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Transition of regular wave fronts to irregular wave fronts in gravity-driven thin films over topography

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Abstract

This article contributes to an understanding of the pathway from regular to chaotic traveling wave fronts over periodically undulated inclines in thin films. In order to investigate the transition from regular to chaotic waves, we used various undulation forms and varied the Reynolds number and the inclination angle in the measurements. Thereby, we revealed the first partially chaotic waves on a gravity-driven thin film channel flow. The wave is subdivided into: (i) the chaotic wave front and (ii) a regular wave tail. The area of the chaotic part can be increased by increasing the inertia of the system. Various phenomena on the flow were revealed: (a) bubble formation, (b) fingering, (c) splashes, and (d) pinch-offs. Our investigation leads to the conclusion that wave breaking over obstacles is a necessary condition for the transition from regular to chaotic wave fronts.

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References

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

    Google Scholar 

  2. 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, 757–772 (2002)

    Article  MATH  Google Scholar 

  3. Valluri, P., Matar, O.K., Hewitt, G.F., Mendes, M.A.: Thin film flow over structured packings at moderate Reynolds numbers. Chem. Eng. Sci. 60, 1965–1975 (2005)

    Article  Google Scholar 

  4. de Santos, J.M., Melli, T.R., Scriven, L.E.: Mechanics of Gas–Liquid flow in packed-bed contactors. Annu. Rev. Fluid Mech. 23, 233–260 (1991)

    Article  Google Scholar 

  5. Fair, J.R., Bravo, J.R.: Distillation columns containing structured packing. Chem. Eng. Prog. 86, 19–29 (1990)

    Google Scholar 

  6. Kistler, S.F., Schweizer, P.M.: Liquid Film Coating. Chapman and Hall, New York (1997)

    Book  Google Scholar 

  7. Weinstein, S.J., Ruschak, K.J.: Coating flows. Annu. Rev. Fluid Mech. 36, 29–53 (2004)

    Article  MATH  Google Scholar 

  8. Gugler, G., Beer, R., Mauron, M.: Operative limits of curtain coating due to edges. Chem Eng Process Process Intensif 50, 462–465 (2011)

    Article  Google Scholar 

  9. Luca, I., Hutter, K., Tai, Y.C., Kuo, C.Y.: A hierarchy of avalanche models on arbitrary topography. Acta Mech. 205, 121–149 (2009)

    Article  MATH  Google Scholar 

  10. Greve, R., Blatter, H.: Dynamics of Ice Sheets and Glaciers. Springer, Berlin (2009)

    Book  Google Scholar 

  11. Kumar, A., Karig, D., Acharya, R., Neethirajan, S., Mukherjee, P.P., Retterer, S., Doktycz, M.J.: Microscale confinement features can affect biofilm formation. Microfluid. Nanofluid. 14, 895–902 (2013)

    Article  Google Scholar 

  12. Hutter, K., Svendsen, B., Rickenmann, D.: Debris flow modeling: a review. Contin. Mech. Thermodyn. 8, 1–35 (1994)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  14. Kapitza, P.L.: Wave flow of thin layers of a viscous fluid. Zh. Eksp. Teor. Fiz. 18, 1–28 (1948)

    Google Scholar 

  15. Kapitza, P.L., Kapitza, S.P.: Wave 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, 554–574 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  17. Yih, C.S.: Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321–334 (1963)

    Article  MATH  Google Scholar 

  18. Alekseenko, S.V., Nakoryakov, V.Y., Pokusaev, B.G.: Wave formation on a vertical falling liquid film. AIChE 31, 1446–1460 (1985)

    Article  MATH  Google Scholar 

  19. Liu, J., Paul, J.D., Gollub, J.P.: Measurements of the primary instabilities of film flows. J. Fluid Mech. 250, 69–101 (1993)

    Article  Google Scholar 

  20. Chang, H.C.: Wave evolution on a falling film. Annu. Rev. Fluid Mech. 26, 103–136 (1994)

    Article  MathSciNet  Google Scholar 

  21. Liu, J., Gollub, J.P.: Solitary wave dynamics of film flows. Phys. Fluids 6, 1702–1712 (1994)

    Article  Google Scholar 

  22. Vlachogiannis, M., Bontozoglou, V.: Observations of solitary wave dynamics of film flows. J. Fluid Mech. 435, 191–215 (2001)

    Article  MATH  Google Scholar 

  23. Gjevik, B.: Occurrence of finite-amplitude surface waves on falling liquid films. Phys. Fluids 13, 1918–1925 (1970)

    Article  MATH  Google Scholar 

  24. Craster, R.V., Matar, O.K.: Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 1131–1198 (2009)

    Article  Google Scholar 

  25. Chang, H.C., Demekhin, E.A.: Complex Wave Dynamics on Thin Films. Elsevier, Amsterdam (2002)

    Google Scholar 

  26. Oron, A., Davis, S.H., Bankoff, S.G.: Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931–980 (1997)

    Article  Google Scholar 

  27. Bontozoglou, V., Papapolymerou, G.: Laminar film flow down a wavy incline. Int. J. Multiph. Flow 23, 69–79 (1997)

    Article  MATH  Google Scholar 

  28. Scholle, M., Aksel, N.: An exact solution of visco-capillary flow in an inclined channel. Zeitschrift für Angewandte Mathematik und Physik ZAMP 52, 749–769 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  29. Pozrikidis, C.: The flow of a liquid film along a periodic wall. J. Fluid Mech. 188, 275–300 (1988)

    Article  MATH  Google Scholar 

  30. Trifonov, Y.Y.: Viscous liquid film flows over a periodic surface. Int. J. Multiph. Flow 24, 1139–1161 (1999)

    Article  MATH  Google Scholar 

  31. Wierschem, A., Bontozoglou, V., Heining, C., Uecker, H., Aksel, N.: Linear resonance in viscous films on inclined wavy planes. Int. J. Multiph. Flow 34, 580–589 (2008)

    Article  Google Scholar 

  32. Trifonov, Y.Y.: Viscous liquid film flows over a vertical corrugated surface and the film free surface stability. Russ. J. Eng. Thermophys. 10(2), 129–145 (2000)

    Google Scholar 

  33. Vlachogiannis, M., Bontozoglou, V.: Experiments on laminar film flow along a periodic wall. J. Fluid Mech. 457, 133–156 (2002)

    Article  MATH  Google Scholar 

  34. Wierschem, A., Aksel, N.: Instability of a liquid film flowing down an inclined wavy plane. Physica D 186, 221–237 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  35. Wierschem, A., Scholle, M., Aksel, N.: Comparison of different theoretical approaches to experiments on film flow down an inclined wavy channel. Exp. Fluids 33, 429–442 (2002)

    Article  Google Scholar 

  36. Wierschem, A., Lepski, C., Aksel, N.: Effect of long undulated bottoms on thin gravity-driven films. Acta Mech. 179, 41–66 (2005)

    Article  MATH  Google Scholar 

  37. Trifonov, Y.Y.: Stability of a viscous liquid film flowing down a periodic surface. Int. J. Multiph. Flow 33, 1186–1204 (2007)

    Article  Google Scholar 

  38. Trifonov, Y.Y.: Stability and nonlinear wavy regimes in downward film flows on a corrugated surface. J. App. Mech. Tech. Phys. 48, 91–100 (2007)

    Article  MATH  Google Scholar 

  39. Dávalos-Orozco, L.A.: Nonlinear instability of a thin film flowing down a smoothly deformed surface. Phys. Fluids 19, 074103 (2007)

    Article  MATH  Google Scholar 

  40. Dávalos-Orozco, L.A.: Instabilities of thin films flowing down flat and smoothly deformed walls. Microgravity Sci. Technol. 20, 225–229 (2008)

    Article  Google Scholar 

  41. Trifonov, Y.Y.: Stability and bifurcations of the wavy film flow down a vertical plate: the results of integral approaches and full-scale computations. Fluid Dyn. Res. 44, 031418 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  42. Heining, C., Aksel, N.: Bottom reconstruction in thin-film flow over topography: steady solution and linear stability. Phys. Fluids 21, 083605 (2009)

