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Stability of the channel flow—new phenomena in an old problem

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Abstract

In this article, we show an unexpected effect of side walls on the stability of gravity-driven viscous film flows over flat substrates in an open channel. Until now, it was assumed that side walls have a stabilizing effect on the flow, but the qualitative structure of the neutral curve remains the same. We show a fragmentation of the stability chart that was only known from undulated substrates. The cause of the fragmentation is a distinct damping of waves with a certain frequency that is independent of the underlying instability of the two-dimensional flow. This paper provides a detailed parameter study that identifies the channel width as the decisive parameter for the damping frequency. In addition, the damping is not an effect of flat side walls exclusively, since corrugated side walls have qualitatively the same impact on the stability chart. Furthermore, we demonstrate that the curvature of the emerging wave’s crest line has a non-monotonous dependency on the frequency and shows a distinct maximum. The frequency of this maximum shows the same behavior as the damping frequency when the channel width is varied. We therefore assume a relation between the wave’s curvature and its growth rate, both massively affected by the side walls.

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References

  1. Kistler, S.F., Schweizer, P.M.: Liquid Film Coating. Springer, Dodrecht (1997). https://doi.org/10.1007/978-94-011-5342-3. ISBN: 978-94-010-6246-6

    Google Scholar 

  2. Weinstein, S.J., Ruschak, K.J.: Coating flows. Ann. Rev. Fluid Mech. 36(1), 29–53 (2004). https://doi.org/10.1146/annurev.fluid.36.050802.122049

    Article  MATH  Google Scholar 

  3. Gugler, G., Beer, R., Mauron, M.: Operative limits of curtain coating due to edges. Chem. Eng. Process. Process Intensif. 50(5), 462–465 (2011). https://doi.org/10.1016/j.cep.2011.01.010

    Article  Google Scholar 

  4. Webb, R.L.: Principles of Enhanced Heat Transfer. Wiley, New York (1994)

    Google Scholar 

  5. Vlasogiannis, P., Karagiannis, G., Argyropoulos, P., Bontozoglou, V.: Air-water two-phase flow and heat transfer in a plate heat exchanger. Int. J. Multiph. Flow. 28(5), 757–772 (2002). https://doi.org/10.1016/S0301-9322(02)00010-1

    Article  MATH  Google Scholar 

  6. Greve, R., Blatter, H.: Dynamics of Ice Sheets and Glaciers. Springer, Berlin (2009). https://doi.org/10.1007/978-3-642-03415-2

    Book  Google Scholar 

  7. Luca, I., Hutter, K., Tai, Y.C., Kuo, C.Y.: A hierarchy of avalanche models on arbitrary topography. Acta Mech. 205(1), 121–149 (2009). https://doi.org/10.1007/s00707-009-0165-4

    Article  MATH  Google Scholar 

  8. Hutter, K., Svendsen, B., Rickenmann, D.: Debris flow modeling: a review. Contin. Mech. Thermodyn. 8(1), 1–35 (1994). https://doi.org/10.1007/BF01175749

    Article  MathSciNet  MATH  Google Scholar 

  9. Kumar, A., Karig, D., Acharya, R., Neethirajan, S., Mukherjee, P.P., Retterer, S., Doktycz, M.J.: Microscale confinement features can affect biofilm formation. Microfluid. Nanofluidics 14(5), 895–902 (2013). https://doi.org/10.1007/s10404-012-1120-6

    Article  Google Scholar 

  10. Braun, R.J.: Dynamics of the tear film. Ann. Rev. Fluid Mech. 44, 267–297 (2012)

    Article  MathSciNet  Google Scholar 

  11. Nusselt, W.: Die Oberflächenkondensation des Wasserdampfes. VDI Z 60, 541–546 (1916)

    Google Scholar 

  12. Scholle, M., Aksel, N.: An exact solution of visco-capillary flow in an inclined channel. Z. Angew. Math. Phys. 52(5), 749–769 (2001). https://doi.org/10.1007/PL00001572

    Article  MathSciNet  MATH  Google Scholar 

  13. Haas, A., Pollak, T., Aksel, N.: Side wall effects in thin gravity-driven film flow—steady and draining flow. Phys. Fluids 23(6), 062107 (2011). https://doi.org/10.1063/1.3604002

    Article  Google Scholar 

  14. Kapitza, P.L.: Wavy flow of thin layers of viscous liquid. Zh. Eksper. Teoret. Fiz. 18, 3–28 (1948)

    Google Scholar 

  15. Kapitza, P.L., Kapitza, S.P.: Wavy flow of thin layers of a viscous fluid. Zh. Eksp. Teor. Fiz. 19, 105–120 (1949)

    MATH  Google Scholar 

  16. Benjamin, T.B.: Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2(06), 554 (1957). https://doi.org/10.1017/S0022112057000373

    Article  MathSciNet  MATH  Google Scholar 

  17. Yih, C.-S.: Stability of liquid flow down an inclined plane. Phys. Fluids 6(3), 321 (1963). https://doi.org/10.1063/1.1706737

    Article  MATH  Google Scholar 

  18. Vlachogiannis, M., Bontozoglou, V.: Experiments on laminar film flow along a periodic wall. J. Fluid Mech. 457, 133–156 (2002). https://doi.org/10.1017/S0022112001007637

    Article  MATH  Google Scholar 

  19. Wierschem, A., Aksel, N.: Instability of a liquid film flowing down an inclined wavy plane. Phys. D Nonlinear Phenomena 186(3), 221–237 (2003). https://doi.org/10.1016/S0167-2789(03)00242-2

