Nonlinear waves in viscoelastic cylindrical shell with a viscous incompressible fluid and surrounded by an elastic medium

Mathematics. Physics. Mechanics


Blinkova A. Y.1*, Ivanov S. V.2**, Kuznetsova E. L.3***, Mogilevich L. I.1****

1. Yuri Gagarin State Technical University of Saratov, 77, Politechnicheskaya str., Saratov, 410054, Russia
2. Saratov State University named after N. G. Chernyshevsky, 83, Astrakhanskaya str., Saratov, 410012, Russia
3. Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia



The equations describing strain waves dynamics for coupled hydroelasticity problems, including the dynamic equation for geometrically nonlinear viscoelastic shells surrounded by elastic media and accounting the dynamic equations for viscous incompressible liquids inside shells, are derived by means of asymptotic methods with appropriate boundary conditions. The obtained equation generalizes the well-known equation of Gardner. Accounting the small radius of the shell midsurface as compared with the strain wavelength, the dynamic equations for viscous incompressible fluids are asymtotically transformed to the classical equation of hydrodynamic lubrication theory.

In this paper, the Cauchy problem for the derived equation is solved numerically taking into account the effect of the fluid and of the surrounding elastic medium. The finite difference schema is based on the construction of an overdetermined differential equations’ system. These ones are approximating the integral conservation laws and the integral relations between the unknown functions and their derivatives. As a result, the finite difference scheme is defined as a compatibility condition for this system and automatically provides the integral conservation laws for the areas composed by the basic finite volumes.

The fluid filling the shell that is immersed in an elastic medium result the increasing or decreasing of the strain wave amplitude, depending on the Poisson ratio of the viscoelastic medium. On the other hand, the elastic medium surrounding the shell increases the velocity of the nonlinear wave deformation.

Use of the proposed model significantly enhances the possibility of the experimental data analysis for the significantly nonlinear dynamical systems such as fuel and cooling systems for aerospace engineering, etc..


non-linear waves, viscous incompressible liquid, viscoelastic cylindrical shell, surrounded by an elastic medium


  1. Zemlyanukhin A.I., Mogilevich L.I. Nelineinye volny v tsilindricheskikh obolochkakh: solitony, simmetrii, evolyutsiya (Nonlinear Waves in Cylindrical Shells: Solitons, Symmetry, Evolution), Saratov, SGAU imeni N.I. Vavilova, 1999, 132 p.
  2. Arshinov G.A., Zemlyanukhin A.I., Mogilevich L.I. Akusticheskii zhurnal, 2000, vol.46, no 1, pp. 116-117.
  3. Arshinov G.A., Mogilevich L.I. Staticheskie i dinamicheskie zadachi vyazkouprugosti (Static and Dynamic Problems of Viscoelasticity), Saratov, SGAU imeni N.I. Vavilova, 2000, 152 p.
  4. Loitsyansky L.G. 4. Loitsyanskii L.G. Mekhanika zhidkosti i gaza (Fluid Mechanics), Moscow, Drofa, 2003, 840 p.
  5. Volmir A.S. Nelineinaya dinamika plastinok i obolochek (Dynamics of Plates and Shells), Moscow, Nauka, 1972, 432 p.
  6. Moskvitin V.V. Soprotivlenie vyzko-uprugikh materialov (Resistance Vyzko-elastic materials), Moscow, Nauka, 1972, 328 p.
  7. Vlasov V.Z. Balki, plity i obolochki na uprugom osnovanii (Beams, plates and shells on elastic foundation), Moscow, Fizmatlit, 1960, 490 p.
  8. Chivilikhin S.A., Popov I.Yu., Gusarov V.V. Doklady RAN, 2007, vol. 412, no. 2, pp. 201-203.
  9. Popov Yu.I., Rozygina O.A., Chivilikhin S. A., Gusarov V.V. Pis’ma v Zhurnal tekhnicheskoi fiziki, 2010, vol. 36, no.18, pp. 42-54.
  10. Blinkov Ju.A., Mozzhilkin V.V. Programmirovanie, 2006. vol. 32, no. 2, pp. 71–74.
  11. Gerdt V.P., Blinkov Yu.A., Mozzhilkin V.V. Gröbner Bases and Generation of Difference Schemes for Partial Differential Equations, Symmetry, Integrability and Geometry: Methods and Applications, 2006, vol. 2, 26 p. available at: (accessed 29.10.2014).
  12. Gerdt V.P., Blinkov Yu.A. Involution and Difference Schemes for the Navier-Stokes Equations, Computer Algebra in Scientific Computing, Springer Berlin, Heidelberg, 2009, vol. 5743 of Lecture Notes in Computer Science. pp. 94–105.

Download — informational site MAI

Copyright © 2000-2024 by MAI