Journal of Engineering Mathematics

, Volume 97, Issue 1, pp 33–48 | Cite as

Falling film on a flexible wall in the presence of insoluble surfactant



This work investigates the effect of an insoluble surfactant on the gravity-driven flow of a liquid film down a vertical flexible wall. The paper builds upon previous work [Matar et al., Phys Rev E 76(5):056301, 2007; Sisoev et al., Chem Eng Sci 65(2):950–961, 2010] to include the Marangoni effect attributable to the gradient of surfactant concentration on a free surface. Here we employ an integral method to derive a set of asymptotic evolution equations valid for a moderate flow rate, based on a long-wave approximation. A normal-mode approach is used to examine the linear stability of the system. Similar to the work presented by Matar et al., the results show that a flexible wall with weak damping acts to stabilize flow, while wall tension plays an unstable role. The insoluble surfactant, which acts to stabilize film flow, can reduce the effects of wall flexibility (wall damping and tension) on flow linear stability. The nonlinear evolution equations for the system are solved numerically for both a given initial perturbation wave packet and a periodic perturbation at the inlet boundary. The equations are mainly concerned with the evolution of the flow stability and wave interaction processes, during which solitary-like waveforms are observed. When wall damping is weak, it tends to deplete the ripples preceding the solitary-like humps. However, as wall damping increases in strength, the ripples intensify; a similar phenomenon is observed with an increase in wall tension. The surfactant, which reduces the amplitude and traveling speed of the solitary-like waveforms, acts to distinctly weaken the dispersion of the interfacial wave.


Falling film Flexible substrate Surfactant 



The authors acknowledge financial support from NSFC Grant No. 11172152 and the National Science and Technology Major Project of the Ministry of Science and Technology of China Grant No. 2011ZX02601.


