Journal of Engineering Mathematics

, Volume 57, Issue 2, pp 145–158 | Cite as

Dynamics and stability of flow down a flexible incline

Original Paper


The flow of a thin liquid film down a flexible inclined wall is examined. Two configurations are studied: constant flux (CF) and constant volume (CV). The former configuration involves constant feeding of the film from an infinite reservoir of liquid. The latter involves the spreading of a drop of constant volume down the wall. Lubrication theory is used to derive a pair of coupled two-dimensional nonlinear evolution equations for the film thickness and wall deflection. The contact-line singularity is relieved by assuming that the underlying wall is pre-wetted with a precursor layer of uniform thickness. Solution of the one-dimensional evolution equations demonstrates the existence of travelling-wave solutions in the CF case and self-similar solutions in the CV case. The effect of varying the wall tension and damping coefficient on the structure of these solutions is elucidated. The linear stability of the flow to transverse perturbations is also examined in the CF case only. The results indicate that the flow, which is already unstable in the rigid-wall limit, is further destabilized as a result of the coupling between the fluid and underlying flexible wall.


Thin film Flexible Lubrication Stability 


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  1. 1.
    Bertozzi AL, Brenner MP (1997) Linear stability and transient growth in driven contact lines. Phys Fluids 9:530–539MATHCrossRefADSMathSciNetGoogle Scholar
  2. 2.
    Eres MH, Schwartz LW, Roy RV (2000) Fingering phenomena for driven coating films. Phys Fluids 12:1278–1295CrossRefADSMathSciNetMATHGoogle Scholar
  3. 3.
    Kataoka DE, Troian SM (1997) A theoretical study of instabilities at the advancing front of thermally driven coating films. J Colloid Interface Sci 192:350–362CrossRefGoogle Scholar
  4. 4.
    Kondic L, Diez J (2003) Flow of thin films on patterned surfaces. Colloids Surf 214:1–11CrossRefGoogle Scholar
  5. 5.
    Oron A, Davis SH, Bankoff SG (1997) Long-scale evolution of thin liquid films. Rev Mod Phys 69:931–980CrossRefADSGoogle Scholar
  6. 6.
    Spaid MA, Homsy GM (1996) Stability of Newtonian and viscoelastic dynamic contact lines. Phys Fluids 8:460–478MATHCrossRefADSMathSciNetGoogle Scholar
  7. 7.
    Riley JJ, el Hak MG, Metcalfe RW (1988) Compliant coatings. Ann Rev Fluid Mech 20:393–420ADSGoogle Scholar
  8. 8.
    Grotberg JB (1994) Pulomnary flow and transport phenomena. Ann Rev Fluid Mech 26:529–571CrossRefADSGoogle Scholar
  9. 9.
    Berger SA, Jou L-D (2000) Flows in stenotic vessels. Ann Rev Fluid Mech 32:347–382CrossRefADSMathSciNetGoogle Scholar
  10. 10.
    Carvalho MS, Scriven LE (1997) Deformable roll coating flows: steady state and linear perturbation analysis. J Fluid Mech 339:143–172MATHCrossRefADSMathSciNetGoogle Scholar
  11. 11.
    Kumaran V, Muralikrishnan R (2000) Spontaneous growth of fluctuations in the viscous flow of a fluid past a soft interface. Phys Rev Lett 84:3310–3313CrossRefADSGoogle Scholar
  12. 12.
    Matar OK, Kumar S (2004) Rupture of a surfactant-covered thin liquid film on a flexible wall. SIAM J Appl Math 6:2144–2166CrossRefMathSciNetGoogle Scholar
  13. 13.
    Halpern D, Grotberg JB (1992) Fluid-elastic instabilities of liquid-lined flexible tubes. J Fluid Mech 244:615–632MATHCrossRefADSGoogle Scholar
  14. 14.
    Halpern D, Grotberg JB (1993) Surfactant effects on fluid-elastic instabilities of liquid-lined flexible tubes: a model for airway closure. J Biomech Eng 115:271–277Google Scholar
  15. 15.
    Atabek HB, Lew SH (1966) Wave propagation through a viscous incompressible fluid contained in an initially stressed elastic tube. Biophys J 6:481CrossRefGoogle Scholar
  16. 16.
    Landau LD, Lifshitz EM (1986) Theory of elasticity. Butterworth-Heineman, DordrechtGoogle Scholar
  17. 17.
    Troian SM, Herbolzheimer E, Safran SA (1989) Fingering instabilities of driven spreading films. Europhys Lett 10:25–30CrossRefADSGoogle Scholar
  18. 18.
    Keast P, Muir PH (1991) Algorithm 6888 EPDCOL—a more efficient PDECOL Code. ACM Trans Math Software 17:153–166MATHCrossRefGoogle Scholar
  19. 19.
    Edmonstone BD, Matar OK, Craster RV (2005) Coating of an inclined plane in the presence of insoluble surfactant. J Colloid Interface Sci 287:261–272CrossRefGoogle Scholar
  20. 20.
    Huppert HE (1982) The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface. J Fluid Mech 121:43–58CrossRefADSGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  1. 1.Department of Chemical EngineeringImperial College LondonLondonUK
  2. 2.Department of Chemical Engineering and Materials ScienceUniversity of MinnesotaMinneapolisUSA

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