# The Study of Wave Propagation in a Shell with Soft Nonlinearity and with a Viscous Liquid Inside

2019, Vol. 15, no. 3, pp.  233-250

Author(s): Mogilevich L., Ivanov S.

This article is devoted to studying longitudinal deformation waves in physically nonlinear elastic shells with a viscous incompressible fluid inside them. The impact of construction damping on deformation waves in longitudinal and normal directions in a shell, and in the presence of surrounding medium are considered.
The presence of a viscous incompressible fluid inside the shell and the impact of fluid movement inertia on the wave velocity and amplitude are taken into consideration. In the case of a shell filled with a viscous incompressible fluid, it is impossible to study deformation wave models by qualitative analysis methods. This makes it necessary to apply numerical methods. The numerical study of the constructed model is carried out by means of a difference scheme analogous to the Crank – Nickolson scheme for the heat conduction equation. The amplitude and velocity do not change in the absence of surrounding medium impact, construction damping in longitudinal and normal directions, as well as in the absence of fluid impact. The movement occurs in the negative direction, which means that the movement velocity is subsonic. The numerical experiment results coincide with the exact solution, therefore, the difference scheme and the modified Korteweg – de Vries – Burgers equation are adequate.
Keywords: nonlinear waves, elastic cylinder shell, viscous incompressible fluid, Crank – Nickolson difference scheme
Citation: Mogilevich L., Ivanov S., The Study of Wave Propagation in a Shell with Soft Nonlinearity and with a Viscous Liquid Inside, Rus. J. Nonlin. Dyn., 2019, Vol. 15, no. 3, pp.  233-250
DOI:10.20537/nd190303

## References

[1] Samarskii, A. A. and Mikhailov, A. P., Principles of Mathematical Modelling: Ideas, Methods, Examples, Numerical Insights, 3, CRC, Boca Raton, Fla., 2002, 360 pp.
[2] Ilyushin, A. A., Continuum Mechanics, 3rd ed., MGU, Moscow, 1990, 310 pp. (Russian)
[3] Kauderer, H., Nichtlineare Mechanik, Springer, Berlin, 1958, 700 pp.
[4] Volmir, A. S., Nonlinear Dynamics of Plates and Shells, Nauka, Moscow, 1972, 492 pp. (Russian)
[5] Zemlyanukhin, A. I. and Mogilevich, L. I., Nonlinear Waves in Cylindrical Shells: Solitons, Symmetry, Evolution, SSU, Saratov, 1999, 132 pp. (Russian)
[6] Loitsyanskiy, L. G., Mechanics of Liquids and Gases, 6th ed., Begell House, New York, 1995, 971 pp.
[7] Zemlyanukhin, A. I. and Mogilevich, L. I., “Nonlinear Deformation Waves in Cylindrical Shells”, Izv. Vyssh. Uchebn. Zaved. Prikl. Nelin. Dinam., 3:1 (1995), 52–58 (Russian)
[8] Erofeev, V. I. and Klyueva, N. V., “Solitons and Nonlinear Periodic Strain Waves in Rods, Plates, and Shells (A Review)”, Acoust. Phys., 48:6 (2002), 643–655    ; Akust. Zh., 48:6 (2002), 725–740 (Russian)
[9] Bochkarev, A. V., Zemlyanukhin, A. I., and Mogilevich, L. I., “Solitary Waves in an Inhomogeneous Cylindrical Shell Interacting with an Elastic Medium”, Acoust. Phys., 63:2 (2017), 145–151    ; Akust. Zh., 63:2 (2017), 145–151 (Russian)
[10] Krysko, V. A., Zhigalov, M. V., and Saltykova, O. A., “Nonlinear Dynamics of Beams of Euler – Bernoulli and Timoshenko Type”, Izv. Vyssh. Uchebn. Zaved. Mashinostr., 2008, no. 6, 7–27 (Russian)
[11] Zemlyanukhin, A. I., Bochkarev, A. V., and Mogilevich, L. I., “Solitary Longitudinal-Bending Waves in Cylindrical Shell Interacting with a Nonlinear Elastic Medium”, Herald of the Bauman Moscow State Technical University. Series Natural Sciences, 2018, no. 1(76), 47–60 (Russian)
[12] Ageev, R. V., Kuznetsova, E. L., Kulikov, N. I., Mogilevich, L. I., and Popov, V. S., “Mathematical Model of Movement of a Pulsing Layer of Viscous Liquid in the Channel with an Elastic Wall”, PNRPU Mech. Bull., 2014, no. 3, 17–35 (Russian)
[13] Lekomtsev, S. V., “Finite-Element Algorithms for Calculation of Natural Vibrations of Three-Dimensional Shells”, Comput. Contin. Mech., 5:2 (2012), 233–243 (Russian)
[14] Bochkarev, S. A. and Matveenko, V. P., “Stability of Coaxial Cylindrical Shells Containing Rotating Fluid Flow”, Comput. Contin. Mech., 6:1 (2013), 94–102 (Russian)
[15] Mamaev, I. S., Tenenev, V. A., and Vetchanin, E. V., “Dynamics of a Body with a Sharp Edge in a Viscous Fluid”, Rus. J. Nonlin. Dyn., 14:4 (2018), 473–494
[16] Borisov, A. V., Mamaev, I. S., and Vetchanin, E. V., “Self-Propulsion of a Smooth Body in a Viscous Fluid under Periodic Oscillations of a Rotor and Circulation”, Regul. Chaotic Dyn., 23:7–8 (2018), 850–874
[17] Kuzenov, V. V. and Ryzhkov, S. V., “Approximate Method for Calculating Convective Heat Flux on the Surface of Bodies of Simple Geometric Shapes”, J. Phys. Conf. Ser., 815 (2017), 012024, 8 pp.
[18] Ryzhkov, S. V. and Kuzenov, V. V., “Analysis of the Ideal Gas Flow over Body of Basic Geometrical Shape”, Int. J. Heat Mass Transf., 132 (2019), 587–592
[19] Vetchanin, E. V., Mamaev, I. S., and Tenenev, V. A., “The Self-Propulsion of a Body with Moving Internal Masses in a Viscous Fluid”, Regul. Chaotic Dyn., 18:1–2 (2013), 100–117
[20] Andrejchenko, K. P. and Mogilevich, L. I., “On the Dynamics of Interaction between a Compressible Layer of a Viscous Incompressible Fluid and Elastic Walls”, Izv. Akad. Nauk. Mekh. Tverd. Tela, 1982, no. 2, 162–172 (Russian)
[21] Gerdt, V. P., Blinkov, Yu. A., and Mozzhilkin, V. V., “Gröbner Bases and Generation of Difference Schemes for Partial Differential Equations”, SIGMA Symmetry Integrability Geom. Methods Appl., 2 (2006), 051, 26 pp.
[22] Ovcharov, A. A. and Brylev, I. S., “Mathematical Model of Deformation of Nonlinear Elastic Reinforced Conical Shells under Dynamic Loading”, Sovremennye Problemy Nauki i Obrazovaniya, 2014, no. 3, 8 pp. (Russian)
[23] Fel'dshtejn, V. A., “Elastic Plastic Deformations of a Cylindrical Shell with a Longitudinal Impact”, Waves in Inelastic Media, Akad. Nauk MSSR, Kishinev, 1970, 199–204 (Russian)