Journal of Engineering Mathematics

, Volume 99, Issue 1, pp 119–136 | Cite as

Stability of an unsupported multi-layer surfactant laden liquid curtain under gravity

  • D. HenryEmail author
  • J. Uddin
  • J. O. Marston
  • S. T. Thoroddsen


The industrial process of curtain coating has long been an important method in coating applications, by which a thin liquid curtain is formed to impinge upon a moving substrate, due to its highly lucrative advantage of being able to coat multiple layers simultaneously. We investigate the linear stability of an unsupported two-layer liquid curtain, which has insoluble surfactants in both liquids, which are widely used in industry to increase the stability of the curtain. We formulate the governing equations, simplified by making a thin film approximation, from which we obtain equations describing the steady-state profiles. We then examine the response of the curtain to small perturbations about this steady state to identify conditions under which the curtain is unstable, finding the addition of surfactants stabilizes the curtain. Our results are then compared to experimental data, showing a favourable trend and thereby extending the works of Brown (J Fluid Mech 10:297–305, 1960) and Dyson et al. (J Eng Math 64:237–250, 2009).


Curtain coating Multi-layer Stability Surfactants 

Mathematics Subject Classification

76E17 76D45 76T99 



D.H. would like to thank the EPSRC for their financial support, and KAUST for funding the experimental work conducted whilst D.H. was on a research visit. The authors would also like to thank the anonymous reviewers who have contributed in improving this paper.


  1. 1.
    Taylor GI (1957) The dynamics of thin sheets of fluid. III. Disintegration of fluid sheets. Proc R Soc Lond A 253:313–321ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brown D (1960) A study of the behaviour of a thin sheet of moving liquid. J Fluid Mech 10:297–305ADSCrossRefzbMATHGoogle Scholar
  3. 3.
    Greiller JF (1972) Method of making photographic elements. U.S. Patent No. 3,632,374Google Scholar
  4. 4.
    Henry D, Uddin J, Thompson J, Blyth MG, Thoroddsen ST, Marston JO (2014) Multi-layer film flow down an inclined plane: experimental investigation. Exp Fluids 55(12):1–14CrossRefGoogle Scholar
  5. 5.
    Blake TD, Bracke M, Shikhmurzaev YD (1999) Experimental evidence of nonlocal hydrodynamic influence on the dynamic contact angle. Phys Fluids 11(8):1995–2007ADSCrossRefzbMATHGoogle Scholar
  6. 6.
    Weinstein SJ, Ruschak KJ (2004) Coating flows. Annu Rev Fluid Mech 36:29–53ADSCrossRefzbMATHGoogle Scholar
  7. 7.
    Durst F, Wagner HG (1997) Slot coating. In: Kistler SF, Schweizer PM (eds) Liquid film coating Ch 11a. Chapman & Hall, LondonGoogle Scholar
  8. 8.
    Hens J, Van Abbenyen W (1997) Slide coating. In: Kistler SF, Schweizer PM (eds) Liquid film coating Ch 11b. Chapman & Hall, LondonGoogle Scholar
  9. 9.
    Hughes DJ (1970) Method for simultaneously applying a plurality of coated layers by forming a stable multilayer free-falling vertical curtain. U.S. Patent No. 3,508,947Google Scholar
  10. 10.
    Lin SP (1981) Stability of a viscous liquid curtain. J Fluid Mech 104:111–118ADSCrossRefzbMATHGoogle Scholar
  11. 11.
    Lin SP, Roberts G (1981) Waves in a viscous liquid curtain. J Fluid Mech 114:443–458ADSCrossRefGoogle Scholar
  12. 12.
    Crapper GD, Dombrowski N, Jepson WP (1975) Wave growth on thin sheets of non-Newtonian liquids. Proc R Soc Lond A 342:225–236ADSCrossRefGoogle Scholar
  13. 13.
    Lin SP, Lian ZW, Creighton BJ (1990) Absolute and convective instability of a liquid sheet. J Fluid Mech 220:673–689ADSCrossRefzbMATHGoogle Scholar
  14. 14.
    De Luca L, Costa M (1997) Instability of a spatially developing liquid sheet. J Fluid Mech 331:127–144ADSCrossRefzbMATHGoogle Scholar
  15. 15.
    Dyson RJ, Brander J, Breward CJW, Howell PD (2009) Long-wavelength stability of an unsupported multilayer liquid film falling under gravity. J Eng Math 64:237–250MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Marston JO, Thoroddsen ST, Thompson J, Blyth MG, Henry D, Uddin J (2014) Experimental investigation of hysteresis in the break-up of liquid curtains. Chem Eng Sci 117:248–263CrossRefGoogle Scholar
  17. 17.
    Finnicum DS, Weinstein SJ, Ruschak KJ (1993) The effect of applied pressure on the shape of a two dimensional liquid curtain falling under the influence of gravity. J Fluid Mech 255:647–665ADSCrossRefGoogle Scholar
  18. 18.
    Roche JS, Le Grand N, Brunet P, Lebon L, Limat L (2006) Perturbations on a liquid curtain near break-up: wakes and free edges. Phys Fluids 18:082101Google Scholar
  19. 19.
    Rosen MJ (1989) Surfactants and interfacial phenomena, 2nd edn. Wiley, New YorkGoogle Scholar
  20. 20.
    Edwards DA, Brenner H, Wasan DT (1991) Interfacial transport processes and rheology. Bufferworth-Heinemann, BostonGoogle Scholar
  21. 21.
    De Luca L, Meola C (1995) Surfactant effects on the dynamics of a thin liquid sheet. J Fluid Mech 300:71–85ADSCrossRefGoogle Scholar
  22. 22.
    Stone HA, Leal LG (1990) The effects of surfactants on drop deformation and breakup. J Fluid Mech 220:161–186ADSCrossRefzbMATHGoogle Scholar
  23. 23.
    Blyth MG, Pozrikidis C (2004) Evolution equations for the surface concentration of an insoluble surfactant; applications to the stability of an elongating thread and a stretched interface. Theor Comput Fluid Dyn 17:147–164CrossRefzbMATHGoogle Scholar
  24. 24.
    Frumkin A (1925) Die Kapillarkurve der höheren Fettsäuren und die Zustandsgleichung der Oberflächenschicht. Z Phys Chem Stoechiom Verwandschaftsl 116:466–484Google Scholar
  25. 25.
    Adamson AW, Grist AP (1997) Physical chemistry of surfaces, 6th edn. Wiley-Interscience, New YorkGoogle Scholar
  26. 26.
    Leal LG (2007) Advanced transport phenomena. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • D. Henry
    • 1
    Email author
  • J. Uddin
    • 1
  • J. O. Marston
    • 2
  • S. T. Thoroddsen
    • 3
  1. 1.School of MathematicsUniversity of BirminghamBirminghamUK
  2. 2.Department of Chemical EngineeringTexas Tech UniversityLubbockUSA
  3. 3.Division of Physical Sciences and EngineeringKing Abdullah University of Science and TechnologyThuwalSaudi Arabia

Personalised recommendations