Journal of Engineering Mathematics

, Volume 94, Issue 1, pp 1–3 | Cite as

Preface to the special issue on “Thin films and fluid interfaces”

  • Thomas P. Witelski

Free-surface problems, where the boundary of a moving body of fluid evolves as part of the dynamics, are one of the most interesting and challenging areas in the field of fluid dynamics. Such problems are ubiquitous in natural settings over a wide range of physical scales, from the propagation of surface waves on oceans to the spreading of raindrops on plant leaves. These systems exhibit intricate behaviors and have inspired many different types of applied studies in physics and engineering such as using fluid wetting to test substrate material properties, improving transport in designs of microfluidic devices, and investigating self-assembly of patterned materials. Likewise, free-surface problems have been a long-standing source of fundamental mathematical questions on how to formulate models, analyze systems of nonlinear partial differential equations, and compute efficient numerical simulations.

Within the broad spectrum of free-surface problems, one of the most developed branches concerns the behaviors produced by slender bodies of fluids, such as jets and thin sheets [1]. These problems are fundamental to many physical situations including coating processing that involve the spreading of thin films of fluids on solid substrates [2]. The governing equations for these flows are derived through the use of longwave asymptotics and lubrication theory. The resulting mathematical models are higher-order nonlinear partial differential equations that include the influences of surface tension and the geometry of the problem. Some notable papers have given overviews of physical [3, 4, 5, 6, 7, 8, 9, 10], mathematical [11, 12, 13, 14], and computational [15, 16] aspects of this field.

Contributing to this rich and active field of research, the motivation for this special issue of the Journal of Engineering Mathematics came from the workshop on “Thin Liquid Films and Fluid Interfaces: Models, Experiments and Applications” held at the Banff International Research Station (BIRS) on December 9–14, 2012 [17]. The workshop added to the growing list of focused meetings organized by various members of this research community. Some of the other conferences in the field were held at BIRS in 2003–2010 [18, 19, 20, 21], UCLA’s IPAM center in 2006 [22], the EUROMECH 490 workshop in London in 2007 [23], two meetings at ICMS in Edinburgh in 1999 and 2009 (EUROMECH 497) [24, 25], at the Fields Institute in 2012 [26], and the 2013 programme at the Newton Institute at Cambridge University [27]. Several of these meetings have inspired special issues of journals containing new research results on thin film problems [28, 29, 30, 31, 32].

An international and interdisciplinary array of researchers in physics, engineering, and applied mathematics attended the 2012 BIRS workshop. Physical problems discussed at the meeting spanned applications from nano-scale to geophysical settings and physiological to industrial contexts. Mathematical questions on modeling of complex fluids and fluid–structure interactions also led to spirited interactions among the conference participants.

This special issue consists of seven articles that touch on several of the computational, analytical, and experimental themes that were represented at the workshop. The first paper, by Afkhami and Kondic, is a computational study of inertial dewetting of nano-scale liquid drops. The article by Sibley et al. gives an overview of different analytical models for the dynamics of moving contact lines for spreading drops. The work by Huth et al. presents an overview and a new approach for computing of the spreading of droplets on top of other fluid layers. The paper by Swanson et al. compares experimental and modeling results for the spreading of surfactants on thin films. The behaviors of thin films subject to the influences of Marangoni effects produced by applied thermal and electrical forcing were studied in the paper by Corbett and Kumar. The remaining two papers in the special issue present different examinations of flows of thin films of non-Newtonian fluids in the classic problem of flow down an inclined plane [33]. In their paper, Lam et al. examine the instabilities in a model of gravity driven flows of nematic liquid crystals. In the final paper, Pritchard et al. derive similarity solutions for generalized families of non-Newtonian models with shear-rate dependent viscosities.



On behalf of all of the organizers of the 2012 BIRS workshop (R. P. Behringer, K. Daniels, R. Levy, O. K. Matar, M. Shearer, T. Witelski), I would like to thank the personnel of BIRS for hosting our workshop. I also want to acknowledge the support of the National Science Foundation, which helped fund the travel of students and junior researchers at the meeting [FRG: Collaborative Research: Dynamics of Thin Liquid Films: Mathematics and Experiments (2010–2013) under Grants DMS #0968258 (Shearer, Daniels), DMS #0968154 (Levy), DMS #0968252 (Witelski)].


