Abstract
The nonlinear theory of motion in a film of liquid flowing down an inclined plane predicts the existence of an interval k0<k<km, inside of which the wave number of periodic wave motion may lie [1]. The condition of the stability of experimentally attained motions imposes a limitation on their wave numbers. In [2] a numerical investigation of the stability of wavy motions was made; in the investigated range of change in the Galileo number and the wave number all the motions were found to be unstable; however, the fastest growing were perturbations imposed on a motion with a determined wave number (“optimal” conditions). In [3] the instability of motions with a wavelength exceeding some limiting value was established in a long-wave approximation. In the present work, within the framework of the two-dimensional problem, an investigation was made of the stability of periodic wavy motions, based on expansion in terms of the small parameter km. It is established that, within the interval k0<k<km, there lies a finite subinterval of wave numbers for which wavy motions are stable. The narrowness of this interval (δk≈0.07 km) may be the reason why, in the experiment, with not too great Galileo numbers for fully established periodic wavy motions, no substantial differences in the wave-length are observed [4].
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Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 28–34, May–June, 1974.
The author thanks E. M. Zhukhovitskii for posing the problem and for his continuing interest in the work, and G. I. Petrov and the participants in his seminar, in particular V. Ya. Shkadov, as well as G. Z. Gershuni, V. A. Briskman, and L. N. Maurin for their invaluable evaluations.
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Nepomnyashchii, A.A. Stability of wavy conditions in a film flowing down an inclined plane. Fluid Dyn 9, 354–359 (1974). https://doi.org/10.1007/BF01025515
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DOI: https://doi.org/10.1007/BF01025515