  1. 1.
    Oron A, Davis SH, Bankoff SG (1997) Long-scale evolution of thin liquid films. Rev Mod Phys 69(3):931–980ADSCrossRefGoogle Scholar
  2. 2.
    Craster RV, Matar OK (2009) Dynamics and the stability of thin liquid films. Rev Mod Phys 81(3):1131–1198ADSCrossRefGoogle Scholar
  3. 3.
    Wong H, Fatt I, Radke CJ (1996) Deposition and thinning of the human tear film. J Colloid Interface Sci 184:44–51CrossRefGoogle Scholar
  4. 4.
    Huppert HE (1982) Flow and instability of a viscous gravity current down a slope. Nature 300:427–429ADSCrossRefGoogle Scholar
  5. 5.
    Kapitza PL (1948) Wave flow of a thin viscous fluid layer. I. Free flow. J Exp Theor Phys 18(1):3–20Google Scholar
  6. 6.
    Kapitza PL, Kapitza SP (1949) Wave flow of thin viscous liquid films. III. Experimental study of wave regime of a flow. J Exp Theor Phys 19(2):105–120Google Scholar
  7. 7.
    Chang HC, Demekhin EA (2002) Complex wave dynamics on thin films, vol 14. Elsevier, New YorkGoogle Scholar
  8. 8.
    Benney DJ (1966) Long waves on liquid films. J Math Phys 45(2):150–155MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Sivashinsky GI, Michelson DM (1980) On irregular wavy flow on liquid film down a vertical plane. Prog Theor Phys 63:2112–2114ADSCrossRefGoogle Scholar
  10. 10.
    Smyrlis YS, Papageorgiou DT (1991) Predicting chaos for the infinite dimensional dynamical systems: the Kuramoto–Sivashinsky equation, a case study. PNAS 88(24):11129–11132ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Smyrlis YS, Papageorgiou DT (1996) Computational study of chaotic and ordered solutions of the Kuramoto–Sivashinsky equation. No. ICASE-96-12Google Scholar
  12. 12.
    Halpern D, Grotberg JB (1993) Surfactant effects on fluid-elastic instablities of liquid-lined flexible tubes: a model of airway closure. J Biomech Eng 115(3):271–277CrossRefGoogle Scholar
  13. 13.
    Grotberg JB (1994) Pulmonary flow and transport phenomena. Annu Rev Fluid Mech 26(1):529–571ADSCrossRefMATHGoogle Scholar
  14. 14.
    Carvalho MS, Scriven LE (1997) Deformable roll coating flows: steady state and linear perturbation analysis. J Fluid Mech 339:143–172ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Matar OK, Kumar S (2007) Dynamics and stability of flow down a flexible incline. J Eng Math 57(2):145–158MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Matar OK, Craster RV, Kumar S (2007) Falling films on flexible inclines. Phys Rev E 76(5):056301ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Sisoev GM, Matar OK, Craster RV, Kumar S (2010) Coherent wave structures on falling fluids films flowing down a flexible wall. Chem Eng Sci 65(2):950–961CrossRefGoogle Scholar
  18. 18.
    Peng J, Zhang YJ, ZhuGe WL (2014) Falling film on flexible wall in the limit of weak viscoelasticity. J Non-Newton Fluid Mech 210:85–95CrossRefGoogle Scholar
  19. 19.
    Edwards DA, Brenner H, Wasan DT (1991) Interfacial transport processes and rheology, vol 40. Butterworth-Heinemann, BostonGoogle Scholar
  20. 20.
    Ji W, Setterwall F (1994) On the instabilities of vertical falling liquid films in the presence of surface-active solute. J Fluid Mech 278:297–323ADSCrossRefMATHGoogle Scholar
  21. 21.
    Shkadov VY, Velarde MG, Shkadova VP (2004) Falling films and Marangoni effect. Phys Rev E 69(5):056310ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Blyth MG, Pozrikidis C (2004) Effect of surfactant on the stability of film flow down an inclined plane. J Fluid Mech 521:241–250ADSMathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Blyth MG, Pozrikidis C (2004) Effect of surfactants on the stability of two-layer channel flow. J Fluid Mech 505:59–86ADSMathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Matar OK, Kumar S (2004) Rupture of a surfactant-covered thin liquid film on a flexible wall. SIAM J Appl Math 64(6):2144–2166MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Pereira A, Kalliadasis S (2008) Dynamics of a falling film with solutal Marangoni effect. Phys Rev E 78(3):036312ADSCrossRefGoogle Scholar
  26. 26.
    Heil M, Hazel AL, Smith JA (2008) The mechanics of airway closure. Resp Physiol Neurobi 163(1):214–221CrossRefGoogle Scholar
  27. 27.
    Halpern D, Frenkel A (2003) Destabilization of a creeping flow by interfacial surfactant: linear theory extended to all wavenumbers. J Fluid Mech 485:191–220ADSCrossRefMATHGoogle Scholar
  28. 28.
    Peng J, Zhu KQ (2010) Linear instability of two-fluid Taylor–Couette flow in the presence of surfactant. J Fluid Mech 651:357–385Google Scholar
  29. 29.
    Atabek HB, Lew HS (1966) Wave propagation through a viscous incompressible fluid contained in an initially stressed elastic tube. Biophys J 6(4):481–503CrossRefGoogle Scholar
  30. 30.
    Ruyer-Quil C, Manneville P (2000) Improved modeling of flows down inclined planes. Eur Phys J B 15:357–369ADSCrossRefMATHGoogle Scholar
  31. 31.
    Ruyer-Quil C, Manneville P (2002) Further accuracy and convergence results on the modeling of flows down inclined planes by weighted-residual approximations. Phys Fluids 14(1):170–183ADSMathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Trevelyan PMJ, Kalliadasis S (2004) Wave dynamics on a thin-liquid film falling down a heated wall. J Eng Math 50(2–3):177–208MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Liu J, Paul JD, Gollub J (1993) Measurements of the primary instabilities of film flows. J Fluid Mech 250:69–101ADSCrossRefGoogle Scholar
  34. 34.
    Hu FQ, Hussaini MY, Manthey JL (1996) Low-dissipation and low-dispersion Runge–Kutta schemes for computational acoustics. J Comput Phys 124(1):177–191ADSMathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Hu FQ (1996) On perfectly matched layer as an absorbing boundary condition. AIAA paper 96–1664Google Scholar
  36. 36.
    Ramaswamy B, Chippada S, Joo SW (1996) A full-scale numerical study of interfacial instabilities in thin-film flows. J Fluid Mech 325:163–194ADSCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • J. Peng
    • 1
  • L. Y. Jiang
    • 1
  • W. L. Zhuge
    • 2
  • Y. J. Zhang
    • 2
  1. 1.Department of Engineering MechanicsTsinghua UniversityBeijingChina
  2. 2.State Key Laboratory of Automotive Safety and EnergyTsinghua UniversityBeijingChina

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