Abstract
Solitary waves in a thin layer of viscous liquid which is running down a vertical surface under the action of gravity are investigated. The existence of such waves was demonstrated in the experiments of [1, 2]. The difficulties that must be faced in a theoretical computation were also noted in these studies. Below a solution of the problem of stationary waves is obtained by the method of expansion in the small parameter in two regions with subsequent matching and also by a numerical integration method. It is shown that in each case a solution of solitary wave type exists along with the single-parameter family of periodic solutions (parameter—the wave number α). On decreasing the wave number, the periodic waves go over into a succession of solitary waves.
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P. L. Kapitsa, “Wave flow of thin layers of a viscous liquid. 1, 2,” Zh. Éksp. Teor. Fiz.,18. No. 1 (1948).
P. L. Kapitsa and S. P. Kapitsa, “Wave flow of thin layers of a viscous liquid. 3,” Zh. Éksp. Teor. Fiz.,19, No. 2 (1949).
V. Ya. Shkadov, “Wave flow regimes of a thin layer of a viscous liquid under the action of gravity,” Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, No. 1 (1967).
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Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 63–66, January–February, 1977.
The authors thank L. N. Maurin for helpful discussions and A. M. Tereshchenko for assisting in the computations.
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Shkadov, V.Y. Solitary waves in a layer of viscous liquid. Fluid Dyn 12, 52–55 (1977). https://doi.org/10.1007/BF01074624
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DOI: https://doi.org/10.1007/BF01074624