Acta Mechanica

, Volume 179, Issue 1–2, pp 41–66

Effect of long undulated bottoms on thin gravity-driven films



We carry out a perturbation analysis for steady gravity-driven film flow over undulations of moderate steepness that are long compared to the film thickness and study the linear stability of the flow in the framework of Floquet analysis. The effect of geometric nonlinearities on the instability becomes relevant for moderate bottom variations. We find that the critical Reynolds number for the onset of surface waves is higher than that for a flat bottom. At higher inclination angles, the theoretical results are in good quantitative agreement with experiment. At inclination angles where the flat part of the undulation is close to being horizontal, the basic solution for the steady flow fails to describe the flow in the flat part, and the linear stability analysis overestimates the critical Reynolds number.


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  1. Kistler S. F., Schweizer P. M. (eds.): Liquid film coating. London: Chapman & Hall 1997.Google Scholar
  2. Webb, R. L. 1994Principles of enhanced heat transferWileyNew YorkGoogle Scholar
  3. Alekseenko, S. V., Ye Nakoryakov, V. , Pokusaev, B. G. 1994Wave flow of liquid filmsBegell HouseNew YorkGoogle Scholar
  4. Wang, C. Y. 1981Liquid film flowing slowly down a wavy inclineAIChE27207212CrossRefGoogle Scholar
  5. Wang, C. Y. 1984Thin film flowing down a curved surfaceZ. Angew. Math. Phys.35533544CrossRefGoogle Scholar
  6. Scholle, M., Wierschem, A., Aksel, N. 2004Creeping films with vortices over strongly undulated bottomsActa Mech.168167193CrossRefGoogle Scholar
  7. Pozrikidis, C. 1988The flow of a liquid film along a periodic wallJ. Fluid Mech.188275300Google Scholar
  8. Bontozoglou, V., Papapolymerou, G. 1997Laminar film flow down a wavy inclineInt. J. Multiphase Flow236979CrossRefGoogle Scholar
  9. Malamataris, N. A., Bontozoglou, V. 1999Computer aided analysis of viscous film flow along an inclined wavy wallJ. Comp. Phys.154372392CrossRefGoogle Scholar
  10. Bontozoglou, V. 2000Laminar film flow along a periodic wallCMES1133142Google Scholar
  11. Zhao, L., Cerro, R. I. 1992Experimental characterization of viscous film flows over complex surfacesInt. J. Multiphase Flows18495516CrossRefGoogle Scholar
  12. Wierschem, A., Scholle, M., Aksel, N. 2002Comparison of different theoretical approaches to experiments on film flow down an inclined wavy channelExp. Fluids33429442Google Scholar
  13. Wierschem, A., Scholle, M., Aksel, N. 2003Vortices in film flow over strongly undulated bottom profiles at low Reynolds numbersPhys. Fluids15426435CrossRefMathSciNetGoogle Scholar
  14. Wierschem, A., Aksel, N. 2004Hydraulic jumps and standing waves in gravity-driven flows of viscous liquids in wavy open channelsPhys. Fluids1638683877CrossRefGoogle Scholar
  15. Wierschem, A., Aksel, N. 2004Influence of inertia on vortices created in films creeping over strongly undulated substratesPhys. Fluids1645664574CrossRefGoogle Scholar
  16. Chang, H.-C., Demekhin, E. A. 2002Complex wave dynamics on thin filmsElsevierAmsterdamGoogle Scholar
  17. Yih, C. S. 1963Stability of liquid flow down an inclined planePhys. Fluids6321334Google Scholar
  18. Tougou, H. 1978Long waves on a film flow of a viscous fluid down an inclined uneven wallJ. Phys. Soc. Jpn.4410141019CrossRefGoogle Scholar
  19. Vlachogiannis, M., Bontozoglou, V. 2002Experiments on laminar film flow along a periodic wallJ. Fluid Mech.457133156CrossRefGoogle Scholar
  20. Wierschem, A., Aksel, N. 2003Instability of a liquid film flowing down an inclined wavy planePhysica D186221237Google Scholar
  21. Schmid, P. J., Henningson, D. S. 2001Stability and transition in shear flowsSpringerNew YorkGoogle Scholar
  22. Liu, J., Paul, J. D., Gollub, J. P. 1993Measurements of the primary instabilities of film flowsJ. Fluid Mech.25069101Google Scholar

Copyright information

© Springer-Verlag Wien 2005

Authors and Affiliations

  1. 1.Department of Applied Mechanics and Fluid DynamicsUniversity of BayreuthBayreuthGermany

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