    Article  MATH  Google Scholar 

  43. D’Alessio, S.J.D., Pascal, J.P., Jasmine, H.A.: Instability in gravity-driven flow over uneven surfaces. Phys. Fluids 21, 062105 (2009)

    Article  MATH  Google Scholar 

  44. Wierschem, A., Scholle, M., Aksel, N.: Vortices in film flow over strongly undulated bottom profiles at low Reynolds numbers. Phys. Fluids 15, 426–435 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  45. 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, 847–857 (2010)

    Article  Google Scholar 

  46. Tseluiko, D., Blyth, M.G., Papageorgiou, D.T.: Stability of film flow over inclined topography based on a long-wave nonlinear model. J. Fluid Mech. 729, 638–671 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  47. Pollak, T., Aksel, N.: Crucial flow stabilization and multiple instability branches of gravity-driven films over topography. Phys. Fluids 25, 024103 (2013)

    Article  Google Scholar 

  48. Trifonov, Y.Y.: Stability of a film flowing down an inclined corrugated plate: the direct Navier–Stokes computations and Floquet theory. Phys. Fluids 26, 114101 (2014)

    Article  Google Scholar 

  49. Schörner, M., Reck, D., Aksel, N.: Stability phenomena far beyond the Nusselt flow–revealed by experimental asymptotics. Phys. Fluids 28, 022102 (2016)

    Article  Google Scholar 

  50. Schörner, M., Reck, D., Aksel, N., Trifonov, Y.Y.: Switching between different types of stability isles in films over topographies. Acta. Mech. 229, 423–436 (2018)

    Article  MathSciNet  Google Scholar 

  51. Cao, Z., Vlachogiannis, M., Bontozoglou, V.: Experimental evidence for a short-wave global mode in film flow along periodic corrugations. J. Fluid Mech. 718, 304–320 (2013)

    Article  MATH  Google Scholar 

  52. Schörner, M., Reck, D., Aksel, N.: Does the topography’s specific shape matter in general for the stability of film flows? Phys. Fluids 27, 042103 (2015)

    Article  Google Scholar 

  53. Aksel, N., Schörner, M.: Films over topography: from creeping flow to linear stability, theory, and experiments, a review. Acta Mech. 229, 1453–1482 (2018)

    Article  MathSciNet  Google Scholar 

  54. Lin, S.P.: Finite-amplitude stability of a parallel flow with a free surface. J. Fluid Mech. 36, 113–126 (1969)

    Article  MATH  Google Scholar 

  55. Chang, H.-C., Demekhin, E.A., Kopelevich, D.I.: Nonlinear evolution of waves on a vertically falling film. J. Fluid Mech. 250, 433–480 (1993)

    Article  MathSciNet  Google Scholar 

  56. Benney, D.J.: Long waves on liquid films. J. Math. Phys. 45, 150–155 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  57. Krantz, W.B., Goren, S.L.: Finite-amplitude, long waves on liquid films flowing down a plane. Ind. Eng. Chem. Fundam. 9, 107–113 (1970)

    Article  Google Scholar 

  58. Trifononv, Y.Y., Tsvelodub, O.Y.: Nonlinear waves on the surface of a falling liquid film. Part 1. Waves of the first family and their stability. J. Fluid Mech. 229, 531–554 (1990)

    Article  Google Scholar 

  59. Yu, L.Q., Wasden, F.K., Dukler, A.E., Balakotaiah, V.: Nonlinear evolution of waves on falling films at high Reynolds numbers. Phys. Fluids 7, 1886–1902 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  60. Ruyer-Quil, C., Manneville, P.: Improved modeling of flows down inclined planes. Eur. Phys. J. B Condens. Matter Complex Syst. 15, 357–369 (2000)

    Article  MATH  Google Scholar 

  61. Rosenau, P., Oron, A.: Evolution and breaking of liquid film flowing on a vertical cylinder. Phys. Fluids A 1, 1763–1766 (1989)

