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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).

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