Skip to main content
SpringerLink
Log in

Friedrichs Inequalities and Sharpened Sufficient Stability Conditions of Plane-Parallel Flows

  • Published:
Moscow University Mechanics Bulletin Aims and scope

Abstract

From the standpoint of the linearized stability theory, two eigenvalue problems for the Orr–Sommerfeld equation with two groups of boundary conditions having a certain mechanical meaning are considered. The stability parameter, which is a real part of the spectral parameter, is estimated on the basis of the integral relations method operating with quadratic functionals. The technique of the method involves the application of the Friedrichs inequality for various classes of complex-valued functions. Using the minimizing property of the first positive eigenvalues in the corresponding problems, the values of the constants in some Friedrichs inequalities are increased, which entails the strengthening of the stability sufficient integral estimates for plane-parallel shear flows in a plane layer.

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

REFERENCES

  1. R. Betchov and W. O. Criminale, Stability of Parallel Flows (Academic Press, New York, 1967).

    MATH  Google Scholar 

  2. D. V. Georgievskii, ‘‘Asymptotics of eigenvalues in the Orr–Sommerfeld problem for low velocities of unperturbed flow,’’ Dokl. Math. 103, 19–22 (2021). https://doi.org/10.1134/S106456242101004X

    Article  MathSciNet  MATH  Google Scholar 

  3. O. R. Kozyrev and Yu. A. Stepanyants, ‘‘Methods of integral relations in the linear theory of hydrodynamic stability,’’ Itogi Nauki Tekh. Ser. Mekh. Zhidk. Gaza 25, 3–89 (1991).

    Google Scholar 

  4. D. D. Joseph, ‘‘Eigenvalue bounds for the Orr–Sommerfeld equation. Part 1,’’ J. Fluid Mech. 33, 617–621 (1968). https://doi.org/10.1017/S0022112068001552

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. D. D. Joseph, ‘‘Eigenvalue bounds for the Orr–Sommerfeld equation. Part 2,’’ J. Fluid Mech. 36, 721–734 (1969). https://doi.org/10.1017/S0022112069001959

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. C.-S. Yih, ‘‘Note on eigenvalue bounds for the Orr–Sommerfeld equation,’’ J. Fluid Mech. 38, 273–278 (1969). https://doi.org/10.1017/S0022112069000164

    Article  ADS  MathSciNet  Google Scholar 

  7. K. Rektorys, Variational Methods in Mathematics, Science and Engineering (Springer, Dordrecht, 1980), 2nd ed.

    MATH  Google Scholar 

  8. D. V. Georgievskii, Selected Problems of Continuum Mechanics (Lenand, Moscow, 2018).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Chair of Elasticity Theory, Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia

    D. V. Georgievskii

Authors
  1. D. V. Georgievskii
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to D. V. Georgievskii.

Ethics declarations

The author declares that he has no conflicts of interest.

Additional information

Translated by E. Oborin

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Georgievskii, D.V. Friedrichs Inequalities and Sharpened Sufficient Stability Conditions of Plane-Parallel Flows. Moscow Univ. Mech. Bull. 77, 61–65 (2022). https://doi.org/10.3103/S0027133022030049

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S0027133022030049

Keywords:

Search

Navigation