#### For citation:

Blinkov Y. A., Mesyanzhin A. V., Mogilevich L. I. Wave Occurrences Mathematical Modeling in Two Geometrically Nonlinear Elastic Coaxial Cylindrical Shells, Containing Viscous Incompressible Liquid. Izv. Sarat. Univ. Math. Mech. Inform., 2016, vol. 16, iss. 2, pp. 184-197. DOI: 10.18500/1816-9791-2016-16-2-184-197

# Wave Occurrences Mathematical Modeling in Two Geometrically Nonlinear Elastic Coaxial Cylindrical Shells, Containing Viscous Incompressible Liquid

The investigation of deformation waves behavior in elastic shells is one of the important trends in the contemporary wave dynamics. There exist mathematical models of wave motions in inﬁnitely long geometrically non-linear shells, containing viscous incompressible liquid, based on the related hydroelasticity problems, which are derived by the shell dynamics and viscous incompressible liquid equations in the form of generalized Korteweg – de Vries equations. In addition, mathematical models of the wave process in inﬁnitely long geometrically non-linear coaxial cylindrical elastic shells are obtained by the perturbation method. These models differ from the known ones by the consideration of incompressible liquid between the shells, based on the related hydroelasticity problems. These problems are described by shell dynamics and viscous incompressible liquid equations with corresponding edge conditions in the form of generalized KdV equation system. The paper presents the investigation of wave occurrences in two geometrically non-linear elastic coaxial cylindrical shells of Kirchhoff-Love type, containing viscous incompressible liquid both between and inside them. The difference schemes of Crank-Nicholson type are obtained for the considered equation system by taking into account liquid and with the help of GroЁbner basis construction. To generate these difference schemes, the basic integral difference correlations, approximating the initial equation system, were used. The usage of GroЁbner basis technology provides generating the schemes, for which it becomes possible to obtain discrete analogs of the laws of preserving the initial equation system. To do that, equivalent transformations were made. Based on the computation algorithm the corresponding software, providing graphs generation and numerical solutions under exact solutions of coaxial shell dynamics equation system obtaining, was developed.

- Gromeka I. S. K teorii dvizheniia zhidkosti v uzkikh tsilindricheskikh trubakh [On the Theory of Fluid Motion in Narrow Cylindrical Tubes]. Collected works. Moscow, Publ. House of the Academy of Sciences of the USSR, 1952, pp. 149–171 (in Russian).
- Zemlyanukhin A. I., Mogilevich L. I. Nonlinear Waves of Deformation in Cylindrical Shells. Izvestiya VUZ. Applied nonlinear dynamics, 1995, vol. 3, no. 1, pp. 52–58 (in Russian).
- Erofeev V. I., Klyueva N. V. Solitons and nonlinear periodic strain waves in rods, plates, and shells (a review). Acoustical Physics, 2002, vol. 48, no. 6, pp. 643–655. DOI: https://doi.org/10.1134/1.1522030.
- Zemlyanukhin A. I., Mogilevich L. I. Nonlinear waves in inhomogeneous cylindrical shells: a new evolution equation. Acoustical Physics, 2001, vol. 47, no. 3, pp. 303–307. DOI: https://doi.org/10.1134/1.1371586.
- Arshinov G. A., Zemlyanukhin A. I., Mogilevich L. I. Two-dimensional solitary waves in a strained nonlinear viscoelastic medium. Acoustical Physics, 2000, vol. 46, no. 1, pp. 100–101.
- Blinkova A. Iu., Ivanov S. V., Kovalev A. D., Mogilevich L. I. Mathematical and Computer Modeling of Nonlinear Waves Dynamics in a Physically Nonlinear Elastic Cylindrical Shells with Viscous Incompressible Liquid inside Them. Izv. Saratov Univ. (N. S.), Ser. Physics, 2012, vol. 12, iss. 2, pp. 12–18 (in Russian).
- Blinkov Yu. A., Kovaleva I. A., Mogilevich L. I. Nonlinear Waves Dynamics Modeling in Coaxial Geometrically and Physically Nonlinear Shell Containing Viscous Incompressible Fluid in between. Bulletin of Peoples’ Friendship University of Russia. Ser. Mathematics. Information Sciences. Physics, 2013, vol. 3, pp. 42–51 (in Russian).
- Loitsianskii L. G. Mekhanika zhidkosti i gaza [And Fluid Mechanics Gas]. Мoscow, Drofa, 2003, 840 p. (in Russian).
- Vallander S. V. Lektsii po gidroaeromekhanike [Lectures on Hydromechanics]. Leningrad, Leningrad Univ. Press, 1978, 296 p. (in Russian).
- Vol’mir A. S. Nelineinaia dinamika plastinok i obolochek [Nonlinear Dynamics of Plates and Shells]. Мoscow. Nauka, 1972, 432 p. (in Russian).
- Vol’mir A. S. Obolochki v potoke zhidkosti i gaza : zadachi gidrouprugosti [Shell in the Liquid and Gas Flow : Tasks Hydroelasticity]. Мoscow, Nauka, 1979, 320 p. (in Russian).
- Schlichting H. Boundary Layer Theory. New York, USA, McCgraw-Hill, 1960, 605 p. (Russ. ed. : Schlichting H. Teoriia pogranichnogo sloia. Moscow, Nauka, 1974, 712 p.)
- Chivilikhin S. A., Popov I. Yu., Gusarov V. V. Dynamics of nanotube twisting in a viscous fluid. Doklady Physics, 2007, vol. 52, no. 1, pp. 60–62.
- Popov I. Yu., Rodygina O. A., Chivilikhin S. A., Gusarov V. V. Soliton in a nanotube wall and Stokes flow in the nanotube. Technical Physics Letters, 2010, vol. 36, no. 9, pp. 852–855. DOI: https://doi.org/10.1134/S1063785010090221.
- Gerdt V. P., Blinkov Yu. A. On Selection of Nonmultiplicative Prolongations in Computation of Janet Bases. Programming and Computer Software, 2007, vol. 33, no. 3, pp. 147–153. DOI: https://doi.org/10.1134/S0361768807030048.
- Blinkov Yu. A., Gerdt V. P. Specialized Computer Algebra System Ginv. Programming and Computer Software, 2008, vol. 34, no. 2, pp. 67–80. DOI: https://doi.org/10.1007/s11086-008-2009-8.
- Gerdt V. P., Blinkov Yu. A. Involution and difference schemes for the Navier – Stokes equations. Computer Algebra in Scientific Computing, vol. 5743 of Lecture Notes in Computer Science, 2009, pp. 94–105. DOI: https://doi.org/10.1007/978-3-642-04103-7_10.

- 282 reads