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Nonlinear evolution of waves on a vertically falling film

  • H.-C. Chang (a1), E. A. Demekhin (a2) and D. I. Kopelevich (a2)


Wave formation on a falling film is an intriguing hydrodynamic phenomenon involving transitions among a rich variety of spatial and temporal structures. Immediately beyond an inception region, short, near-sinusoidal capillary waves are observed. Further downstream, long, near-solitary waves with large tear-drop humps preceded by short, front-running capillary waves appear. Both kinds of waves evolve slowly downstream such that over about ten wavelengths, they resemble stationary waves which propagate at constant speeds and shapes. We exploit this quasi-steady property here to study wave evolution and selection on a vertically falling film. All finite-amplitude stationary waves with the same average thickness as the Nusselt flat film are constructed numerically from a boundary-layer approximation of the equations of motion. As is consistent with earlier near-critical analyses, two travelling wave families are found, each parameterized by the wavelength or the speed. One family γ1 travels slower than infinitesimally small waves of the same wavelength while the other family γ2 and its hybrids travel faster. Stability analyses of these waves involving three-dimensional disturbances of arbitrary wavelength indicate that there exists a unique nearly sinusoidal wave on the slow family γ1 with wavenumber αs (or α2) that has the lowest growth rate. This wave is slightly shorter than the fastest growing linear mode with wavenumber αm and approaches the wave on γ1 with the highest flow rate at low Reynolds numbers. On the fast γ2 family, however, multiple bands of near-solitary waves bounded below by αf are found to be stable to two-dimensional disturbances. This multiplicity of stable bands can be interpreted as a result of favourable interaction among solitary-wave-like coherent structures to form a periodic train. (All waves are unstable to three-dimensional disturbances with small growth rates.) The suggested selection mechanism is consistent with literature data and our numerical experiments that indicate waves slow down immediately beyond inception as they approach the short capillary wave with wavenumber α2 of the slow γ1 family. They then approach the long stable waves on the γ2 family further downstream and hence accelerate and develop into the unique solitary wave shapes, before they succumb to the slowly evolving transverse disturbances.



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Armbruster, D., Guckenheimer, J. & Holmes, P. 1988 Heteroclinic cycles and modulated travelling waves in systems with O(2) symmetry. Physica 29D, 257282.
Alekseenko, S. V., Nakoryakov, V. E. & Pokusaev, B. G. 1985 Wave formation on a vertical falling liquid film. AIChE J. 31, 14461460.
Bach, P. & Villadsen, J. 1984 Simulation of the vertical flow of a thin wavy film using a finite element method. Intl J. Heat Mass Transfer 27, 815827.
Benjamin, T. B. 1957 Wave formation in laminar flow down an incline plane. J. Fluid Mech. 2, 554574.
Benney, B. J. 1966 Long waves in liquid films. J. Math. Phys. 45, 150155.
Brauner, H. & Maron, D. M. 1983 Modeling of wavy flow inclined thin films. Chem. Engng Sci. 38, 775788.
Bunov, A. V., Demekhin, E. A. & Shkadov, V. Ya 1984 On the non-uniqueness of nonlinear wave solutions in a viscous layer. Prikh. Mat. Mekh. 48, 691696.
Chang, H.-C. 1986 Traveling waves in fluid interfaces: Normal form analysis of the Kuramoto–Sivashinsky equation. Phys. Fluids 29, 31423147.
Chang, H.-C. 1989 Onset of nonlinear waves on falling films. Phys. Fluids A 1, 13141327.
Chang, H.-C., Demekhin, E. A. & Kopelevich, D. I. 1993 Laminarizing effects of dispersion in an active-dissipative nonlinear medium. Physica D (in press).
Cheng, M. & Chang, H.-C. 1990 A generalized sideband stability theory via center manifold projection. Phys. Fluids A 2, 13641379.
Cheng, M. & Chang, H.-C. 1992a Stability of axisymmetric waves on liquid films flowing down a vertical column to azimuthal and streamwise disturbances. Chem. Engng Commun. 118, 327340.
Cheng, M. & Chang, H.-C. 1992b Subharmonic instabilities of finite-amplitude monochromatic waves. Phys. Fluids A 4, 505523.
Demekhin, E. A. 1983 Bifurcation of the solution to the problem of steady traveling waves in a layer of viscous liquid on an inclined plane. Izv. Akad. Nauk SSSR Mekh. Zhidk i Gaza 5, 3644.
Demekhin, E. A., Demekhin, I. A. & Shkadov, V. Ya 1983 Solitons in flowing layer of a viscous fluid. Izv. Akad. Nauk SSSR, Mekh Zhid i Gaza 4, 916.
Demekhin, E. A. & Kaplan, M. A. 1989 Stability of stationary traveling waves on the surface of a vertical film of viscous fluid. Izv. Akad. Nauk SSSR, Mekh. Zhidk. i Gaza 3, 3441.
Demekhin, E. A. & Shkadov, V. Ya 1985 Two-dimensional wave regimes of a thin liquid films. Izv. Akad. Nauk SSSR, Mekh. Zhidk i Gaza 3, 6367.
Demekhin, E. A. & Shkadov, V. Ya 1986 Theory of solitons in systems with dissipation. Izv. Akad. Nauk SSSR, Mekh. Zhidk i Gaza 3, 9197.
Demekhin, E. A., Tokarev, G. Yu & Shkadov, V. Ya 1991 Hierarchy of bifurcations of space-periodic structures in a nonlinear model of active dissipative media. Physica D 52, 338361.
Elphick, C., Meron, E. & Spiegel, E. A. 1988 Spatiotemporal complexity in traveling patterns. Phys. Rev. Lett. 61, 496499.
Fujimura, K. 1991 Method of center manifold and multiple scales for weakly nonlinear stability of fluid motions. Proc. R. Soc. Lond. A 434, 719733.
Gjevik, B. 1970 Occurrence of finite-amplitude surface waves on falling liquid films. Phys. Fluids 13, 19181925.
Glendinning, P. & Sparrow, C. 1984 Local and global behavior near homoclinic orbits. J. Statist. Phys. 35, 645696.
Ho, L.-W. & Patera, A. T. 1990 A Legendre spectral element method for simulation of unsteady incompressible viscous free-surface flow. Comput. Meth. Appl. Mech. Engng 80, 355366.
Hwang, S.-H. & Chang, H.-C. 1987 Turbulent and inertial roll waves in inclined film flow. Phys. Fluids 30, 12591268.
Joo, S. W., Davis, S. H. & Bankoff, S. G. 1991 Long-wave instabilities of heated falling films: two-dimensional theory of uniform layers. J. Fluid Mech. 230, 117146.
Kapitza, P. L. 1948 Wave flow of thin viscous fluid layers. Zh. Eksp. Teor. Fiz. 18, 1, 328; also in Collected Works, Pergamon (1965.)
Kapitza, P. L. & Kapitza, S. P. 1949 Wave flow of thin fluid layers of liquid. Zh. Eksp. Teor. Fiz. 19, 105120; also in Collected Works, Pergamon (1965.)
Kevrekides, I., Nicolaenko, B. & Scovel, J. C. 1990 Back in the saddle again: a computer-assisted study of the Kuramoto-Sivashinskey equation. SIAM J. Appl. Maths 50, 760790.
Kheshgi, H. A. & Scriven, L. E. 1987 Disturbed film flow on a vertical plate. Phys. Fluids 30, 990997.
Lin, S. P. 1974 Finite amplitude side-band stability of a viscous film. J. Fluid Mech. 63, 417429.
Lin, S. P. 1983 In Film Waves on Fluid Interfaces (ed. R. E. Meyer), pp. 261290. Academic.
Nakaya, C. 1975 Long waves on thin fluid layer flowing down an incline plane. Phys. Fluids 18, 14071412.
Nakoryakov, V. E., Pokusaev, B. G. & Radev, K. B. 1985 Influence of waves on convective gas diffusion in a falling down liquid film. In Hydrodynamics and Heat and Mass Transfer of Free-Surface Flows, pp. 532. Institute of Heat Physics, Siberian Branch of the USSR Academy of Science, Novosibirsk, (in Russian.)
Nepomnyaschy, A. A. 1974 Stability of wave regimes in a film flowing down an inclined plane. Izv. Akad. Nauk SSSR, Mekh. Zhidk i Gaza 3, 2834.
Pierson, F. W. & Whitaker, S. 1977 Some theoretical and experimental observation of wave structure of falling liquid films. Ind. Engng Chem. Fundam. 16, 401408.
Prokopiou, Th. Cheng, M. & Chang, H.-C. 1991 Long waves on inclined films at high Reynolds number. J. Fluid Mech. 222, 665691.
Pugh, J. D. & Saffman, P. G. 1988 Two-dimensional superharmonic stability of finite-amplitude waves in plane Poiseuille flow. J. Fluid Mech. 194, 295307.
Pumir, A., Manneville, P. & Pomeau, Y. 1983 On solitary waves running down an inclined plane. J. Fluid Mech. 135, 2750.
Shkadov, V. Ya 1967 Wave conditions in the flow of thin layer of a viscous liquid under the action of gravity. Izv. Akad. Nauk SSSR, Mekh. Zhidk i Gaza 1, 4350.
Shkadov, V. Ya 1968 Theory of wave flows of a thin layer of a viscous liquid. Izv. Akad. Nauk SSSR, Mekh. Zhidk i Gaza 2, 20.
Shkadov, V. Ya 1973 Some Methods and problems of the Theory of Hydrodynamic Stability. Moscow: Izd. MGU (in Russian.)
Stainthorp, F. P. & Allen, J. M. 1965 The development of ripples on the surface of liquid film flowing inside a vertical tube. Trans. Inst. Chem. Engrs 43, 8591.
Tougou, H. 1981 Deformation of supercritical stable waves on a viscous liquid film down an inclined wall with the decrease of wave numbers. J. Phys. Soc. Japan 50, 10171024.
Trifonov, Yu Ya & Tsvelodub, O. Yu 1985 Nonlinear waves on the surface of liquid films flowing down vertical wall. Zh. Prikl. Mekh. Tekhn. Fiz. 5, 1519.
Trifonov, Yu Ya & Tsvelodub, O. Yu 1991 Nonlinear waves on the surface of a falling liquid film. J. Fluid Mech. 229, 531554.
Tsvelodub, O. Yu 1980 Steady traveling waves on a vertical film of fluid. Izv. Akad. Nauk SSSR, Mekh. Zhidk i Gaza 4, 142146.
Tsvelodub, O. Yu & Trifonov, Yu Ya 1989 On steady-state travelling solutions of an evolution equation describing the behavior of disturbances in an active dissipative media. Physica D 39, 336351.
Tuck, E. O. 1983 Continuous coating with gravity and jet stripping. Phys. Fluids 26, 23522358.
Yih, C.-S. 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321334.
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Nonlinear evolution of waves on a vertically falling film

  • H.-C. Chang (a1), E. A. Demekhin (a2) and D. I. Kopelevich (a2)


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