    Article  MathSciNet  MATH  Google Scholar 

  20. D’Alessio, S.J.D., Pascal, J.P., Jasmine, H.A.: Instability in gravity-driven flow over uneven surfaces. Phys. Fluids 21(6), 062105 (2009). https://doi.org/10.1063/1.3155521

    Article  MATH  Google Scholar 

  21. Heining, C., Aksel, N.: Effects of inertia and surface tension on a power-law fluid flowing down a wavy incline. Int. J. Multiph. Flow 36(11), 847–857 (2010). https://doi.org/10.1016/j.ijmultiphaseflow.2010.07.002

    Article  Google Scholar 

  22. Pollak, T., Aksel, N.: Crucial flow stabilization and multiple instability branches of gravity-driven films over topography. Phys. Fluids 25(2), 024103 (2013). https://doi.org/10.1063/1.4790434

    Article  Google Scholar 

  23. Schörner, M., Reck, D., Aksel, N., Trifonov, Y.: Switching between different types of stability isles in films over topographies. Acta Mech. 229(2), 423–436 (2018). https://doi.org/10.1007/s00707-017-1979-0

    Article  MathSciNet  Google Scholar 

  24. Schörner, M., Reck, D., Aksel, N.: Does the topography’s specific shape matter in general for the stability of film flows? Phys. Fluids 27(4), 042103 (2015). https://doi.org/10.1063/1.4917026

    Article  Google Scholar 

  25. Schörner, M., Aksel, N.: The stability cycle—a universal pathway for the stability of films over topography. Phys. Fluids 30(1), 012105 (2018). https://doi.org/10.1063/1.5003449

    Article  Google Scholar 

  26. Aksel, N., Schörner, M.: Films over topography: from creeping flow to linear stability, theory, and experiments, a review. Acta Mech. 229(4), 1453–1482 (2018). https://doi.org/10.1007/s00707-018-2146-y

    Article  MathSciNet  Google Scholar 

  27. Vlachogiannis, M., Samandas, A., Leontidis, V., Bontozoglou, V.: Effect of channel width on the primary instability of inclined film flow. Phys. Fluids 22(1), 012106 (2010). https://doi.org/10.1063/1.3294884

    Article  MATH  Google Scholar 

  28. Georgantaki, A., Vatteville, J., Vlachogiannis, M., Bontozoglou, V.: Measurements of liquid film flow as a function of fluid properties and channel width: evidence for surface-tension-induced long-range transverse coherence. Phys. Rev. E 84(2), 026325 (2011)

    Article  Google Scholar 

  29. Pollak, T., Haas, A., Aksel, N.: Side wall effects on the instability of thin gravity-driven films—from long-wave to short-wave instability. Phys. Fluids 23(9), 094110 (2011). https://doi.org/10.1063/1.3634042

    Article  Google Scholar 

  30. Kögel, A., Aksel, N.: Massive stabilization of gravity-driven film flows with corrugated side walls. Phys. Fluids 30(11), 114105 (2018). https://doi.org/10.1063/1.5055931

    Article  Google Scholar 

  31. Leontidis, V., Vatteville, J., Vlachogiannis, M., Andritsos, N., Bontozoglou, V.: Nominally two-dimensional waves in inclined film flow in channels of finite width. Phys. Fluids 22(11), 112106 (2010). https://doi.org/10.1063/1.3484250

    Article  Google Scholar 

  32. Scholle, M., Haas, A., Aksel, N., Wilson, M.C.T., Thompson, H.M., Gaskell, P.H.: Competing geometric and inertial effects on local flow structure in thick gravity-driven fluid films. Phys. Fluids 20(12), 123101 (2008). https://doi.org/10.1063/1.3041150

    Article  MATH  Google Scholar 

  33. Chang, H., Demekhin, E.A.: Complex Wave Dynamics on Thin Films, vol. 14. Elsevier, Amsterdam (2002). ISBN 0-08-052953-4

  34. Liu, J., Paul, J.D., Gollub, J.P.: Measurements of the primary instabilities of film flows. J. Fluid Mech. 250, 69–101 (1993). https://doi.org/10.1017/S0022112093001387

    Article  Google Scholar 

  35. Wierschem, A., Lepski, C., Aksel, N.: Effect of long undulated bottoms on thin gravity-driven films. Acta Mech. 179(1), 41–66 (2005). https://doi.org/10.1007/s00707-005-0242-2

    Article  MATH  Google Scholar 

  36. Schörner, M., Reck, D., Aksel, N.: Stability phenomena far beyond the Nusselt flow–revealed by experimental asymptotics. Phys. Fluids 28(2), 022102 (2016). https://doi.org/10.1063/1.4941000

    Article  Google Scholar 

  37. Dauth, M., Schörner, M., Aksel, N.: What makes the free surface waves over topographies convex or concave? A study with Fourier analysis and particle tracking. Phys. Fluids 29(9), 092108 (2017). https://doi.org/10.1063/1.5003574

    Article  Google Scholar 

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Acknowledgements

The authors acknowledge Marion Märkl and Stephan Eißner for their help in carrying out the experiments.

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Authors and Affiliations

  1. Department of Applied Mechanics and Fluid Mechanics, University of Bayreuth, 95440, Bayreuth, Germany

    Armin Kögel & Nuri Aksel

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  1. Armin Kögel
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  2. Nuri Aksel
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Correspondence to Armin Kögel.

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Kögel, A., Aksel, N. Stability of the channel flow—new phenomena in an old problem. Acta Mech 231, 1063–1082 (2020). https://doi.org/10.1007/s00707-019-02568-8

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  • DOI: https://doi.org/10.1007/s00707-019-02568-8

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