Skip to main content
SpringerLink
Log in

Stability of wavy conditions in a film flowing down an inclined plane

  • Published:
Fluid Dynamics Aims and scope Submit manuscript

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].

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Literature cited

  1. V. Ya. Shkadov, “Wavy flow conditions of a thin layer of a viscous liquid under the action of the force of gravity,” Izv. Akad. Nauk SSSR, Mekhan. Zhidk. Gaza, No. 1 (1967).

  2. V. Ya. Shkadov, “The theory of wavy flows of a thin layer of viscous liquid,” Izv. Akad. Nauk SSSR, Mekhan. Zhidk. Gaza, No. 2 (1968).

  3. Yu. P. Ivanilov and L. V. Pashinina, “The stability of long waves in a flow of a viscous incompressible liquid,” Izv. Akad. Nauk SSSR, Mekhanika, No. 1 (1965).

  4. P. L. Kapitsa and S. P. Kapitsa, “Wavy flow of thin layers of a viscous liquid. III. Experimental investigation of wavy flow conditions,” Zh. Éksperim. Tekh. Fiz.,19, No. 2 (1949).

  5. C. S. Yih, “Stability of liquid flow down an inclined plane,” Phys. Fluids,6, No. 3 (1963).

  6. S. P. Lin, “Instability of a liquid film flowing down an inclined plane,”10, No. 2 (1967).

  7. P. L. Kapitsa, “Wavy flow of thin layers of a viscous liquid. 1. Free flow,” Zh. Éksperim. Tekh. Fiz.,18, No. 1 (1948).

  8. Yu. P. Ivanilov, “Rolling waves in an inclined channel,” Zh. Vychislit. Mat. Mat. Fiz.,1, No. 6 (1961).

  9. L. N. Maurin and V. S. Sorokin, “The wavy flow of thin layers of a viscous liquid,” Zh. Prikl. Mekhan. Tekh. Fiz., No. 4 (1962).

  10. V. Ya. Shkadov, L. P. Kholpanov, V. A. Malyusov, and N. M. Zhavoronkov, “The nonlinear theory of wavy flows of a film of liquid,” Teor. Osnovy Khim. Tekhnol.,4, No. 6 (1970).

  11. Gjevik, “Occurence of finite-amplitude surface waves on falling liquid films,” Phys. Fluids,13, No. 8 (1970).

  12. V. Penev, V. S. Krylov, Kh. Boyadzhiev (Ch. Boyadjiev), and V. R. Vorotilin, “Wavy flow of thin liquid films,” Internal. J. Heat Mass Transfer,15, No. 7 (1972).

  13. W. Eckhaus, Studies in Non-Linear Stability Theory, Springer, Berlin (1965).

    Google Scholar 

  14. S. Kogelman and R. C. Di Prima, “Stability of spatially periodic supercritical flow in hydrodynamics,” Phys. Fluids,13, No. 1 (1970).

  15. A. Schlüter, D. Lortz, and F. Busse, “On the stability of steady finite-amplitude convection,” J. Fluid. Mech.,23, Part 1 (1965).

Download references

Author information

Authors and Affiliations

  1. Perm'

    A. A. Nepomnyashchii

Authors
  1. A. A. Nepomnyashchii
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

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.

Rights and permissions

Reprints and permissions

About this article

Cite this article

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

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01025515

Keywords

Search

Navigation