Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia1898610.22363/2312-9735-2018-26-3-203-215Research ArticleWave Processes Modeling in Two Coaxial Shells Filled with a Viscous Liquid and Surrounded by Elastic MediumBlinkovY A<p>Doctor of of Physical and Mathematical Sciences, Chair of Department of Mathematic and Computer Modeling, Saratov State University</p>blinkovua@info.sgu.ruEvdokimovaE V<p>Postgraduate of Department of Applied Mathematics and System Analysis, Saratov State Technical University</p>eev2106@mail.ruMogilevichL I<p>Doctor of Engineering, Professor of Department of Applied Mathematics and System Analysis, Saratov State Technical University</p>mogilevich@sgu.ruRebrinaA Y<p>Candidate of Physics and Mathematics, Associate Professor of Department of Applied Mathematics and System Analysis, Saratov State Technical University</p>anblinkova26@gmail.comSaratov State UniversityYuri Gagarin State Technical University of Saratov1512201826320321504082018Copyright © 2018, Blinkov Y.A., Evdokimova E.V., Mogilevich L.I., Rebrina A.Y.2018<p>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 shells dynamics and viscous incompressible liquid equations in the form of centralized Korteweg-de Vries (KdV) equations. In addition, mathematical models or 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 equations 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 in between and inside, and surrounded by an elastic medium, acting in both normal and longitudinal directions. The difference schemes of Crank-Nicholson type are obtained for the considered equations system by taking into account liquid impact and with the help of Grobner basis construction. To generate these difference schemes, the basic integral difference correlations, approximating initial equations system, were used.</p>nonlinear wavesviscous incompressible liquidelastic cylindrical shellsGrobner basisнелинейные волнывязкая несжимаемая жидкостьцилиндрические упругие оболочкибазис Грёбнера[M. P. Paidoussis, V. B. Nguyen, A. K. Misra, A Theoretical Study of the Stability of Cantilevered Coaxial Cylindrical Shells Conveying Fluid, Journal of Fluids and Structures 5 (2) (1991) 127–164. doi:10.1016/0889-9746(91)90454-W.][M. Amabili, Nonlinear Vibrations and Stability of Shells and Plates, Cambridge University Press, 2008. doi:10.1017/CBO9780511619694.][L. I. Mogilevich, V. S. Popov, Dynamics of the Interaction of an Elastic Cylinder with a Layer of a Viscous Incompressible Fluid, Mechanics of Solids (5) (2004) 179–190, in Russian.][S. A. Bochkarev, V. P. Matveenko, Stability of Coaxial Cylindrical Shells Containing a Rotating Fluid Flow, Computational Continuum Mechanics 6 (1) (2013) 94–102, in Russian. doi:10.7242/1999-6691/2013.6.1.12.][A. G. Bagdoev, V. I. Erofeev, S. F. Sheshenin, Linear and Nonlinear Waves in Dispersive Continuous Media, Fizmatlit, Moscow, 2009, in Russian.][V. I. Erofeev, V. V. Kazhaev, I. S. Pavlov, Inelastic Interaction and Splitting of Strain Solitons Propagating in a Granular Medium, Computational Continuum Mechanics 6 (2) (2013) 140–150, in Russian. doi:10.7242/1999-6691/2013.6.2.17.][Y. A. Blinkov, S. V. Ivanov, L. I. Mogilevich, Mathematical and Computer Modeling of Non-linear Deformation Waves in Shell with Viscous Liquid Inside, Bulletin of Peoples’ Friendship University of Russia. Series: Mathematics. Information Sciences. Physics 3 (2012) 52–60, in Russian.][Y. A. Blinkov, I. A. Kovaleva, L. I. Mogilevich, Nonlinear Waves Dynamics Modeling in Coaxial Geometrically And Physically Nonlinear Shell Containing Viscous Incompressible Fluid in between, Bulletin of Peoples’ Friendship University of Russia. Series: Mathematics. Information Sciences. Physics 3 (2013) 42–51, in Russian.][Y. A. Blinkov, A. V. Mesyazhin, L. I. Mogilevich, Propagation of Nonlinear Waves in Coaxial Physically Nonlinear Cylindrical Shells Filled with a Viscous Fluid, RUDN Journal of Mathematics, Information Sciences and Physics 25 (1) (2017) 19–35, in Russian. doi:10.22363/2312-9735-2017-25-1-19-35.][G. Cowderer, Nonlinear Mechanics, Foreign Literature, Moscow, 1961, in Russian.][L. G. Loytsiansky, Mechanics of Liquid and Gas, Drofa, Moscow, 2003, in Russian.][S. V. Vallander, Lectures on Hydroaeromechanics, Ed. Leningrad State University, Leningrad, 1978, in Russian.][A. S. Volmir, Nonlinear Dynamics of Plates and Shells, Nauka, Moscow, 1972, in Russian.][A. S. Volmir, Shells in a Fluid and Gas Flow: Hydroelasticity Problems, Nauka, Moscow, 1979, in Russian.][V. Z. Vlasov, N. N. Leontiev, Beams, Plates and Shells on an Elastic Base, Gos. ed. fiz.-mat. literature, Moscow, 1960, in Russian.][V. I. Erofeev, V. V. Kazhaev, E. E. Lisenkova, N. P. Semerikova, Nonsinusoidal Bending Waves in Timoshenko Beam Lying on Nonlinear Elastic Foundation, Journal of Machinery Manufacture and Reliability 36 (3) (2008) 230–235, in Russian.][G. Mikhasev, A. Sheiko, On the Influence of the Elastic Nonlocality Parameter on the Natural Frequencies of Vibrations of a Carbon Nanotube in an Elastic Medium, Vol. 153, BSTU, Minsk, 2012, pp. 41–44, in Russian.][A. V. Bochkarev, A. I. Zemlyanukhin, L. I. Mogilevich, Solitary Waves in an Inhomogeneous Cylindrical Shell Interacting with an Elastic Medium, Akusticheskij Zhurnal 63 (2) (2017) 145–151, in Russian.][A. Y. Blinkova, S. V. Ivanov, A. D. Kovalev, L. I. Mogilevich, Mathematical and Computer Modeling of Nonlinear Waves Dynamics in a Physically Nonlinear Elastic Cylindrical Shells with Viscous Incompressible Liquid inside Them, Izvestiya of Saratov University. New series. Series: Physics 12 (2) (2012) 12–18, in Russian. doi:10.18500/1816-9791-2016-16-2-184-197.][V. Y. Belashov, S. V. Vladimirov, Solitary Waves in Dispersive Complex Media: Theory, Simulation, Applications, Springer-Verlag, Berlin, 2005.][A. A. Samarskii, The Theory of Difference Schemes, Marcel Dekker Inc., N.-Y., 2001.][E. E. Rosinger, Nonlinear Equivalence, Reduction of PDEs to ODEs and Fast Convergent Numerical Methods, Pitman, London, 1983. doi:10.1137/1026088.][V. P. Gerdt, Y. A. Blinkov, V. V. Mozzhilkin, Gr¨obner Bases and Generation of Difference Schemes for Partial Differential Equations, Symmetry, Integrability and Geometry: Methods and Applications 2 (2006) 26. doi:10.3842/SIGMA.2006.051.][V. P. Gerdt, Consistency Analysis of Finite Difference Approximations to PDE Systems, Vol. 7125, MMCP. Lecture Notes in Computer Science, 2011, pp. 28–42.]