholar
Yatsyshin, P., Savva, N. & Kalliadasis, S. 2015 Wetting of prototypical one-and two-dimensional systems: thermodynamics and density functional theory. J. Chem. Phys. 142, 034708.Google Scholar
Zhang, W. W. & Lister, J. R. 1999 Similarity solutions for van der Waals rupture of a thin film on a solid substrate. Phys. Fluids 9, 2454–2462.Google Scholar
Zheng, Z., Christov, I. C. & Stone, H. A. 2014 Influence of heterogeneity on second-kind self-similar solutions for viscous gravity currents. J. Fluid Mech. 747, 218–246.Google Scholar
Zheng, Z., Shin, S. & Stone, H. A. 2015 Converging gravity currents over a permeable substrate. J. Fluid Mech. 778, 669–690.Google Scholar