Longitudinal Waves Modelling in a Shell with Physically Quadratic Nonlinearity Filled with Liquid and Enveloped by Elastic MediumLongitudinal Waves Modelling in a Shell with Physically Quadratic Nonlinearity Filled with Liquid and Enveloped by Elastic Medium


DOI: 10.34759/trd-2020-111-3

Аuthors

Bykova T. V.*, Evdokimova E. V.**, Mogilevich L. I.***, Popov V. S.****

Yuri Gagarin State Technical University of Saratov, 77, Politechnicheskaya str., Saratov, 410054, Russia

*e-mail: tbykova69@mail.ru
**e-mail: eev2106@mail.ru
***e-mail: mogilevichli@gmail.com
****e-mail: vic_p@bk.ru

Abstract

A perturbation technique for deformation waves simulation based on considering the coupled problem of hydroelasticity in elastic cylinder shell with quadratic physical non-linearity is developing in the presented article. The shell is encompassed by the elastic medium and filled with viscous uncompressing liquid, which inertia of movement is accounted fort while considering its dynamics. The article demonstrates that the presence of encompassing medium leads to integral-differential equation, generalizing Korteweg-de Vries equation, possessing solution in the form of a solitary wave, called solitron. It does not hold an arbitrary constant wave number, in contrast to the Korteweg-de Vries equation solution. The uncompressing liquid behavior inside the shell are being described by the Navier- Stokes equations and continuity equation. They are being solved together with boundary conditions of adhesion to the shell wall. Solution is being presented by direct expansion of the sought functions by the small parameter of the hydroelesticity problem and reduced to liquid dynamics in the framework of the hydrodynamic lubrication theory. As the result, tensions from the liquid side, acting on the shell in longitudinal direction and normally, are being determined. The presence of liquid in the shell adds into the longitudinal waves equation a term, which does not allow finding the exact solution. Thus, numerical study is being realized using up-to-date approach, based on universal algorithm of commutative algebra. As the result of Gröbner differential basis construction, difference schemes of Crank-Nicolson type, obtained using basic integral difference relations, approximating the initial system of equations were generated. Numerical experiment revealed that liquid movement inertia reduced the wave velocity, and liquid viscous friction decreases the wave amplitude.

Keywords:

nonlinear waves, viscous incompressible liquid, cylindrical physically nonlinear shell, elastic medium

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