  1. 1.
    Lin SP (2003) Breakup of liquid sheets and jets. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  2. 2.
    Friedman A (1988) Unresolved mathematical issues in coating flow mechanics. In: Mathematics in industrial problems. The IMA volumes in mathematics and its applications, vol 16. Springer, New York, pp 20–31Google Scholar
  3. 3.
    de Gennes PG (1985) Wetting—statics and dynamics. Rev Mod Phys 57(3):827–863CrossRefADSGoogle Scholar
  4. 4.
    Oron A, Davis SH, Bankoff SG (1997) Long-scale evolution of thin liquid films. Rev Mod Phys 69(3):931–980CrossRefADSGoogle Scholar
  5. 5.
    Eggers J (1997) Nonlinear dynamics and breakup of free-surface flows. Rev Mod Phys 69(3):865–929zbMATHCrossRefADSGoogle Scholar
  6. 6.
    Bonn D, Eggers J, Indekeu J, Meunier J, Rolley E (2009) Wetting and spreading. Rev Mod Phys 81(2):739–805CrossRefADSGoogle Scholar
  7. 7.
    Eggers J, Villermaux E (2008) Physics of liquid jets. Rep Prog Phys 71(3):036601CrossRefADSGoogle Scholar
  8. 8.
    Craster RV, Matar OK (2009) Dynamics and stability of thin liquid films. Rev Mod Phys 81(3):1131–1198CrossRefADSGoogle Scholar
  9. 9.
    Pomeau Y, Villermaux E (2006) Two hundred years of capillarity research. Phys Today 59(3):39–44CrossRefGoogle Scholar
  10. 10.
    de Gennes P-G, Brochard-Wyart F, Quere D (2003) Capillarity and wetting phenomena: drops, bubbles, pearls, waves. Springer, New YorkGoogle Scholar
  11. 11.
    Bertozzi AL (1998) The mathematics of moving contact lines in thin liquid films. Not Am Math Soc 45(6):689–697zbMATHMathSciNetGoogle Scholar
  12. 12.
    Myers TG (1998) Thin films with high surface tension. SIAM Rev 40(3):441–462zbMATHMathSciNetCrossRefADSGoogle Scholar
  13. 13.
    Eggers J, Fontelos MA (2009) The role of self-similarity in singularities of partial differential equations. Nonlinearity 22(1):R1–R44zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Kondic L (2003) Instabilities in gravity driven flow of thin fluid films. SIAM Rev 45(1):95–115zbMATHMathSciNetCrossRefADSGoogle Scholar
  15. 15.
    Witelski TP (2002) Computing finite-time singularities in interfacial flows. In: Modern methods in scientific computing and applications (Montréal, QC, 2001). NATO Science Series II: Mathematics, Physics and Chemistry, vol 75. Kluwer Academic Publishers, Dordrecht, pp 451–487Google Scholar
  16. 16.
    Beltrame P, Thiele U (2010) Time integration and steady-state continuation for 2D lubrication equations. SIAM J Appl Dyn Syst 9(2):484–518zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Behringer RP, Daniels K, Levy R, Matar OK, Shearer M, Witelski TP (2010) Thin liquid films and fluid interfaces: models, experiments and applications. BIRS workshop, 12w5035.
  18. 18.
    Almgren R, Behringer RP, Bertozzi AL, Pugh MC, Shearer M, Witelski TP (2003) Nonlinear dynamics of thin films and fluid interfaces. BIRS workshop, 03w5021.
  19. 19.
    Feng JF, Liu C (2006) Interfacial dynamics in complex fluids. BIRS workshop, 06w5047.
  20. 20.
    Balmforth N, Frigaard I (2005) Visco-plastic fluids: from theory to application. BIRS workshop, 05w5028.
  21. 21.
    Craster R, Homsy GM, Papageorgiou DT (2010) Small scale hydrodynamics: microfluidics and thin films. BIRS workshop, 10w5035.
  22. 22.
    Behringer RP, Bertozzi AL, Witelski TP, Shearer M, Slepcev D (2006) Thin films and fluid interfaces. UCLA–IPAM–NSF workshop.
  23. 23.
    Matar OK, Craster RV, Muench A, Witelski TP (2007) Dynamics and stability of thin liquid films and slender jets. EUROMECH 490 workshop.
  24. 24.
    Wilson SK, Duffy BR, Grinfeld M (1999) The dynamics of thin fluid films. ICMS workshop.
  25. 25.
    Duffy BR, Homsy GM, Wilson SK (2009) Recent developments and new directions in thin-film flow. ICMS/EUROMECH colloquium 497.
  26. 26.
    Chugunova M, Levy R, Smolka LB (2012) Workshop on surfactant driven thin film flows. Fields Institute workshop.
  27. 27.
    Holm D, Kruse K, Olmsted P, Pismen L, Thiele U (2013) Mathematical modelling and analysis of complex fluids and active media in evolving domains. Newton Institute programme.
  28. 28.
    Wilson SK (2001) Editorial—the dynamics of thin fluid films. Eur J Appl Math 12(3):193–194zbMATHGoogle Scholar
  29. 29.
    Behringer RP, Shearer M (2005) Preface [Non-linear dynamics of thin films and fluid interfaces]. Phys D 209(1–4):vii–viiiGoogle Scholar
  30. 30.
    Crowdy DG, Lawrence CJ, Wilson SK (2004) Preface [The dynamics of thin liquid films]. J Eng Math 50(2–3):95–97Google Scholar
  31. 31.
    Bertozzi AL (2006) Editorial to “UCLA–IPAM–NSF workshop on thin films and fluid interfaces”. Appl Math Res Express 2006(34149):1–2MathSciNetGoogle Scholar
  32. 32.
    Wilson SK, Duffy BR (2012) Preface to the special issue on “Recent developments and new directions in thin-film flow”. J Eng Math 73(1):1–2MathSciNetCrossRefGoogle Scholar
  33. 33.
    Huppert HE (1982) Flow and instability of a viscous current down a slope. Nature 300(5891):427–429CrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of MathematicsDuke UniversityDurhamUSA

Personalised recommendations