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Blinkov Y. A., Evdokimova E. V., Mogilevich L. I. Nonlinear waves in cylinder shell containing viscous liquid, under the impact of surrounding elastic medium and structural damping in longitudinal direction. *Izvestiya VUZ. Applied Nonlinear Dynamics*, 2018, vol. 26, iss. 6, pp. 32-47. DOI: 10.18500/0869-6632-2018-26-6–32-47

# Nonlinear waves in cylinder shell containing viscous liquid, under the impact of surrounding elastic medium and structural damping in longitudinal direction

Subject of the study. The present article deals with further developing of perturbation method for deformation non-linear waves in an elastic cylinder shell, filled with viscous incompressible liquid, surrounded by an elastic media and under construction damping in longitudial direction. Surrounding medium presence leads to integro-differential equation, to generalizing Korteweg–de Vries ones and possessing the same soliton in the form of a solitary wave – a soliton. It does not contain an arbitrary constant number unlike Korteweg–de Vries equation solution. The viscous incompressible liquid presence inside the shell behavior is described by means of dynamics and continuity equation, is solved together with boundary conditions liquid adhesion to a shell wall. Methods. The solution is presented by direct expansion of unknown function by small parameter of hydroelasticity problem and reduced to the problem for hydrodynamics lubrication theory equations. The equations solution defines the tensions on the part of the liquid, the tensions influence the shell longitudinal and normal directions. The liquid presence in the shell adds to longitudial deformation waves equations one more equation member, which does not allow to find exact solution. Construction damping in a longitudial direction adds the same equation member, like liquid presence does. They posses opposite signs in the case of shell Poisson coefficient being smaller than 1/2. In contrary case signs coincide. Liquid presence in the shell and construction damping demand for numerical research. The liquid presence leads to the equation, generalizing Korteveg–de Vries equation, lacking the exact solution and demanding numerical investigation. The numerical investigation is carried out with the use of the modern approach, relying on the universal algorithm of commutative algebra for integro-interpolation method. Results. As a result of difference Grobner basis construction, the difference Crank–Nicolson type schemes are generalized. The schemes were obtained due to the use of basic integral difference correlations, approximating the initial equations system. Acknowledgements. This work was supported by the Russian Foundation for Basic Research (project 16-01-00175-a).

- Kligman E.P., Kligman, I.E., Matvienko V.P. Spectral problem for shells with fluid. Journal of Applied Mechanics and Technical Physics, 2005, vol. 46, iss. 6, pp. 876–882. DOI:10.1007/s10808-005-0147-9
- Bochkarev S.A., Matveenko V.P. Stability analysis of cylindrical shells containing a fluid with axial and circumferential velocity components. Journal of Applied Mechanics and Technical Physics, 2012, vol. 53, iss. 5, pp. 768–776. DOI:10.1134/S0021894412050161
- Gromeka I.S. To the Theory of Fluid Flow in Narrow Cylindrical Tubes. Moscow: AS USSR, 1952, pp. 149–171 (in Russian).
- Kondratov D.V., Mogilevich L.I. Mathematical modeling of the interaction of two cylindrical shells with a fluid layer between them in the absence of an outward flow under vibrations. Vestnik Saratov State Technical University, 2007, vol. 3, iss. 2 (27), pp. 15–23 (in Russian).
- Kondratov D.V., Kondratova N.Yu., Mogilevich L.I. Studies of the amplitude frequency characteristics of oscillations of the tube elastic walls of a circular profile during pulsed motion of a viscous fluid under the conditions of rigid jamming on the butt-ends. J. Mach. Manuf. Reliab., 2009, vol. 38, iss. 3, pp. 229–234. DOI:10.3103/S1052618809030030
- 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., 1991, vol. 5, iss. 2, pp. 127–164. DOI:10.1016/0889-9746(91)90454-W
- 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., 2002, vol. 16, iss. 6, pp. 795–809. DOI: 10.1006/jfls.2002.0446
- Amabili M. Nonlinear Vibrations and Stability of Shells and Plates. Cambridge University Press, 2008. 374 p. DOI:10.1017/CBO9780511619694
- Mogilevich L.I., Popov V.S. Dynamics of the interaction between an elastic cylinder and a viscous incompressible fluid layer. Mechanics of Solids, 2004, iss. 5, pp. 179–190.
- Bochkarev S.A. Natural vibrations of a rotating circular cylindrical shell containing fluid. Computational Continuum Mechanics, 2010, vol. 3, iss. 2, pp. 24–33. DOI:10.7242/1999- 6691/2010.3.2.14
- Lekomtsev S.V. Finite-element algorithms for calculation of natural vibrations of three-dimensional shells. Computational Continuum Mechanics, 2012, vol. 5, iss. 2, pp. 233–243. DOI:10.7242/ 1999-6691/2012.5.2.28
- Bochkarev S.A., Matveenko V.P. Stability of coaxial cylindrical shells containing a rotating fluid. Computational Continuum Mechanics, 2013, vol. 6, iss. 1, pp. 94–102. DOI:10.7242/1999- 6691/2013.6.1.12
- Nariboli G.A. Nonlinear longitudinal dispersive waves in elastic rods. J. Math. Phys. Sci, 1970, vol. 4, 64–73.
- Nariboli G.A., Sedov A. Burger’s–Korteweg–De Vries equation for viscoelastic rods and plates. J. Math. Anal. and Appl., 1970, vol. 32, pp. 661–667.
- Erofeev V.I., Kazhaev V.V. Inelastic interaction and splitting of deformation solitons propagating in the rod. Computational Continuum Mechanics, 2017, vol. 10, iss. 2, pp. 127–137. DOI:10.7242/ 1999-6691/2017.10.2.11
- Zemlyanukhin A.I., Mogilevich L.I. Nonlinear waves of deformations in cylindrical shells. Izvestiya VUZ, Applied Nonlinear Dynamics, 1995, iss. 1, pp. 52–58 (in Russian).
- Erofeev V.I., Klyueva N.V. Solitons and nonlinear periodic strain waves in rods, plates and shells: Review. Acoustical Physics, 2002, vol. 48, iss. 6, pp. 643–655. DOI:1063-7710/02/4806
- Erofeev V.I., Zemlyanukhin A.I., Katson V.M., Sheshenin S.F. Formation of deformation solitons in the Cosserat continuum with constrained rotation. Computational Continuum Mechanics, 2009, iss. 4, pp. 67–75. DOI:10.7242/1999-6691/2009.2.4.32
- Bagdoev A.G., Erofeev V.I., Shekoyan A.V. Linear and Nonlinear Waves in Dispersive Continuous Media. Moscow: Fizmatlit, 2009. 320 p. (in Russian).
- Erofeev V.I., Kazhaev V.V., Pavlov I.S. Inelastic interaction and splitting of strain solitons propagating in a granular medium. Computational Continuum Mechanics, 2013, vol. 6, iss. 2, pp. 140–150. DOI:10.7242/1999-6691/2013.6.2.17
- Zemlyanukhin A.I., Bochkarev A.V. The perturbation method and exact solutions of nonlinear dynamics equations for media with microstructure. Computational Continuum Mechanics, 2016, vol. 9, iss. 2, pp. 182–191 (in Russian). DOI:10.7242/1999-6691/2016.9.2.16
- Zemlyanukhin A.I., Bochkarev A.V. Continued fractions, the perturbation method and exact solutions to nonlinear evolution equations. Izvestiya VUZ. Applied Nonlinear Dynamics. 2016. Vol. 24, iss. 4, pp. 71–85 (in Russian). DOI:10.18500/0869-6632-2016-24-4-71-85
- Zemlyanukhin A.I., Bochkarev A.V. Newton’s method of constructing exact solutions to nonlinear differential equations and non-integrable evolution equations. Izvestiya VUZ. Applied Nonlinear Dynamics. 2017. Vol. 25. iss. 1, pp. 64–83. DOI:10.18500/0869-6632-2017-25-1-64-83
- Blinkova A.Y., Blinkov Y.A., Mogilevich L.I. Non-linear waves in coaxial cylinder shells containing viscous liquid inside with consideration for energy dispersion. Computational Continuum Mechanics, 2013, vol. 6, iss. 3, pp. 336–345. DOI:10.7242/1999-6691/2013.6.3.38
- Blinkova A.Yu., 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. Proceedings of Saratov University. New Ser. Ser. Physics, 2012. vol. 12, iss. 2, pp. 12–18 (in Russian). DOI:10.18500/1816-9791-2016-16-2-184-197
- Blinkova A.Yu., Blinkov Yu.A., Ivanov S.V.,Mogilevich L.I. Nonlinear Deformation Waves in a Geometrically and Physically Nonlinear Viscoelastic Cylindrical Shell Containing Viscous Incompressible Fluid and Surrounded by an Elastic Medium. Antisymmetric Higher Order Edge Waves in Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2015, vol. 15, iss. 2, pp. 193–202. DOI:10.18500/1816-9791-2015-15-2-193-202
- Loytsiansky L.G. Mechanics of Liquid and Gas. Moscow. Drofa, 2003. 840 p. (in Russian).
- Volmir A.S. Shells in a Fluid and Gas Flow: Hydroelasticity Problems. Moscow, Science, 1979. 320 p.
- Vlasov V.Z., Leontiev N.N. Beams, Plates and Shells on an Elastic Base. Moscow. Gos. Izd. Fiz.-Mat. Lit., 1960. 490 p.
- Mikhasev G.I., Sheiko A.N. On the influence of the elastic nonlocality parameter on the natural frequencies of vibrations of a carbon nanotube in an elastic medium. Proceedings of BSTU. Minsk: BSTU, 2012, iss. 6 (153), pp. 41–44.
- Popov I.Yu., Rodygina O.A., Chivilikhin S.A., Gusarov V.V. Soliton in a nanotube wall and Stokes current in nanotube. Technical Physical Letters, 2010, vol. 36, iss. 9, pp. 852–875, DOI: 10.1134/S1063785010090221.
- Blinkov Y.A., Gerdt V.P. Specialized computer algebra system GINV. Programming and Computer Software, 2008, vol. 34, iss. 2, pp. 112–123. DOI: 10.1134/S0361768808020096
- Gerdt V.P., Blinkov Yu.A. Involution and difference schemes for the Navier-Stokes equations. CASC. Lecture Notes in Computer Science, 2009, vol. 5743, pp. 94–105. DOI: 10.1007/978-3- 642-04103-7_10
- Amodio P., Blinkov Yu.A., Gerdt V.P., La Scala R. On consistency of finite difference approximations to the Navier–Stokes equations. CASC. Lecture Notes in Computer Science, 2013, vol. 8136, pp. 46–60. DOI: 10.1007/978-3-319-02297-0_4

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