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Nonlinear Instability of Nonparallel Flows pp 436–448Cite as

Long-Disturbance Stability of Nonlinear Wave Regimes in a Falling Liquid Film

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Abstract

So called Kuramoto-Sivashinsky equation is considered. The nonlinear stability of periodic steady-state travelling solutions with respect to long length disturbances is investigated. The method of multiple scales is used.

It is shown that if their wave numbers are near the neutral wave number, then the nonlinear evolution of the initially infinitesimal waves are described by equation that is similar to the Ginzburg-Landau equation. But as far as stability is concerned there is essential difference between them. It follows from this equation that all of these waves are unstable to the sideband disturbances, while the same solutions of Ginzburg-Landau equation are stable to ones.

If their wave numbers are far away from the neutral wave number, another nonlinear evolution equation is obtained that allow us to determine the conditions of the double periodic regime arising.

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References

  • Lin, S.P. 1974 Finite amplitude side-band stability of a viscous film, J. Fluid Mech., 63, 417–429.

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  • Nepomnyshchy, A.A. 1974 Stability of wave regimes in a film flowing down along the inclined plane, Izv: Akad. Nauk SSSR, Mekh.Zhid. i Gasa,3, 28–34.

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  • Trifonov, Yu.Ya. & Tsvelodub, O.Yu. 1991 Nonlinear waves on the surface of a falling liquid film. Part L Waves of the first family and their stability, J. Fluid Mech., 229, 531–554.

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  • Tsvelodub, O.Yu. & Trifonov, Yu.Ya. 1989 On Steady-state travelling solutions of an evolution equation describing the behaviour of disturbances in active dissipative media, Physica D 39, 336–351.

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Author information

Authors and Affiliations

  1. Siberian Branch of Russian Academy of Sciences, Institute of Thermophysics, Prospect Lavrentyev 1, Novosibirsk, 630090, Russia

    O. Yu. Tsvelodub

Authors
  1. O. Yu. Tsvelodub
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Editor information

Editors and Affiliations

  1. Dept. of Mechanical and Aeronautical Engineering, Clarkson University, Potsdam, NY, 13699-5725, USA

    S. P. Lin , W. R. C. Phillips  & D. T. Valentine ,  & 

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© 1994 Springer-Verlag, Berlin Heidelberg

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Tsvelodub, O.Y. (1994). Long-Disturbance Stability of Nonlinear Wave Regimes in a Falling Liquid Film. In: Lin, S.P., Phillips, W.R.C., Valentine, D.T. (eds) Nonlinear Instability of Nonparallel Flows. International Union of Theoretical and Applied Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-85084-4_36

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  • DOI: https://doi.org/10.1007/978-3-642-85084-4_36

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-85086-8

  • Online ISBN: 978-3-642-85084-4

  • eBook Packages: Springer Book Archive

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