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Stability of laminar boundary layer on an elastic surface in an incompressible fluid

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Interest in the present problem arose after the publication of the results of the experiments of Kramer [1–3]. In addition to the studies indicated in [4], the articles [5–8] are devoted to the question of the interaction of a flexible elastic surface with the boundary layer. In the present paper the problem of the interaction of an elastic surface with disturbances arising in the boundary layer is posed as in [4]. The approximate nature of the methods of solving the problem of the hydrodynamic stability of the laminar boundary layer leads to a difference in the final computational formulas even in the case when authors use the same Heisenberg-Tollmien-Schlichting-Lin scheme. Therefore, in what follows we present a comparison of the data on the stability of the boundary layer on a solid wall obtained by several authors with the calculations using the formulas, which are then generalized to the case of the elastic surface.

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  1. 1.

    M. O. Kramer, “Boundary layer stabilization by distributed damping,” J. Aeronaut Sci.,24, no. 6, 1957.

  2. 2.

    M. O. Kramer, “Boundary layer stabilization by distributed damping,” Naval Engrs. J.,72, no. 1, 1960.

  3. 3.

    M. O. Kramer, “Boundary layer stabilization by distributed damping,” Naval Engrs. J.,74, no. 2, 1962.

  4. 4.

    A. I. Korotkin, “Stability of plane Poiseuille flow with elastic boundaries,” PMM,29, no. 6, 1965.

  5. 5.

    T. B. Benjamin, “The threefold classification on unstable disturbances in flexible surfaces bounding inviscid flows,” J. Fluid Mech.,16, parts, 1963.

  6. 6.

    T. Nonweiler, “Qualitative solutions of the stability equations for a boundary layer in contact with various forms of flexible surface,” Aeronaut. Res. Council Current Papers, no. 622, 1963.

  7. 7.

    E. Becker, Die laminare incompressible Grenzschicht an einer durch laufende Wellen deformierten ebenen Wand, Dtsch. Versuchanstalt Luftfahrt, no. 132, 1960.

  8. 8.

    N. Gregory and E. M. Love, “Progress report on an experiment on the effect of surface flexibility on stability of laminar flow.” Aeronaut. Res. Council, C. P., no. 602, 1962.

  9. 9.

    C. C. Lin, Hydrodynamic Stability [Russian translation], Moscow, Izd. inostr. lit., 1958.

  10. 10.

    H. Schlichting, Boundary Layer Theory [Russian translation], Izd. inostr. lit., Moscow, 1956.

  11. 11.

    J. A. Zaat, Numerische Beiträge zur Stabilitätstheorie der Grenzschichten, Grenzschichtforschung, Symposium, Freiburg, 1957.

  12. 12.

    G. B. Schubauer and H. K. Skramstad, “Laminar boundary oscillations and stability of laminar flow,” JAS, 14, no. 2, 1947.

  13. 13.

    J. G. Burns, W. H. Childs, A. A. Nicol, and M. A. Ross, “Development and use of a vane device for boundary layer measurements,” J. Fluid Mech.,6, no. 1, 1959.

  14. 14.

    F. R. Hama, “Boundary layer transition induced by a vibrating ribbon on a flat plate,” Proceedings of the 1960 Heat Transfer and Fluid Mechanics Institute, Stanford University Press, 1960.

  15. 15.

    F. K. Wortman, “Study of unstable oscillations of the boundary layer in a water channel using the tellurium method,” in collection: The Boundary Layer Problem and Questions of Heat Transfer [Russian translation], Moscow-Leningrad, Gosenergoizdat, 1960.

  16. 16.

    H. Schlichting and A. Ulrich, Zur Berechnung des Umschlages laminar-turbulent, Jb. d. dt. Luftfahrtforschung, 1942, S. 18.

  17. 17.

    E. Hahneman, J. C. Freeman, and M. Finston, “Stability of boundary layers and of flow in entrance section of a channel,” JAS,15, no. 8, 1948.

  18. 18.

    J. Pretsch, Die Stabilität einer ebenen laminarströmung bei Druckgefälle und Druckanstieg, Jb. d. dt. Luftfahrtforschung, 58, 1941.

  19. 19.

    N. Tetervin, “A study of the stability of the incompressible boundary layer on infinite wedges,” NACA Techn. Notes, 2976, 1953.

  20. 20.

    I. P. Soprunenko, “Calculation of flow stability of a laminar layer with positive pressure gradient,” Izv. AN SSSR, Mekhan. i mashinostr, no. 5, 1964.

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The author wishes to thank G. I. Petrov and V. A. Medvedev for discussions of the present study.

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Korotkin, A.I. Stability of laminar boundary layer on an elastic surface in an incompressible fluid. Fluid Dyn 1, 25–28 (1966). https://doi.org/10.1007/BF01106866

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  • Boundary Layer
  • Fluid Dynamics
  • Incompressible Fluid
  • Present Problem
  • Solid Wall