Skip to main content
SpringerLink
Log in

Singular Nonlinear Problems for Phase Trajectories of Some Self-Similar Solutions of Boundary Layer Equations: Correct Formulation, Analysis, and Calculations

  • ORDINARY DIFFERENTIAL EQUATIONS
  • Published:
Computational Mathematics and Mathematical Physics Aims and scope Submit manuscript

Abstract

We study a singular initial value problem for a nonlinear non-autonomous ordinary differential equation of the second order, defined on a semi-infinite interval and degenerating in the initial data for the phase variable. The problem arises in the dynamics of a viscous incompressible fluid as an auxiliary problem in the study of self-similar solutions of the boundary layer equations for a stream function with a zero pressure gradient (plane-parallel laminar flow in a mixing layer). It is also of independent mathematical interest. Using the previously obtained results on singular nonlinear Cauchy problems and parametric exponential Lyapunov series, a correct formulation and a complete mathematical analysis of this singular initial value problem are given. Restrictions on the “self-similarity parameter” for the global existence of solutions are formulated, two-sided estimates of solutions, and results of calculations of the phase trajectories of solutions for different values of this parameter are given.

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

Fig. 3.
Fig. 4.

REFERENCES

  1. N. B. Konyukhova and S. V. Kurochkin, “Singular nonlinear problems for self-similar solutions of boundary-layer equations with zero pressure gradient: Analysis and numerical solution,” Comput. Math. Math. Phys. 61 (10), 1603–1629 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  2. V. N. Diyesperov, “Investigation of self-similar solutions describing flows in mixing layers,” J. Appl. Math. Mech. 50 (3), 303–312 (1986).

    Article  MathSciNet  Google Scholar 

  3. V. N. Diyesperov, “Behavior of self-similar solutions of the boundary layer equation with a zero pressure gradient,” Reports on Applied Mathematics (Vychisl. Tsentr Akad. Nauk SSSR, Moscow, 1986) [in Russian].

  4. V. N. Diyesperov, “A problem in the theory of mixing layers,” J. Appl. Math. Mech. 60 (6), 987–998 (1996).

    Article  MathSciNet  Google Scholar 

  5. H. Schlichting, Boundary Layer Theory (McGraw-Hill, New York, 1968).

    MATH  Google Scholar 

  6. O. A. Oleinik and V. N. Samokhin, Mathematical Methods in Boundary Layer Theory (Nauka, Moscow, 1997) [in Russian].

    MATH  Google Scholar 

  7. L. G. Loitsyanskii, Mechanics of Liquids and Gases (Pergamon, Oxford, 1972).

    MATH  Google Scholar 

  8. A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maier, Qualitative Theory of Second-Order Dynamic Systems (Wiley, New York, 1973).

    MATH  Google Scholar 

  9. A. M. Lyapunov, General Problem of the Stability of Motion (Princeton Univ. Press, Princeton, 1947).

    Google Scholar 

  10. N. B. Konyukhova, “On the stationary Lyapunov problem for a system of first-order quasilinear partial differential equations,” Differ. Equations 30 (8), 1284–1294 (1994).

    MathSciNet  MATH  Google Scholar 

  11. N. B. Konyukhova, “Stable Lyapunov manifolds for autonomous systems of nonlinear ordinary differential equations,” Comput. Math. Math. Phys. 34 (10), 1179–1195 (1994).

    MathSciNet  MATH  Google Scholar 

  12. N. B. Konyukhova, “Smooth Lyapunov manifolds and singular boundary value problems,” in Reports on Applied Mathematics of Computing Center of the Russian Academy of Sciences (Vychisl. Tsentr Ross. Akad. Nauk, Moscow, 1996) [in Russian].

    Google Scholar 

  13. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955).

    MATH  Google Scholar 

  14. W. R. Wasow, Asymptotic Expansions for Ordinary Differential Equations (Wiley, New York, 1965).

    MATH  Google Scholar 

  15. N. B. Konyukhova, “Singular Cauchy problems for systems of ordinary differential equations,” USSR Comput. Math. Math. Phys. 23 (3), 72–82 (1983).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Federal Research Center “Computer Science and Control,” Russian Academy of Sciences, 119333, Moscow, Russia

    N. B. Konyukhova & S. V. Kurochkin

Authors
  1. N. B. Konyukhova
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. V. Kurochkin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to N. B. Konyukhova or S. V. Kurochkin.

Ethics declarations

The authors declare that they have no conflicts of interest.

Additional information

Translated by E. Chernokozhin

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Konyukhova, N.B., Kurochkin, S.V. Singular Nonlinear Problems for Phase Trajectories of Some Self-Similar Solutions of Boundary Layer Equations: Correct Formulation, Analysis, and Calculations. Comput. Math. and Math. Phys. 63, 202–217 (2023). https://doi.org/10.1134/S0965542523020082

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542523020082

Keywords:

Search

Navigation