Skip to main content
SpringerLink
Log in

Self-similar solutions to Lin-Reissner-Tsien equation

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

The Lin-Reissner-Tsien equation describes unsteady transonic flows under the transonic approximation. In the present paper, the equation is reduced to an ordinary differential equation via a similarity transformation. The resulting equation is then solved analytically and even exactly in some cases. Numerical simulations are provided for the cases in which there is no exact solution. Travelling wave solutions are also obtained.

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. Lin, C. C., Reissner, E., and Tsien, H. S. On two-dimensional non-steady motion of a slender body in a compressible fluid. Journal of Mathematical Physics, 27, 220–231 (1948)

    MathSciNet  MATH  Google Scholar 

  2. Ames, W. F. and Nucci, W. N. Analysis of fluid equations by group methods. Journal of Engineering Mechanics, 20, 181–187 (1985)

    MathSciNet  Google Scholar 

  3. Kuz’min, A. G. and Ivanova, A. V. The structural instability of transonic flow associated with amalgamation/splitting of supersonic regions. Fluid Dynamics, 18, 335–344 (2004)

    MathSciNet  MATH  Google Scholar 

  4. Bogdanov, A. N. and Duesperov, V. N. The properties of asymptotic expansions for the parameters of unsteady transonic flow with axial symmetry. Journal of Applied Mathematics and Mechanics, 72, 33–35 (2008)

    Article  MathSciNet  Google Scholar 

  5. Bibik, Y. V., Duesperov, V. N., and Popov, S. P. Structure of time dependent transonic flows in pane channels. Fluid Dynamics, 40, 315–325 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Glazatov, S. N. On solvability of a spatial periodic problem for the Lin-Reissner-Tsien equation of transonic gas dynamics. Mathematical Notes, 87, 130–134 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bogdanov, A. N. and Duesperov, V. N. Higher approximations of the transonic expansion in problems of unsteady transonic flow. Journal of Applied Mathematics and Mechanics, 61, 775–786 (1997)

    Article  MathSciNet  Google Scholar 

  8. Diesperov, V. N. and Korolev, G. L. Triple-deck analysis of formation and evolution of supersonic zones and local separation zones in unsteady transonic flow over a surface roughness element. Computational Mathematics and Mathematical Physics, 45, 516–523 (2005)

    MathSciNet  Google Scholar 

  9. Bogdanov, A. N. and Duesperov, V. N. Tollmien-Schlichting waves in a transonic boundary layer: excitation from the outer flow and from the surface. Journal of Applied Mathematics and Mechanics, 71, 258–268 (2007)

    Article  MathSciNet  Google Scholar 

  10. Titov, S. S. A method of finite-dimensional rings for solving nonlinear equations of mathematical physics. Aerodynamics (ed. Ivanova, T. P.), Saratov University Press, Saratov, 104–110 (1988)

    Google Scholar 

  11. Svirshchevskii, S. R. Lie-Bäcklund symmetries of linear ODEs and generalized separation of variables in nonlinear equations. Physics Letters A, 199, 344–348 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  12. Polyanin, A. D. and Zaitsev, V. F. Handbook of Nonlinear Partial Differential Equations, Chapman and Hall/CRC, Boca Raton (2004)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Central Florida, P. O. Box 161364, Orlando, Florida, 32816-1364, USA

    J. Haussermann, K. Vajravelu & R. A. Van Gorder

Authors
  1. J. Haussermann
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. Vajravelu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. R. A. Van Gorder
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. A. Van Gorder.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Haussermann, J., Vajravelu, K. & Van Gorder, R.A. Self-similar solutions to Lin-Reissner-Tsien equation. Appl. Math. Mech.-Engl. Ed. 32, 1447–1456 (2011). https://doi.org/10.1007/s10483-011-1514-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-011-1514-6

Key words

Chinese Library Classification

2010 Mathematics Subject Classification

Search

Navigation