Skip to main content
SpringerLink
Log in

Dynamics and stability of flow down a flexible incline

  • Original Paper
  • Published:
Journal of Engineering Mathematics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bertozzi AL, Brenner MP (1997) Linear stability and transient growth in driven contact lines. Phys Fluids 9:530–539

    Article  MATH  ADS  MathSciNet  Google Scholar 

  2. Eres MH, Schwartz LW, Roy RV (2000) Fingering phenomena for driven coating films. Phys Fluids 12:1278–1295

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Kataoka DE, Troian SM (1997) A theoretical study of instabilities at the advancing front of thermally driven coating films. J Colloid Interface Sci 192:350–362

    Article  Google Scholar 

  4. Kondic L, Diez J (2003) Flow of thin films on patterned surfaces. Colloids Surf 214:1–11

    Article  Google Scholar 

  5. Oron A, Davis SH, Bankoff SG (1997) Long-scale evolution of thin liquid films. Rev Mod Phys 69:931–980

    Article  ADS  Google Scholar 

  6. Spaid MA, Homsy GM (1996) Stability of Newtonian and viscoelastic dynamic contact lines. Phys Fluids 8:460–478

    Article  MATH  ADS  MathSciNet  Google Scholar 

  7. Riley JJ, el Hak MG, Metcalfe RW (1988) Compliant coatings. Ann Rev Fluid Mech 20:393–420

    ADS  Google Scholar 

  8. Grotberg JB (1994) Pulomnary flow and transport phenomena. Ann Rev Fluid Mech 26:529–571

    Article  ADS  Google Scholar 

  9. Berger SA, Jou L-D (2000) Flows in stenotic vessels. Ann Rev Fluid Mech 32:347–382

    Article  ADS  MathSciNet  Google Scholar 

  10. Carvalho MS, Scriven LE (1997) Deformable roll coating flows: steady state and linear perturbation analysis. J Fluid Mech 339:143–172

    Article  MATH  ADS  MathSciNet  Google Scholar 

  11. Kumaran V, Muralikrishnan R (2000) Spontaneous growth of fluctuations in the viscous flow of a fluid past a soft interface. Phys Rev Lett 84:3310–3313

    Article  ADS  Google Scholar 

  12. Matar OK, Kumar S (2004) Rupture of a surfactant-covered thin liquid film on a flexible wall. SIAM J Appl Math 6:2144–2166

    Article  MathSciNet  Google Scholar 

  13. Halpern D, Grotberg JB (1992) Fluid-elastic instabilities of liquid-lined flexible tubes. J Fluid Mech 244:615–632

    Article  MATH  ADS  Google Scholar 

  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 115:271–277

    Google Scholar 

  15. Atabek HB, Lew SH (1966) Wave propagation through a viscous incompressible fluid contained in an initially stressed elastic tube. Biophys J 6:481

    Article  Google Scholar 

  16. Landau LD, Lifshitz EM (1986) Theory of elasticity. Butterworth-Heineman, Dordrecht

    Google Scholar 

  17. Troian SM, Herbolzheimer E, Safran SA (1989) Fingering instabilities of driven spreading films. Europhys Lett 10:25–30

    Article  ADS  Google Scholar 

  18. Keast P, Muir PH (1991) Algorithm 6888 EPDCOL—a more efficient PDECOL Code. ACM Trans Math Software 17:153–166

    Article  MATH  Google Scholar 

  19. Edmonstone BD, Matar OK, Craster RV (2005) Coating of an inclined plane in the presence of insoluble surfactant. J Colloid Interface Sci 287:261–272

    Article  Google Scholar 

  20. Huppert HE (1982) The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface. J Fluid Mech 121:43–58

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Chemical Engineering, Imperial College London, South Kensington Campus, London, SW7 2AZ, UK

    Omar K. Matar

  2. Department of Chemical Engineering and Materials Science, University of Minnesota, 151 Amundson Hall 421 Washington Ave. SE, Minneapolis, MN, 55455, USA

    Satish Kumar

Authors
  1. Omar K. Matar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Satish Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Omar K. Matar.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Matar, O.K., Kumar, S. Dynamics and stability of flow down a flexible incline. J Eng Math 57, 145–158 (2007). https://doi.org/10.1007/s10665-006-9069-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10665-006-9069-7

Keywords

Search

Navigation