ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Mogilevich L. I., Popova E. V. Longitudinal waves in the walls of an annular channel filled with liquid and made of a material with fractional nonlinearity. Izvestiya VUZ. Applied Nonlinear Dynamics, 2023, vol. 31, iss. 3, pp. 365-376. DOI: 10.18500/0869-6632-003040, EDN: RKTVVT

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
Full text PDF(En):
Article type: 

Longitudinal waves in the walls of an annular channel filled with liquid and made of a material with fractional nonlinearity

Mogilevich Lev Ilyich, Yuri Gagarin State Technical University of Saratov
Popova Elizaveta Viсtorovna, Yuri Gagarin State Technical University of Saratov

Purpose of this paper is to study the evolution of longitudinal strain waves in the walls of an annular channel filled with a viscous incompressible fluid. The walls of the channel were represented as coaxial shells with fractional physical nonlinearity. The viscosity of the fluid and its influence on the wave process was taken into account within the study.

Metods. The system of two evolutionary equations, which are generalized Schamel equations, was obtained by the two-scale asymptotic expansion method. The fractional nonlinearity of the channel wall material leads to the necessity to use a computational experiment to study the wave dynamics in them. The computational experiment was conducted based on obtaining new difference schemes for the governing equations. These schemes are analogous to the Crank–Nicholson scheme for modeling heat propagation.

Results. Numerical simulation showed that over time, the velocity and amplitude of the deformation waves remain unchanged, and the wave propagation direction concurs with the positive direction of the longitudinal axis. The latter specifies that the velocity of the waves is supersonic. For a particular case, the coincidence of the computational experiment with the exact solution is shown. This substantiates the adequacy of the proposed difference scheme for the generalized Schamel equations. In addition, it was shown that solitary deformation waves in the channel walls are solitons.

The study was funded by Russian Science Foundation (RSF) according to the project No. 23-29-00140.
  1.  Nariboli GA. Nonlinear longitudinal dispersive waves in elastic rods. J. Math. Phys. Sci. 1970; 4:64–73.
  2. Nariboli GA, Sedov A. Burgers’s-Korteweg-De Vries equation for viscoelastic rods and plates. J. Math. Anal. Appl. 1970;32(3):661–677. DOI: 10.1016/0022-247X(70)90290-8.
  3. Erofeev VI, Klyueva NV. Solitons and nonlinear periodic strain waves in rods, plates, and shells (a review). Acoustical Physics. 2002;48(6):643–655. DOI: 10.1134/1.1522030.
  4. Zemlyanukhin AI, Mogilevich LI. Nonlinear waves in inhomogeneous cylindrical shells: A new evolution equation. Acoustical Physics. 2001;47(3):303–307. DOI: 10.1007/BF03353584.
  5. Zemlyanukhin AI, Andrianov IV, Bochkarev AV, Mogilevich LI. The generalized Schamel equation in nonlinear wave dynamics of cylindrical shells. Nonlinear Dynamics. 2019;98(1): 185–194. DOI: 10.1007/s11071-019-05181-5.
  6. Bochkarev SA, Matveenko VP. Stability of coaxial cylindrical shells containing a rotating fluid. Computational Continuum Mechanics. 2013;6(1):94–102. DOI: 10.7242/1999-6691/2013.6.1.12.
  7. Mogilevich L, Ivanov S. Longitudinal waves in two coaxial elastic shells with hard cubic nonlinearity and filled with a viscous incompressible fluid. In: Dolinina O, Bessmertny I, Brovko A, Kreinovich V, Pechenkin V, Lvov A, Zhmud V, editors. Recent Research in Control Engineering and Decision Making. ICIT 2020. Vol. 337 of Studies in Systems, Decision and Control. Cham: Springer; 2021. P. 14–26. DOI: 10.1007/978-3-030-65283-8_2.
  8. Paıdoussis MP. Fluid-Structure Interactions: Slender Structures and Axial Flow. 2nd edition. London: Academic Press; 2014. 867 p. DOI: 10.1016/C2011-0-08057-2.
  9. Amabili M. Nonlinear Vibrations and Stability of Shells and Plates. New York: Cambridge University Press; 2008. 374 p. DOI: 10.1017/CBO9780511619694.
  10. Samarskii AA. The Theory of Difference Schemes. Boca Raton: CRC Press; 2001. 786 p. DOI: 10.1201/9780203908518.
  11. Il’yushin AA. Continuum Mechanics. Moscow: Moscow University Press; 1990. 310 p. (in Russian).
  12. Jones RM. Deformation Theory of Plasticity. Blacksburg: Bull Ridge Publishing; 2009. 622 p.
  13. Kauderer H. Nichtlineare Mechanik. Berlin: Springer-Verlag; 1958. 684 s. (in German). DOI: 10.1007/978-3-642-92733-1.
  14. Zemlyanukhin AI, Bochkarev AV, Andrianov IV, Erofeev VI. The Schamel-Ostrovsky equation in nonlinear wave dynamics of cylindrical shells. Journal of Sound and Vibration. 2021;491:115752. DOI: 10.1016/j.jsv.2020.115752.
  15. Loitsyanskii LG. Mechanics of Liquids and Gases. Vol. 6 of International Series of Monographs in Aeronautics and Astronautics. Oxford: Pergamon Press; 1966. 804 p. DOI: 10.1016/C2013-0- 05328-5.
  16. Gerdt VP, Blinkov YA, Mozzhilkin VV. Grobner bases and generation of difference schemes for partial differential equations. Symmetry, Integrability and Geometry: Methods and Applications. 2006;2:051. DOI: 10.3842/SIGMA.2006.051. 
Available online: