Skip to main content
SpringerLink
Log in

Effect of the pipe curvature on internal elastic wave propagation

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

Abstract

A mathematical model of internal elastic wave propagation in a bent pipe is developed. For slightly bent pipes, a method is devised for reducing the equations to a lower dimension problem. It is shown that the wave dynamics are described by the Korteweg-de Vries and Klein-Gordon-Fock equations. Thus, the class of problems to which these equations apply is expanded.

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. N. E. Zhukovskii, Hydraulic Shock in Water Pipelines (Gostekhteorizdat, Moscow, 1949) [in Russian].

    Google Scholar 

  2. R. I. Nigmatulin, Dynamics of Multiphase Media (Nauka, Moscow, 1987), part 2 [in Russian].

    Google Scholar 

  3. I. A. Charnyi, Unsteady Motion of a Real Fluid in Pipes (Nedra, Moscow, 1975) [in Russian].

    Google Scholar 

  4. V. A. Rukavishnikov and O. P. Tkachenko, “The Korteweg-de Vries Equation in a Cylindrical Pipe,” Zh. Vychisl. Mat. Mat. Fiz. 48, 146–153 (2008) [Comput. Math. Math. Phys. 48, 139–146 (2008)].

    MATH  MathSciNet  Google Scholar 

  5. V. A. Rukavishnikov and O. P. Tkachenko, “Numerical and Asymptotic Solution to the Equations of Propagation of Hydroelastic Vibrations in a Curved Pipe,” Prikl. Mekh. Tekh. Fiz. 41(6), 161–169 (2000).

    MATH  Google Scholar 

  6. L. I. Sedov, A Course in Continuum Mechanics (Wolters-Noordhoff, Groningen, 1971; Lan’, St. Petersburg, 2004), Vol. 1.

    MATH  Google Scholar 

  7. M. A. Il’gamov, Introduction to Nonlinear Hydroelasticity (Nedra, Moscow, 1991) [in Russian].

    MATH  Google Scholar 

  8. L. G. Loitsyanskii, Mechanics of Liquids and Gases (Pergamon, Oxford, 1972; Drofa, Moscow, 2003).

    MATH  Google Scholar 

  9. A. C. Newell, Solitons in Mathematics and Physics (SIAM, Philadelphia, Pa., 1985; Mir, Moscow, 1989).

    Google Scholar 

  10. A. A. Samarskii and A. V. Gulin, Numerical Methods (Nauka, Moscow, 1989) [in Russian].

    MATH  Google Scholar 

  11. A. D. Polyanin, Handbook of Linear Equations of Mathematical Physics (Fizmatlit, Moscow, 2001) [in Russian].

    MATH  Google Scholar 

  12. V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitaevskii, Relativistic Quantum Theory (Pergamon, Oxford, 1971; Nauka, Moscow, 1968), Part 1.

    Google Scholar 

  13. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1972; Nauka, Moscow, 1979).

    MATH  Google Scholar 

  14. A. H. Nayfeh, Perturbation Methods (Wiley, New York, 1973; Mir, Moscow, 1976).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computing Center, Far East Division, Russian Academy of Sciences, ul. Kim Yu Chena 65, Khabarovsk, 680063, Russia

    V. A. Rukavishnikov & O. P. Tkachenko

Authors
  1. V. A. Rukavishnikov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. O. P. Tkachenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. A. Rukavishnikov.

Additional information

Original Russian Text © V.A. Rukavishnikov, O.P. Tkachenko, 2010, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2010, Vol. 50, No. 11, pp. 1988–1997.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rukavishnikov, V.A., Tkachenko, O.P. Effect of the pipe curvature on internal elastic wave propagation. Comput. Math. and Math. Phys. 50, 1886–1894 (2010). https://doi.org/10.1134/S0965542510110126

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542510110126

Keywords

Search

Navigation