Minimization of quadratic functionals ratio in eigenvalue problems for the Orr-Sommerfeld equation

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription Access

Abstract

In eigenvalue problems for the Orr–Sommerfeld equation, in cases of no-slip conditions or the assignment of shear stress on one of the boundaries, upper estimates for the real parts of the eigenvalues responsible for stability are analytically obtained. To evaluate more accurate estimates than the known ones, it is necessary to minimize the ratios of certain combinations of quadratic functionals arising from the application of the integral relations method. The exact minima of the ratios are calculated and compared with the estimated minima obtained based on well-known Friedrichs inequalities.

About the authors

D. V. Georgievskii

Lomonosov Moscow State University; Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences; Moscow Center for Fundamental and Applied Mathematics

Email: georgiev@mech.math.msu.su
Moscow, Russia; Moscow, Russia; Moscow, Russia

References

  1. Joseph D.D. Eigenvalue bounds for the Orr–Sommerfeld equation. Pt. 1 // J. Fluid Mech., 1968, vol. 33, no. 3, pp. 617–621. https://doi.org/10.1017/S0022112068001552
  2. Joseph D.D. Eigenvalue bounds for the Orr–Sommerfeld equation. Pt. 2 // J. Fluid Mech., 1969, vol. 36, no. 4, pp. 721–734. https://doi.org/10.1017/S0022112069001959
  3. Synge J.L. Hydrodynamical stability // Semicentenn. Publ. Amer. Math. Soc., 1938, vol. 2, pp. 227–269.
  4. Lin C.C. The Theory of Hydrodynamic Stability. Cambridge: Univ. Press, 1955.
  5. Rektorys K. Variational Methods in Mathematics, Science and Engineering. Dordrect — Boston: Reidel, 1980. https://doi.org/10.1007/978-94-011-6450-4
  6. Collatz L. Eigenwertaufgaben mit Technischen Anwendungen. Leipzig: Academische Verlag., 1963.
  7. Mikhlin S.G. Variational Methods in Mathematical Physics. Moscow: Nauka, 1970. (In Russian).
  8. Yih C.-S. Note on eigenvalue bounds for the Orr –Sommerfeld equation. // J. Fluid Mech., 1969, vol. 38, no. 2, pp. 273–278. https://doi.org/10.1017/S0022112069000164
  9. Georgescu A. Note on Joseph’s inequalities in stability theory // ZAMP, 1970, vol. 21, no. 1, pp. 258–260. https://doi.org/10.1007/BF01590652
  10. Miklavčič M. Eigenvalues of the Orr– Sommerfeld equation in an unbounded domain // Arch. Rat. Mech.&Analysis, 1983, vol. 83, pp. 221–228. https://doi.org/10.1007/BF00251509
  11. Banerjee M.B., Shandil R.G., Gourla M.G.et al. Eigenvalue bounds for the Orr–Sommerfeld equation and their relevance to the existence of backward wave motion // Studies in Appl. Math., 1999, vol. 103, no. 1, pp. 43–50. http://dx.doi.org/10.1111/1467-9590.00119
  12. Banerjee M.B., Shandil R.G., Chauhan S.S. et al. Eigenvalue bounds for the Orr–Sommerfeld equation and their relevance to the existence of backward wave motion. Pt. II // Studies in Appl. Math., 2000, vol. 105, no. 1, pp. 31–34. http://dx.doi.org/10.1111/1467-9590.00140
  13. Puri P. Stability and eigenvalues bounds of the flow of a dipolar fluid between two parallel plates // Proc. Roy. Soc. A., 2005, vol. 461, pp. 1401–1421. http://dx.doi.org/10.1098/rspa.2004.1434
  14. Watanabe Y., Plum M., Nakao M.T. A computer-assisted instability proof for the Orr — Sommerfeld problem with Poiseuille flow // ZAMM, 2009, vol. 89, no. 1, pp. 5–18. http://dx.doi.org/10.1002/zamm.200700158
  15. Ding S., Lin Z. Stability for two-dimensional plane Couette flow to the incompressible Navier–Stokes equations with Navier boundary conditions // Commun. Math. Sci., 2020, vol. 18, no. 5, pp. 1233–1258. https://doi.org/10.48550/arXiv.1710.04855
  16. Bras e Silva P., Carvalho J. Stability and eigenvalue bounds for micropolar shear flows // ZAMM, 2024, vol. 104, no. 12, pp. 1–10. https://doi.org/10.48550/arXiv.2409.11584
  17. Kozyrev O.R., Stepanyants Yu.A. The method of integral relations in the linear theory of hydrodynamic stability // Advances in Science and Technology. Ser. Fluid Mechanics. Moscow.: VINITI, 1991, vol. 25, pp. 3–89. (in Russian)
  18. Georgievskii D.V. Selected Problems of Continuum Mechanics. 2-nd ed. Moscow: URSS, 2020. (in Russian)
  19. Georgievskii D.V. New estimates of the stability of one-dimensional planeparallel flows of a viscous incompressible fluid // J. Appl. Math.&Mech., 2010, vol. 74, no. 4, pp. 452–459. http://dx.doi.org/10.1016/j.jappmathmech.2010.09.011
  20. Georgievskii D.V. Friedrichs inequalities and sharpened sufficient stability conditions of plane-parallel flows // Moscow Univ. Mech. Bull., 2022, vol. 77, no. 3, pp. 61–65. https://doi.org/10.3103/S0027133022030049

Supplementary files

Supplementary Files
Action
1. JATS XML
Download

Copyright (c) 2025 Russian Academy of Sciences

Согласие на обработку персональных данных

 

Используя сайт https://journals.rcsi.science, я (далее – «Пользователь» или «Субъект персональных данных») даю согласие на обработку персональных данных на этом сайте (текст Согласия) и на обработку персональных данных с помощью сервиса «Яндекс.Метрика» (текст Согласия).