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

  • Markus DauthEmail author
  • Nuri Aksel
Original Paper


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.



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.

Supplementary material


  1. 1.
    Webb, R.L.: Principles of Enhanced Heat Transfer. Wiley, New York (1994)Google Scholar
  2. 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)CrossRefzbMATHGoogle Scholar
  3. 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)CrossRefGoogle Scholar
  4. 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)CrossRefGoogle Scholar
  5. 5.
    Fair, J.R., Bravo, J.R.: Distillation columns containing structured packing. Chem. Eng. Prog. 86, 19–29 (1990)Google Scholar
  6. 6.
    Kistler, S.F., Schweizer, P.M.: Liquid Film Coating. Chapman and Hall, New York (1997)CrossRefGoogle Scholar
  7. 7.
    Weinstein, S.J., Ruschak, K.J.: Coating flows. Annu. Rev. Fluid Mech. 36, 29–53 (2004)CrossRefzbMATHGoogle Scholar
  8. 8.
    Gugler, G., Beer, R., Mauron, M.: Operative limits of curtain coating due to edges. Chem Eng Process Process Intensif 50, 462–465 (2011)CrossRefGoogle Scholar
  9. 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)CrossRefzbMATHGoogle Scholar
  10. 10.
    Greve, R., Blatter, H.: Dynamics of Ice Sheets and Glaciers. Springer, Berlin (2009)CrossRefGoogle Scholar
  11. 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)CrossRefGoogle Scholar
  12. 12.
    Hutter, K., Svendsen, B., Rickenmann, D.: Debris flow modeling: a review. Contin. Mech. Thermodyn. 8, 1–35 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Nusselt, W.: Die Oberflächenkondensation des Wasserdampfes. VDI Z. 60, 541–546 (1916)Google Scholar
  14. 14.
    Kapitza, P.L.: Wave flow of thin layers of a viscous fluid. Zh. Eksp. Teor. Fiz. 18, 1–28 (1948)Google Scholar
  15. 15.
    Kapitza, P.L., Kapitza, S.P.: Wave flow of thin layers of a viscous fluid. Zh. Eksp. Teor. Fiz. 19, 105–120 (1949)zbMATHGoogle Scholar
  16. 16.
    Benjamin, T.B.: Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2, 554–574 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Yih, C.S.: Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321–334 (1963)CrossRefzbMATHGoogle Scholar
  18. 18.
    Alekseenko, S.V., Nakoryakov, V.Y., Pokusaev, B.G.: Wave formation on a vertical falling liquid film. AIChE 31, 1446–1460 (1985)CrossRefzbMATHGoogle Scholar
  19. 19.
    Liu, J., Paul, J.D., Gollub, J.P.: Measurements of the primary instabilities of film flows. J. Fluid Mech. 250, 69–101 (1993)CrossRefGoogle Scholar
  20. 20.
    Chang, H.C.: Wave evolution on a falling film. Annu. Rev. Fluid Mech. 26, 103–136 (1994)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Liu, J., Gollub, J.P.: Solitary wave dynamics of film flows. Phys. Fluids 6, 1702–1712 (1994)CrossRefGoogle Scholar
  22. 22.
    Vlachogiannis, M., Bontozoglou, V.: Observations of solitary wave dynamics of film flows. J. Fluid Mech. 435, 191–215 (2001)CrossRefzbMATHGoogle Scholar
  23. 23.
    Gjevik, B.: Occurrence of finite-amplitude surface waves on falling liquid films. Phys. Fluids 13, 1918–1925 (1970)CrossRefzbMATHGoogle Scholar
  24. 24.
    Craster, R.V., Matar, O.K.: Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 1131–1198 (2009)CrossRefGoogle Scholar
  25. 25.
    Chang, H.C., Demekhin, E.A.: Complex Wave Dynamics on Thin Films. Elsevier, Amsterdam (2002)Google Scholar
  26. 26.
    Oron, A., Davis, S.H., Bankoff, S.G.: Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931–980 (1997)CrossRefGoogle Scholar
  27. 27.
    Bontozoglou, V., Papapolymerou, G.: Laminar film flow down a wavy incline. Int. J. Multiph. Flow 23, 69–79 (1997)CrossRefzbMATHGoogle Scholar
  28. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Pozrikidis, C.: The flow of a liquid film along a periodic wall. J. Fluid Mech. 188, 275–300 (1988)CrossRefzbMATHGoogle Scholar
  30. 30.
    Trifonov, Y.Y.: Viscous liquid film flows over a periodic surface. Int. J. Multiph. Flow 24, 1139–1161 (1999)CrossRefzbMATHGoogle Scholar
  31. 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)CrossRefGoogle Scholar
  32. 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. 33.
    Vlachogiannis, M., Bontozoglou, V.: Experiments on laminar film flow along a periodic wall. J. Fluid Mech. 457, 133–156 (2002)CrossRefzbMATHGoogle Scholar
  34. 34.
    Wierschem, A., Aksel, N.: Instability of a liquid film flowing down an inclined wavy plane. Physica D 186, 221–237 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 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)CrossRefGoogle Scholar
  36. 36.
    Wierschem, A., Lepski, C., Aksel, N.: Effect of long undulated bottoms on thin gravity-driven films. Acta Mech. 179, 41–66 (2005)CrossRefzbMATHGoogle Scholar
  37. 37.
    Trifonov, Y.Y.: Stability of a viscous liquid film flowing down a periodic surface. Int. J. Multiph. Flow 33, 1186–1204 (2007)CrossRefGoogle Scholar
  38. 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)CrossRefzbMATHGoogle Scholar
  39. 39.
    Dávalos-Orozco, L.A.: Nonlinear instability of a thin film flowing down a smoothly deformed surface. Phys. Fluids 19, 074103 (2007)CrossRefzbMATHGoogle Scholar
  40. 40.
    Dávalos-Orozco, L.A.: Instabilities of thin films flowing down flat and smoothly deformed walls. Microgravity Sci. Technol. 20, 225–229 (2008)CrossRefGoogle Scholar
  41. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Heining, C., Aksel, N.: Bottom reconstruction in thin-film flow over topography: steady solution and linear stability. Phys. Fluids 21, 083605 (2009)CrossRefzbMATHGoogle Scholar
  43. 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)CrossRefzbMATHGoogle Scholar
  44. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 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)CrossRefGoogle Scholar
  46. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Pollak, T., Aksel, N.: Crucial flow stabilization and multiple instability branches of gravity-driven films over topography. Phys. Fluids 25, 024103 (2013)CrossRefGoogle Scholar
  48. 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)CrossRefGoogle Scholar
  49. 49.
    Schörner, M., Reck, D., Aksel, N.: Stability phenomena far beyond the Nusselt flow–revealed by experimental asymptotics. Phys. Fluids 28, 022102 (2016)CrossRefGoogle Scholar
  50. 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)MathSciNetCrossRefGoogle Scholar
  51. 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)CrossRefzbMATHGoogle Scholar
  52. 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)CrossRefGoogle Scholar
  53. 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)MathSciNetCrossRefGoogle Scholar
  54. 54.
    Lin, S.P.: Finite-amplitude stability of a parallel flow with a free surface. J. Fluid Mech. 36, 113–126 (1969)CrossRefzbMATHGoogle Scholar
  55. 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)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Benney, D.J.: Long waves on liquid films. J. Math. Phys. 45, 150–155 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 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)CrossRefGoogle Scholar
  58. 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)CrossRefGoogle Scholar
  59. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 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)CrossRefzbMATHGoogle Scholar
  61. 61.
    Rosenau, P., Oron, A.: Evolution and breaking of liquid film flowing on a vertical cylinder. Phys. Fluids A 1, 1763–1766 (1989)MathSciNetCrossRefGoogle Scholar
  62. 62.
    Oron, A., Gottlieb, O.: Nonlinear dynamics of temporally excited falling liquid films. Phys. Fluids 14, 2622–2636 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Malamataris, N.A., Vlachogiannis, M., Bontozoglou, V.: Solitary waves on inclined films: flow structure and binary interactions. Phys. Fluids 14, 1082–1094 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Argyriadi, K., Serifi, K., Bontozoglou, V.: Nonlinear dynamics of inclined films under low-frequency forcing. Phys. Fluids 16, 2457–2468 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Nosoko, T., Miyara, A.: The evolution and subsequent dynamics of waves on a vertically falling liquid film. Phys. Fluids 16, 1118–1126 (2004)CrossRefzbMATHGoogle Scholar
  66. 66.
    Chang, H.-C., Demekhin, E., Kalaidin, E.: Interaction dynamics of solitary waves on a falling film. J. Fluid Mech. 294, 123–154 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    Reck, D., Aksel, N.: Recirculation areas underneath solitary waves on gravity-driven film flows. Phys. Fluids 27, 112107 (2015)CrossRefGoogle Scholar
  68. 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)CrossRefzbMATHGoogle Scholar
  69. 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)CrossRefGoogle Scholar
  70. 70.
    Reck, D., Aksel, N.: Experimental study on the evolution of traveling waves over an undulated incline. Phys. Fluids 25, 102101 (2013)CrossRefGoogle Scholar
  71. 71.
    Trifonov, Y.Y.: Nonlinear waves on a liquid film falling down an inclined corrugated surface. Phys. Fluids 29, 054104 (2017)CrossRefGoogle Scholar
  72. 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)CrossRefGoogle Scholar
  73. 73.
    Dauth, M., Aksel, N.: Breaking of waves on thin films over topographies. Phys. Fluids 30, 082113 (2018)CrossRefGoogle Scholar
  74. 74.
    Schörner, M., Aksel, N.: The stability cycle—a universal pathway for the stability of films over topography. Phys. Fluids 30, 012105 (2018)CrossRefGoogle Scholar
  75. 75.
    Eggers, J.: Drop formation—an overview. ZAMM-Zeitschrift für angewandte Mathematik und Mechanik 179, 400–410 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  76. 76.
    Ockendon, H., Ockendon, J.R.: Viscous Flow. Cambridge Texts in Applied Mathematics, Cambridge (1995)CrossRefzbMATHGoogle Scholar

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Authors and Affiliations

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

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