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Mathematical Modeling of Waves in a Non-linear Shell with Wiscous Liquid Inside It, Taking into Account Its Movement Inertia

  • Lev MogilevichEmail author
  • Yury Blinkov
  • Dmitry Kondratov
  • Sergey Ivanov
Conference paper
Part of the Studies in Systems, Decision and Control book series (SSDC, volume 199)

Abstract

New mathematical models of wave movements in infinitely long physically non-linear shell are constructed. The shell contains viscous incompressible liquid, considering its movement inertia. The models are based on the hydrodynamics related problems, described by shells and viscous incompressible liquid dynamics equations in the form of the generalized MKdV equations. The effective numerical algorithm with the use of the Grobner bases technology was proposed. It was aimed at constructing differential schemes to solve the generalised MKdV equation, obtained in the given article. The algorithm was used to analyze deformation non-linear waves propagation in elastic and non-elastic cylinder shells with viscous incompresible liquid inside them. The numerical experiments were carried out on the basis of the obtained numerical algorithm. The experiments made it possible to reveal new viscosity and incompressible liquid inertia effects on the deformation wave behavior in the shell, depending on Poisson ratio of the shell material. In particular, the exponential growth of wave amplitude under the liquid presence in the shell made of non-organic materials (various pipelines and technological constructions) is revealed. In the case of organic materials (blood vessels) viscous liquid impact leads to wave quick going out. Liquid movement inertia presence leads to deformation wave velocity transformation. Supported by grants RFBR 19-01-00014-a and the President of the Russian Federation MD-756.2018.8.

Keywords

Non-linear shell Viscous incompressible liquid Deformation waves Numerical experiment 

References

  1. 1.
    Gromeka, I.S.: K teorii dvizheniya zhidkosti v uzkikh tsilindricheskikh trubakh. [On the Theory of liquid motion on narrow cylindrical tubes]/Gromeka I. S. [B. m.] : M.: Izd-vo AN SSSR, pp. 149–171 (1952)Google Scholar
  2. 2.
    Nariboli, G.A.: Nonlinear longitudinal dispersive waves in elastic rods. J. Math. Phys. Sci 4, 64–73 (1970)zbMATHGoogle Scholar
  3. 3.
    Nariboli, G.A., Sedov, A.: Burgers-Korteweg-De Vries equation for viscoelastic rods and plates. J. Math. Anal. Appl. 32, 661–667 (1970)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Yerofeyev, V.I., Klyuyeva, N.V.: Solitony i nelineynyye periodicheskiye volny deformatsii v sterzhnyakh, plastinakh i obolochkakh (obzor). Isvesiya VUZ, AND, vol. 48, no. 6, pp. 725–740 (2002)Google Scholar
  5. 5.
    Zemlyanukhin, A.I., Mogilevich, L.I.: Nelineynyye volny v neodnorodnykh tsilindricheskikh obolochkakh: novoye evolyutsionnoye uravneniye. Isvesiya VUZ, AND, vol. 47, no. 3, pp. 359–363 (2001)Google Scholar
  6. 6.
    Bochkarev, S.A.: Sobstvennyye kolebaniya vrashchayushcheysya krugovoy tsilindri-cheskoy obolochki s zhidkost’yu. VMSS. T. 3, no. 2, pp. 24–33 (2010)Google Scholar
  7. 7.
    Paidoussis, M.P., Nguyen, V.B., Misra, A.K.: A theoretical study of the stability of cantilevered coaxial cylindrical shells conveying fluid. J. Fluids Struct. 5(2), 127–164 (1991).  https://doi.org/10.1016/0889-9746(91)90454-WCrossRefGoogle Scholar
  8. 8.
    Amabili, M., Garziera, R.: Vibrations of circular cylindrical shells with nonuniform constraints, elastic bed and added mass. Part III: steady viscous effects on shells conveying fluid. J. Fluids Struct. 16(6), 795–809 (2002).  https://doi.org/10.1006/jfls.2002.0446CrossRefGoogle Scholar
  9. 9.
    Amabili, M.: Nonlinear Vibrations and Stability of Shells and Plates, p. 374. Cambridge University Press, Cambridge (2008).  https://doi.org/10.1017/CBO9780511619694CrossRefzbMATHGoogle Scholar
  10. 10.
    Blinkov, Yu.A., Blinkova, A.Yu., Evdokimova, E.V., Mogilevich, L.I.: Mathematical modeling of nonlinear waves in an elastic cylindrical shell surrounded by an elastic medium and containing a viscous incompressible liquid. In: Acoustical Physics, vol. 64, no. 3, pp. 283–288 (2018). ISSN 1063-7710CrossRefGoogle Scholar
  11. 11.
    Kauderer, H.: Nichtlineare Mechanik, p. 685. Springer-Verlag, Berlin (1958) (Russ. ed.: Nelineynaya mekhanika, p. 778. Jnostrannaya literatura Publications, Moskow (1961))CrossRefGoogle Scholar
  12. 12.
    Vol’mir, A.S.: Obolochki v potoke zhidkosti i gaza: zadachi gidrouprugosti, S. 320. Nauka (1979)Google Scholar
  13. 13.
    Vol’mir, A.S.: Nelineynaya dinamika plastinok i obolochek. Nauka (1972)Google Scholar
  14. 14.
    Loytsyanskiy, L.G.: Mekhanika zhidkosti i gaza [Fluid Mechanics].: M.: Drofa, p. 840 (2003)Google Scholar
  15. 15.
    Gerdt, V.P., Blinkov, Yu.A.: Involution and difference schemes for the Navier-Stokes Equations. In: CASC. Lecture Notes in Computer Science, vol. 5743, pp. 94–105 (2009).  https://doi.org/10.1007/978-3-642-04103-7-10
  16. 16.
    Amodio, P., Blinkov, Yu.A., Gerdt, V.P., La Scala R.: On consistency of finite difference approximations to the Navier-Stokes equations. In: CASC. Lecture notes in Computer Science, vol. 8136, pp. 46–60 (2013).  https://doi.org/10.1007/978-3-319-02297-0-4

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Yuri Gagarin State Technical University of SaratovSaratovRussia
  2. 2.Saratov State UniversitySaratovRussia

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