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Stationary travelling waves on a film flowing down an inclined plane

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Abstract

At small flow rates, the study of long-wavelength perturbations reduces to the solution of an approximate nonlinear equation that describes the change in the film thickness [1–3]. Steady waves can be obtained analytically only for values of the wave numbers α close to the wave number αn that is neutral in accordance with the linear theory [1, 2]. Periodic solutions were constructed numerically for the finite interval of wave numbers 0.5αn ⩽ α ⩽ αn in [4]. In the present paper, these solutions are found in almost the complete range of wave numbers 0 ⩽ α ⩽ αn that are unstable in the linear theory. In particular, soliton solutions of this equation are obtained. The results were partly published in [5].

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Literature cited

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  3. A. A. Nepomnyashchii, “Stability of wave regimes in a film flowing down an inclined plane,” Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 3 (1974).

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  5. 0. Yu. Tsvelodub, “Stationary plane waves on a flowing film of liquid,” in: Thermophysical Investigations [in Russian], Novosibirsk (1977).

  6. Chia-Shun Yih, “Stability of liquid flow down an inclined plane,” Phys. Fluids,6, No. 3 (1963).

  7. V. I. Petviashvili, “The equation of an unusual soliton,” Fiz. Plazmy,2, No. 3 (1976).

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  9. V. E. Nakoryakov, B. G. Pokusaev, and S. V. Alekseenko, “Stationary two-dimensional rolling waves on a vertical liquid film,“ Inzh.-Fiz. Zh.,30, No. 5 (1976).

  10. V. Ya. Shkadov, “Solitons in a viscous liquid film,” Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 1 (1977).

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  1. Novosibirsk

    O. Yu. Tsvelodub

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  1. O. Yu. Tsvelodub
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Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 142–146, July–August, 1980.

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Tsvelodub, O.Y. Stationary travelling waves on a film flowing down an inclined plane. Fluid Dyn 15, 591–594 (1980). https://doi.org/10.1007/BF01089622

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  • DOI: https://doi.org/10.1007/BF01089622

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