    Article  MathSciNet  Google Scholar 

  62. Oron, A., Gottlieb, O.: Nonlinear dynamics of temporally excited falling liquid films. Phys. Fluids 14, 2622–2636 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  63. Malamataris, N.A., Vlachogiannis, M., Bontozoglou, V.: Solitary waves on inclined films: flow structure and binary interactions. Phys. Fluids 14, 1082–1094 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  64. Argyriadi, K., Serifi, K., Bontozoglou, V.: Nonlinear dynamics of inclined films under low-frequency forcing. Phys. Fluids 16, 2457–2468 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  65. Nosoko, T., Miyara, A.: The evolution and subsequent dynamics of waves on a vertically falling liquid film. Phys. Fluids 16, 1118–1126 (2004)

    Article  MATH  Google Scholar 

  66. Chang, H.-C., Demekhin, E., Kalaidin, E.: Interaction dynamics of solitary waves on a falling film. J. Fluid Mech. 294, 123–154 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  67. Reck, D., Aksel, N.: Recirculation areas underneath solitary waves on gravity-driven film flows. Phys. Fluids 27, 112107 (2015)

    Article  Google Scholar 

  68. Trifonov, Y.Y.: Stability and nonlinear wavy regimes in downward film flows on a corrugated surface. J. Appl. Mech. Tech. Phys. 48, 4851–4866 (2007)

    Article  MATH  Google Scholar 

  69. Argyriadi, K., Vlachogiannis, M., Bontozoglou, V.: Experimental study of inclined film flow along periodic corrugations: the effect of wall steepness. Phys. Fluids 18, 012102 (2006)

    Article  Google Scholar 

  70. Reck, D., Aksel, N.: Experimental study on the evolution of traveling waves over an undulated incline. Phys. Fluids 25, 102101 (2013)

    Article  Google Scholar 

  71. Trifonov, Y.Y.: Nonlinear waves on a liquid film falling down an inclined corrugated surface. Phys. Fluids 29, 054104 (2017)

    Article  Google Scholar 

  72. 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, 092108 (2017)

    Article  Google Scholar 

  73. Dauth, M., Aksel, N.: Breaking of waves on thin films over topographies. Phys. Fluids 30, 082113 (2018)

    Article  Google Scholar 

  74. Schörner, M., Aksel, N.: The stability cycle—a universal pathway for the stability of films over topography. Phys. Fluids 30, 012105 (2018)

    Article  Google Scholar 

  75. Eggers, J.: Drop formation—an overview. ZAMM-Zeitschrift für angewandte Mathematik und Mechanik 179, 400–410 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  76. Ockendon, H., Ockendon, J.R.: Viscous Flow. Cambridge Texts in Applied Mathematics, Cambridge (1995)

    Book  MATH  Google Scholar 

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Acknowledgements

The authors acknowledge Stephan Eißner for his help in carrying out parts of the experiments. Furthermore, we want to thank Mario Schörner for the helpful discussions.

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

    Markus Dauth & Nuri Aksel

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  1. Markus Dauth
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  2. Nuri Aksel
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Correspondence to Markus Dauth.

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Appendices

A Origin of the dents

See Fig. 12.

Fig. 12
figure 12

Part of a traveling free surface wave which is already corrected by the steady-state free surface with a sketch of the underlying substrate. Reprinted with permission from M. Dauth and N. Aksel, “Breaking of waves on thin films over topographies” Phys. Fluids 30, 082113 (2018). Copyright 2017 AIP Publishing LLC

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B Wave front over flat incline

See Fig. 13.

Fig. 13
figure 13

The front view on a wave front of a wave flowing over a flat substrate with the inclination angle \(\alpha =30^\circ \) and \(Re=18\)

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C Pathway to chaos

See Fig. 14.

Fig. 14
figure 14

Schematic response diagram for transition to turbulence. In the lower part, characteristic diagrams are plotted. In case of the regular wave front, the amplitude of the traveling wave is plotted over the downstream position

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Dauth, M., Aksel, N. Transition of regular wave fronts to irregular wave fronts in gravity-driven thin films over topography. Acta Mech 230, 2475–2490 (2019). https://doi.org/10.1007/s00707-019-02417-8

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  • DOI: https://doi.org/10.1007/s00707-019-02417-8

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