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Interaction dynamics of solitary waves on a falling film

Published online by Cambridge University Press:  26 April 2006

H.-C. Chang ,
E. Demekhin  and
E. Kalaidin
H.-C. Chang
Affiliation:
Department of Chemical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA
E. Demekhin
Affiliation:
Department of Chemical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA Permanent Address: Department of Applied Mathematics, Kuban State Technological University, Krasnodar, 3500072, Russia.
E. Kalaidin
Affiliation:
Department of Chemical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA Permanent Address: Department of Applied Mathematics, Kuban State Technological University, Krasnodar, 3500072, Russia.
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Abstract

Beyond a short transition region near the inlet, waves on a falling film evolve into distinct pulse-like solitary waves that dominate all subsequent interfacial dynamics. Numerical and physical experiments indicate that these localized structures can attract and repel each other. Attractive interaction through the capillary ripples of the pulses causes two pulses to coalesce into a bigger pulse which accelerates and precipitates further coalescence. This binary interaction between an ‘excited’ pulse after coalescence and its smaller front neighbour is the key mechanism that drives the observed wave dynamics. From symmetry arguments, two dominant modes for a solitary pulse are obtained and used to develop an inelastic coherent structure theory for binary interaction between an excited pulse and its front neighbour. The theory offers a simple dynamical system that quantitatively describes the binary interaction and promises to elucidate the complex wave dynamics on a falling film.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 294 , 10 July 1995 , pp. 123 - 154
Copyright
© 1995 Cambridge University Press

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References

Balmforth, N. J., Ierley, G. R. & Spiegel, E. A. 1994 Chaotic pulse trains. SIAM J. Appl. Maths 54, 1291–1304.Google Scholar
Chang, H.-C. 1986 Traveling waves on fluid interfaces: normal form analysis of the Kuramoto-Sivashinsky equation. Phys. Fluids 29, 3142–3147.CrossRefGoogle Scholar
Chang, H.-C. 1994 Wave evolution on a falling film. Ann. Rev. Fluid Mech. 26, 103–136.Google Scholar
Chang, H.-C., Cheng, M., Demekhin, E. A. & Kopelevich, D. I. 1994 Secondary and tertiary excitation of three-dimensional patterns on a falling film. J. Fluid Mech. 270, 251–275.Google Scholar
Chang, H.-C., Demekhin, E. A. & Kopelevich, D. I. 1993a Nonlinear evolution of waves on a vertically falling film.. J. Fluid Mech. 250, 433–480.Google Scholar
Chang, H.-C., Demekhin, E. A. & Kopelevich, D. I. 1993b Laminarizing effects of dispersion in an active-dissipative nonlinear medium. Physica D 63, 299–320.Google Scholar
Cheng, M. & Chang, H.-C. 1995 Competition between sideband and subharmonic secondary instability on a falling film. Phys. Fluids 7, 34–54.Google Scholar
Demekhin, E. A., Kaplan, M. A. & Shkadov, V. Ya. 1987 Mathematical models of the theory of viscous liquid films. Izv. Akad. Nauk SSSR Mekh. Zhidk. i Gaza 6, 73–81.Google Scholar
Demekhin, E. A. & Shkadov, V. Ya. 1985 Two-dimensional wave regimes of thin liquid films. Izv. Akad. Nauk SSSR. Mekh. Zhidk. i Gaza 3, 63–97.Google Scholar
Demekhin, E. A. & Shkadov, V. Ya. 1986 Theory of solitons in systems with dissipation. Izv. Akad. Nauk SSSR. Mekh. Zhidk. i Gaza 3, 91–97.Google Scholar
Elphick, C., Ierley, G. R., Regev, O. & Spiegel, E. A. 1991 Interacting localized structures with Galilean invariance. Phys. Rev. A 44, 1110–1122.Google Scholar
Frisch, U., Zhen, S. S. & Thual, O. 1986 Viscoelastic behaviour of cellur solutions to the Kuramoto-Sivashinsky model. J. Fluid Mech. 168, 221–240.Google Scholar
Glendinning, P. & Sparrow, C. T. 1984 Local and global behaviour near homoclinic orbits. J. Statist. Phys. 35, 645–696.Google Scholar
Jayaprakash, C., Hayot, F. & Pandit, R. 1993 Universal properties of the two-dimensional Kuramoto-Sivashinsky equation. Phys. Rev. Lett. 71, 12–15.Google Scholar
Kalliadasis, S. & Chang, H.-C. 1994 Drop formation during coating of vertical fibres. J. Fluid Mech. 261, 135–168.Google Scholar
Kawahara, T. & Toh, S. 1988 Pulse interaction in an unstable-dissipative nonlinear system. Phys. Fluids 31, 2103–2111.Google Scholar
Liu, J. & Gollub, J. P. 1993 Onset of spatially chaotic waves on a falling film. Phys. Rev. Lett. 70, 2289–2292.Google Scholar
Liu, J. & Gollub, J. P. 1994 Solitary wave dynamics of film flows. Phys. Fluids 6, 1702–1712.Google Scholar
Nakaya, C. 1989 Waves on a viscous fluid film down a vertical wall. Phys. Fluids A 1, 1143–1154.Google Scholar
Prokopiou, Th., Cheng, M. & Chang, H.-C. 1991 Long waves on inclined films at high Reynolds number. J. Fluid Mech. 222, 665–691.Google Scholar
Pumir, A., Manneville, P. & Pomeau, Y. 1983 On solitary waves running down an inclined plane. J. Fluid Mech. 135, 27–50.Google Scholar
Salamon, T. R., Armstrong, R. C. & Brown, R. A. 1994 Traveling waves on inclined films: numerical analysis by the finite element method. Phys. Fluids 6, 2202–2220.Google Scholar
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, 43–50.Google Scholar
Stainthorp, F. P. & Allen, J. M. 1965 The development of ripples on the surface of a liquid film flowing inside a vertical tub. Trans. Inst. Chem. Engrs 43, 85–91.Google Scholar
Wilson, S. D. R. & Jones, A. F. 1983 The entry of a falling film into a pool and the air-entrainment problem. J. Fluid Mech. 128, 219–230.Google Scholar