Skip to main content Accessibility help
Hostname: page-component-55597f9d44-zdfhw Total loading time: 0.449 Render date: 2022-08-15T00:39:06.314Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

Healing capillary films

Published online by Cambridge University Press:  16 January 2018

Zhong Zheng*
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*
Instituto de Ciencias Matemáticas, C/ Nicolás Cabrera, Madrid, 28049, Spain
Sangwoo Shin
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
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK
Dmitri Tseluiko
Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK
Serafim Kalliadasis
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK
Howard A. Stone*
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA


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 Papers
© 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.)


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.


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, 25502558.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, 239259.CrossRefGoogle Scholar
Barenblatt, G. I.1979 Similarity, Self-Similarity, and Intermediate Asymptotics. Consultants Bureau.Google 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. 155208. 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.CrossRefGoogle ScholarPubMed
Blossey, R. 2003 Self-cleaning surfaces: virtual realities. Nat. Mater. 2, 301306.CrossRefGoogle ScholarPubMed
Bonn, D., Eggers, J., Indekeu, J., Meunier, J. & Rolley, E. 2009 Wetting and spreading. Rev. Mod. Phys. 81, 739805.CrossRefGoogle Scholar
Bostwick, J. B., Dijksman, J. A. & Shearer, M. 2017 Wetting dynamics of a collapsing fluid hole. Phys. Rev. Fluids 2, 014006.CrossRefGoogle Scholar
Brenner, M. P., Lister, J. R. & Stone, H. A. 1996 Pinching threads, singularities and the number 0.0304. Phys. Fluids 8, 28272836.CrossRefGoogle Scholar
Bretherton, F. P. 1961 The motion of long bubbles in a tube. J. Fluid Mech. 10, 166188.CrossRefGoogle Scholar
Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 11311198.CrossRefGoogle 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, 26472666.CrossRefGoogle 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, 11481155.CrossRefGoogle 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 ScholarPubMed
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; 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, 6368.CrossRefGoogle Scholar
Eggers, J. 1993 Universal pinching of 3D axisymmetric free surface flow. Phys. Rev. Lett. 71, 34583460.CrossRefGoogle ScholarPubMed
Eggers, J. 2004 Toward a description of contact line motion at higher capillary numbers. Phys. Fluids 16, 34913494.CrossRefGoogle Scholar
Eggers, J. & Fontelos, M. A. 2009 The role of self-similarity in singularities of partial differential equations. Nonlinearity 22, R1R44.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, 606612.CrossRefGoogle Scholar
de Gennes, P. G., Hua, X. & Levinson, P. 1990 Dynamics of wetting: local contact angle. J. Fluid Mech. 212, 5563.CrossRefGoogle Scholar
Gratton, J. & Minotti, F. 1990 Self-similar viscous gravity currents: phase plane formalism. J. Fluid Mech. 210, 155182.CrossRefGoogle Scholar
Herminghaus, S., Brinkmann, M. & Seemann, R. 2008 Wetting and dewetting of complex surface geometries. Annu. Rev. Mater. Sci. 38, 101121.CrossRefGoogle Scholar
Hocking, L. M. 1983 The spreading of a thin drop by gravity and capillarity. Q. J. Mech. Appl. Maths 36 (Feb), 5569.CrossRefGoogle Scholar
Huppert, H. E. & Woods, A. W. 1995 Gravity driven flows in porous layers. J. Fluid Mech. 292, 5569.CrossRefGoogle Scholar
Jensen, O. E. 1994 Self-similar, surfactant-driven flows. Phys. Fluids 6, 10841094.CrossRefGoogle Scholar
Kalliadasis, S., Bielarz, C. & Homsy, G. M. 2000 Steady free-surface thin film flows over topography. Phys. Fluids 12, 18891898.CrossRefGoogle 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, 350362.CrossRefGoogle ScholarPubMed
Landau, L. & Levich, B. 1942 Dragging of a liquid by a moving plate. Acta Physicochim. URSS 17, 4254.Google Scholar
Levy, R. & Shearer, M. 2004 Comparison of two dynamic contact line models for driven thin liquid films. Eur. J. Appl. Maths 15, 625642.CrossRefGoogle 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, 215249.CrossRefGoogle Scholar
Lister, J. R. & Stone, H. A. 1998 Capillary breakup of a viscous thread surrounded by another viscous fluid. Phys. Fluids 11, 27582764.CrossRefGoogle Scholar
López, P. G., Miksis, M. J. & Bankoff, S. G. 2001 Stability and evolution of a dry spot. Phys. Fluids 13, 16011614.CrossRefGoogle 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.CrossRefGoogle ScholarPubMed
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, 335342.CrossRefGoogle Scholar
Myers, T. G. 1998 Thin films with high surface tension. SIAM Rev. 40, 441462.CrossRefGoogle Scholar
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931980.CrossRefGoogle Scholar
Padday, J. F. 1971 The profile of axially symmetric menisci. Phil. Trans. R. Soc. Lond. A 269, 265293.CrossRefGoogle Scholar
Papageorgiou, D. T. 1995 On the breakup of viscous liquid threads. Phys. Fluids 7, 15291544.CrossRefGoogle Scholar
Savva, N. & Kalliadasis, S. 2009 Two-dimensional droplet spreading over topographical substrates. Phys. Fluids 21, 092102.CrossRefGoogle Scholar
Savva, N. & Kalliadasis, S. 2011 Dynamics of moving contact lines: a comparison between slip and precursor film models. Europhys. Lett. 94, 64004.CrossRefGoogle Scholar
Sharma, A. & Ruckenstein, E. 1990 Energetic criteria for the breakup of liquid films on nonwetting solid surfaces. J. Colloid Interface Sci. 137, 433445.CrossRefGoogle Scholar
Snoeijer, J. H. & Andreotti, B. 2013 Moving contact lines: scales, regimes, and dynamical transitions. Annu. Rev. Fluid Mech. 45, 269292.CrossRefGoogle 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, 14731485.CrossRefGoogle 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, 762776.CrossRefGoogle Scholar
Voinov, O. V. 1976 Hydrodynamics of wetting. Fluid Dyn. 11, 714721.CrossRefGoogle Scholar
Witelski, T. P. & Bernoff, A. J. 1999 Stability of self-similar solutions for van der Waals driven thin film rupture. Phys. Fluids 11, 24432445.CrossRefGoogle Scholar
Yatsyshin, P., Parry, A. O. & Kalliadasis, S. 2016 Complete prewetting. J. Phys.: Condens. Matter 28, 275001.Google ScholarPubMed
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 ScholarPubMed
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, 24542462.CrossRefGoogle 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, 218246.CrossRefGoogle Scholar
Zheng, Z., Shin, S. & Stone, H. A. 2015 Converging gravity currents over a permeable substrate. J. Fluid Mech. 778, 669690.CrossRefGoogle Scholar
Cited by

Save article to Kindle

To save this article to your Kindle, first ensure 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 saving to your Kindle.

Note you can select to save to either the or variations. ‘’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘’ 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.

Healing capillary films
Available formats

Save article to Dropbox

To save 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 used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Healing capillary films
Available formats

Save article to Google Drive

To save 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 used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Healing capillary films
Available formats

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *