Isothermal Case: Two-Dimensional Flow

  • S. KalliadasisEmail author
  • C. Ruyer-Quil
  • B. Scheid
  • M. G. Velarde
Part of the Applied Mathematical Sciences book series (AMS, volume 176)


We analyze the linear and nonlinear stage of the instability of a falling liquid film by using the average models developed in Chap. 6. Their linear stability characteristics, e.g. their description of spatially growing disturbances in relation to the convective nature of the instability, are shown to be in good agreement with the Orr–Sommerfeld eigenvalue problem (Chap. 3). By using the average models, the mechanism of the primary instability, already discussed in Chap. 3, is then re-investigated within the framework of the wave hierarchy analysis proposed by Whitham. We emphasize the similarities between roll waves in open channels and solitary waves in film flows at large Reynolds numbers. In particular, two-equation models of film flows have a structure similar to the Saint-Venant equations for shallow-water flows. In both cases, the mechanism of the primary instability can be understood in terms of a wave hierarchy as the competition between kinematic and dynamic waves. We scrutinize the influence of dispersive effects associated with the stream-wise second-order viscous terms, a phenomenon we refer to as “viscous dispersion,” onto the kinematic waves: viscous damping of high-frequency waves reduces the kinematic wave speed which in turn reduces the gap in speed between kinematic and dynamic waves. As far as the nonlinear stage of the dynamics of a falling liquid film is concerned, it is dominated by a competition between the primary instability of the Nusselt flat film flow and the secondary instabilities of the traveling waves with saturated amplitudes. This competition is characterized by a variety of nonlinear processes (e.g., spatial and temporal modulations, phase locking) which are still not fully understood. Applying a periodic forcing at the inlet may regularize the flow, leading further downstream to regular periodic wave-trains whose properties can be obtained using elements from dynamical systems theory. We construct bifurcation diagrams of permanent-form traveling waves including solitary waves. Particular attention is given to the role of stream-wise viscous effects on the properties, such as shape, speed and solution branches of the traveling waves. Taking into account these effects is crucial for a proper description of the dynamics of wavy film flows.


Solitary Wave Hopf Bifurcation Bifurcation Diagram Travel Wave Solution Homoclinic Orbit 
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  1. 3.
    Alekseenko, S.V., Nakoryakov, V.E., Pokusaev, B.G.: Wave Flow in Liquid Films, 3rd edn. Begell House, New York (1994) Google Scholar
  2. 4.
    Alekseenko, S.V., Nakoryakov, V.E., Pokusaev, B.G.: Wave formation on a vertical falling liquid film. AIChE J. 31, 1446–1460 (1985) CrossRefGoogle Scholar
  3. 13.
    Bach, P., Villadsen, J.: Simulation of the vertical flow of a thin, wavy film using a finite-element method. Int. J. Heat Mass Transf. 27, 815–827 (1984) zbMATHCrossRefGoogle Scholar
  4. 15.
    Balmforth, N.J., Ierley, G.R., Spiegel, E.A.: Chaotic pulse trains. SIAM J. Appl. Math. 54(5), 1291–1334 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 31.
    Brevdo, L., Laure, P., Dias, F., Bridges, T.J.: Linear pulse structure and signaling in a film flow on an inclined plane. J. Fluid Mech. 396, 37–71 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 32.
    Briggs, R.J.: Electron-Stream Interaction with Plasmas. MIT Press, Cambridge (1964) Google Scholar
  7. 33.
    Brock, R.R.: Periodic permanent roll waves. J. Hydrol. Eng. 96(12), 2565–2580 (1970) Google Scholar
  8. 34.
    Bunov, A.V., Demekhin, E.A., Shkadov, V. Ya.: On the non uniqueness of nonlinear waves on a falling film. J. Appl. Math. Mech. 48(4), 691–696 (1984) MathSciNetGoogle Scholar
  9. 35.
    Burns, J.C.: Long waves in running water. Proc. Camb. Philol. Soc. 49, 695–706 (1953) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 37.
    Carbone, F., Aubry, N., Liu, J., Gollub, J.P., Lima, R.: Space-time description of the splitting and coalescence of wave fronts in film flows. Physica D 96, 182–199 (1996) CrossRefGoogle Scholar
  11. 39.
    Champneys, A.R., Kuznetsov, Y.A.: Numerical detection and continuation of codimension-two homoclinic bifurcations. Int. J. Bifurc. Chaos 4, 785–822 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 44.
    Chang, H.-C., Demekhin, E.A.: Complex Wave Dynamics on Thin Films. D. Möbius and R. Miller. Elsevier, Amsterdam (2002) Google Scholar
  13. 45.
    Chang, H.-C., Demekhin, E., Kalaidin, E.: Interaction dynamics of solitary waves on a falling film. J. Fluid Mech. 294, 123–154 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  14. 46.
    Chang, H.-C., Demekhin, E.A., Kalaidin, E.: Simulation of noise-driven wave dynamics on a falling film. AIChE J. 42, 1553–1568 (1996) CrossRefGoogle Scholar
  15. 49.
    Chang, H.-C., Demekhin, E.A., Kopelevitch, D.I.: Laminarizing effects of dispersion in an active-dissipative nonlinear medium. Physica D 63, 299–320 (1993) zbMATHCrossRefGoogle Scholar
  16. 50.
    Chang, H.-C., Demekhin, E.A., Kopelevitch, D.I.: Nonlinear evolution of waves on a vertically falling film. J. Fluid Mech. 250, 433–480 (1993) MathSciNetCrossRefGoogle Scholar
  17. 73.
    Demekhin, E.A., Tokarev, G.Yu., Shkadov, V.Ya.: Hierarchy of bifurcations of space-periodic structures in a nonlinear model of active dissipative media. Physica D 52, 338–361 (1991) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 79.
    Doedel, E.J.: AUTO07p continuation and bifurcation software for ordinary differential equations. Montreal Concordia University (2008) Google Scholar
  19. 84.
    Dressler, R.F.: Mathematical solution of the problem of roll-waves in inclined open channels. Commun. Pure Appl. Math. 2, 149–194 (1949) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 87.
    Duprat, C., Giorgiutti-Dauphiné, F., Tseluiko, D., Saprykin, S., Kalliadasis, S.: Liquid film coating a fiber as a model system for the formation of bound states in active dispersive-dissipative nonlinear media. Phys. Rev. Lett. 103, 234501 (2009) CrossRefGoogle Scholar
  21. 90.
    Elphick, C., Ierley, G.R., Regev, O., Spiegel, E.A.: Interacting localized structures with Galilean invariance. Phys. Rev. A 44, 1110–1123 (1991) CrossRefGoogle Scholar
  22. 101.
    Gaspard, P.: Local birth of homoclinic chaos. Physica D 62, 94–122 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  23. 103.
    Glendinning, P., Sparrow, C.: Local and global behavior near homoclinic orbits. J. Stat. Phys. 35, 645–696 (1984) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 111.
    Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer, New York (1983) zbMATHGoogle Scholar
  25. 116.
    Ho, L.W., Patera, A.T.: A Legendre spectral element method for simulation of unsteady incompressible viscous free-surface flows. Comput. Methods Appl. Mech. Eng. 80, 355–366 (1990) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 120.
    Huerre, P., Monkewitz, P.: Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473–537 (1990) MathSciNetCrossRefGoogle Scholar
  27. 121.
    Huerre, P., Rossi, M.: Hydrodynamic instabilities in open flows. In: Godrèche, C., Manneville, P. (Eds.) Hydrodynamic and Nonlinear Instabilities, pp. 81–294. Cambridge University Press, London (1998). Especially §8, 9 CrossRefGoogle Scholar
  28. 125.
    Johnson, R.S.: Shallow water waves on a viscous fluid—the undular bore. Phys. Fluids 15(10), 1693–1699 (1972) zbMATHCrossRefGoogle Scholar
  29. 126.
    Johnson, R.S.: A Modern Introduction to the Mathematical Theory of Water Waves, 1st edn. Cambridge Texts in Applied Mathematics (No. 19), p. 445. Cambridge University Press, Cambridge (1997) zbMATHCrossRefGoogle Scholar
  30. 128.
    Joo, S.W., Davis, S.H., Bankoff, S.G.: Long-wave instabilities of heated falling films: two-dimensional theory of uniform layers. J. Fluid Mech. 230, 117–146 (1991) zbMATHCrossRefGoogle Scholar
  31. 140.
    Kapitza, P.L.: Wave flow of thin layers of a viscous fluid: I. Free flow. II. Fluid flow in the presence of continuous gas flow and heat transfer. In: ter Haar, D. (Ed.) Collected Papers of P.L. Kapitza (1965), pp. 662–689. Pergamon, Oxford (1948). (Original paper in Russian: Zh. Eksp. Teor. Fiz. 18, I. 3–18, II. 19–28) Google Scholar
  32. 141.
    Kapitza, P.L., Kapitza, S.P.: Wave flow of thin layers of a viscous fluid: III. Experimental study of undulatory flow conditions. In: ter Haar, D. (Ed.) Collected Papers of P.L. Kapitza (1965), pp. 690–709. Pergamon, Oxford (1949). (Original paper in Russian: Zh. Eksp. Teor. Fiz. 19, 105–120) Google Scholar
  33. 144.
    Kawahara, T.: Formation of saturated solitons in a nonlinear dispersive system with instability and dissipation. Phys. Rev. Lett. 51, 381–383 (1983) CrossRefGoogle Scholar
  34. 161.
    Lee, J.-J., Mei, C.C.: Stationary waves on an inclined sheet of viscous fluid at high Reynolds and moderate Weber numbers. J. Fluid Mech. 307, 191–229 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  35. 167.
    Liu, J., Gollub, J.P.: Onset of spatially chaotic waves on flowing films. Phys. Rev. Lett. 70, 2289–2292 (1993) CrossRefGoogle Scholar
  36. 168.
    Liu, J., Gollub, J.P.: Solitary wave dynamics of film flows. Phys. Fluids 6, 1702–1712 (1994) CrossRefGoogle Scholar
  37. 169.
    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
  38. 170.
    Liu, J., Schneider, J.B., Gollub, J.P.: Three-dimensional instabilities of film flows. Phys. Fluids 7, 55–67 (1995) MathSciNetCrossRefGoogle Scholar
  39. 172.
    Liu, Q.Q., Chen, L., Li, J.C., Singh, V.P.: Roll waves in overland flow. J. Hydrol. Eng. 10(2), 110–117 (2005) CrossRefGoogle Scholar
  40. 176.
    Malamataris, N.A., Vlachogiannis, M., Bontozoglou, V.: Solitary waves on inclined films: Flow structure and binary interactions. Phys. Fluids 14(3), 1082–1094 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  41. 177.
    Manneville, P.: Dissipative Structures and Weak Turbulence. Academic Press, New York (1990) zbMATHGoogle Scholar
  42. 189.
    Nepomnyashchy, A.A., Velarde, M.G., Colinet, P.: Interfacial Phenomena and Convection. Chapman & Hall/CRC, London (2002) zbMATHGoogle Scholar
  43. 195.
    Oldeman, B.E., Champneys, A.R., Krauskopf, B.: Homoclinic branch switching: a numerical implementation of Lin’s method. Int. J. Bifurc. Chaos 13, 2977–2999 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  44. 196.
    Ooshida, T.: Surface equation of falling film flows with moderate Reynolds number and large but finite Weber number. Phys. Fluids 11, 3247–3269 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  45. 198.
    Oron, A., Gottlieb, O.: Nonlinear dynamics of temporally excited falling liquid films. Phys. Fluids 14, 2622–2636 (2002) MathSciNetCrossRefGoogle Scholar
  46. 212.
    Pradas, M., Tseluiko, D., Kalliadasis, S.: Rigorous coherent-structure theory for falling liquid films: Viscous dispersion effects on bound-state formation and self-organization. Phys. Fluids 23, 044104 (2011) CrossRefGoogle Scholar
  47. 215.
    de Saint-Venant, A.J.C.: Théorie du mouvement non-permanent des eaux, avec applications aux crues des rivières et à l’introduction des marées dans leur lit. C. R. Acad. Sci. Paris 73, 147–154 (1871) Google Scholar
  48. 216.
    Pumir, A., Manneville, P., Pomeau, Y.: On solitary waves running down an inclined plane. J. Fluid Mech. 135, 27–50 (1983) zbMATHCrossRefGoogle Scholar
  49. 218.
    Ramaswamy, B., Chippada, S., Joo, S.W.: A full-scale numerical study of interfacial instabilities in thin-film flows. J. Fluid Mech. 325, 163–194 (1996) zbMATHCrossRefGoogle Scholar
  50. 226.
    Ruyer-Quil, C., Manneville, P.: Modeling film flows down inclined planes. Eur. Phys. J. B 6, 277–292 (1998) CrossRefGoogle Scholar
  51. 229.
    Ruyer-Quil, C., Manneville, P.: On the speed of solitary waves running down a vertical wall. J. Fluid Mech. 531, 181–190 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  52. 232.
    Salamon, T.R., Armstrong, R.C., Brown, R.A.: Traveling waves on vertical films: Numerical analysis using the finite element method. Phys. Fluids 6, 2202 (1994) zbMATHCrossRefGoogle Scholar
  53. 233.
    Saprykin, S., Demekhin, E.A., Kalliadasis, S.: Self-organization of two-dimensional waves in an active dispersive-dissipative nonlinear medium. Phys. Rev. Lett. 94, 224101 (2005) CrossRefGoogle Scholar
  54. 234.
    Saprykin, S., Demekhin, E.A., Kalliadasis, S.: Two-dimensional wave dynamics in thin films. I. Stationary solitary pulses. Phys. Fluids 17, 117105 (2005) MathSciNetCrossRefGoogle Scholar
  55. 235.
    Saprykin, S., Demekhin, E.A., Kalliadasis, S.: Two-dimensional wave dynamics in thin films. II. Formation of lattices of interacting stationary solitary pulses. Phys. Fluids 17, 117106 (2005) MathSciNetCrossRefGoogle Scholar
  56. 239.
    Scheid, B., Oron, A., Colinet, P., Thiele, U., Legros, J.C.: Nonlinear evolution of non-uniformly heated falling liquid films. Phys. Fluids 14, 4130–4151 (2002). Erratum: Phys. Fluids 15, 583 (2003) MathSciNetCrossRefGoogle Scholar
  57. 248.
    Shkadov, V.Ya.: Wave flow regimes of a thin layer of viscous fluid subject to gravity. Izv. Akad. Nauk SSSR, Meh. židk. Gaza 1, 43–51 (1967). (English translation in Fluid Dynamics 2, 29–34 (Faraday Press, New York, 1970)) Google Scholar
  58. 250.
    Shkadov, V.Ya., Sisoev, G.M.: Waves induced by instability in falling films of finite thickness. Fluid Dyn. Res. 35, 357–389 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  59. 253.
    Sisoev, G.M., Shkadov, V.Ya.: A two-parameter manifold of wave solutions to an equation for a falling film of viscous fluid. Dokl. Phys. 44, 454–459 (1999) MathSciNetGoogle Scholar
  60. 256.
    Smith, M.K.: The mechanism for the long-wave instability in thin liquid films. J. Fluid Mech. 217, 469–485 (1990) zbMATHCrossRefGoogle Scholar
  61. 270.
    Thomas, H.A.: The propagation of waves in steep prismatic conduits. In: Proc. Hydraulics Conf., pp. 214–229. Univ. of Iowa, Iowa City (1939) Google Scholar
  62. 273.
    Tihon, J., Serifi, K., Argyiriadi, K., Bontozoglou, V.: Solitary waves on inclined films: their characteristics and the effects on wall shear stress. Exp. Fluids 41, 79–89 (2006) CrossRefGoogle Scholar
  63. 281.
    Trifonov, Yu.Ya., Tsvelodub, O.Yu.: 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 (1991) MathSciNetzbMATHCrossRefGoogle Scholar
  64. 283.
    Tseluiko, D., Kalliadasis, S.: Coherent-structure theory for 3d active dispersive-dissipative nonlinear media (2011, in preparation) Google Scholar
  65. 284.
    Tseluiko, D., Saprykin, S., Kalliadasis, S.: Interaction of solitary pulses in active dispersive-dissipative media. Proc. Est. Acad. Sci. 59, 139–144 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  66. 286.
    Tseluiko, D., Saprykin, S., Duprat, C., Giorgiutti-Dauphiné, F., Kalliadasis, S.: Pulse dynamics in low-Reynolds-number interfacial hydrodynamics: Experiments and theory. Physica D 239, 2000–2010 (2010) zbMATHCrossRefGoogle Scholar
  67. 287.
    Tsvelodub, O.Yu., Trifonov, Yu.Ya.: Nonlinear waves on the surface of a falling liquid film. Part 2. Bifurcations of the first-family waves and other types of nonlinear waves. J. Fluid Mech. 244, 149–169 (1992) MathSciNetCrossRefGoogle Scholar
  68. 288.
    Usha, R., Uma, B.: Modeling of stationary waves on a thin viscous film down an inclined plane at high Reynolds numbers and moderate Weber numbers using energy integral method. Phys. Fluids 16(7), 2679–2696 (2004) CrossRefGoogle Scholar
  69. 292.
    Verhulst, F.: Nonlinear Differential Equations and Dynamical Systems. Springer, Berlin (1990) zbMATHGoogle Scholar
  70. 294.
    Vlachogiannis, M., Bontozoglou, V.: Observations of solitary wave dynamics of film flows. J. Fluid Mech. 435, 191–215 (2001) zbMATHCrossRefGoogle Scholar
  71. 299.
    Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1974) zbMATHGoogle Scholar
  72. 301.
    Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (1990) zbMATHGoogle Scholar
  73. 305.
    Yoshimura, P.N., Nosoko, T., Nagata, T.: Enhancement of mass transfer into a falling laminar liquid film by two-dimensional surface waves—some experimental observations and modeling. Chem. Eng. Sci. 51(8), 1231–1240 (1996) CrossRefGoogle Scholar

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© Springer-Verlag London Limited 2012

Authors and Affiliations

  • S. Kalliadasis
    • 1
    Email author
  • C. Ruyer-Quil
    • 2
  • B. Scheid
    • 3
  • M. G. Velarde
    • 4
  1. 1.Department of Chemical EngineeringImperial College LondonLondonUK
  2. 2.Laboratoire FASTUniversité Pierre et Marie Curie (UPMC)ParisFrance
  3. 3.TIPs—Fluid Physics UnitUniversité Libre de BruxellesBrusselsBelgium
  4. 4.Instituto PluridisciplinarUniversidad ComplutenseMadridSpain

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