Nonlinear hydroelastic deformation waves in the walls of an annular channel made of material with fractional and quadratic physical nonlinearity surrounded by Winkler elastic medium


Аuthors

Popova E. V.1*, Mogilevich L. I.1**, Evdokimova E. V.1***, Popova M. V.2****

1. Yuri Gagarin State Technical University of Saratov, 77, Politechnicheskaya str., Saratov, 410054, Russia
2. Saratov State University named after N. G. Chernyshevsky, 83, Astrakhanskaya str., Saratov, 410012, Russia

*e-mail: elizaveta.popova.97@bk.ru
**e-mail: mogilevichli@gmail.com
***e-mail: eev2106@mail.ru
****e-mail: mari.popova.2004@internet.ru

Abstract

In this paper, a mathematical model in the form of a system of two evolutionary equations generalizing the Korteweg-de Vries-Schamel equation for the study of longitudinal deformation waves in the walls of an annular channel containing a viscous fluid was carried out. Initially, the axisymmetric hydroelasticity problem for two coaxial cylindrical shells of Kirchhoff-Love type, between which there is a fluid and surrounded by a Winkler elastic medium was formulated. The derivation of the equations of dynamics for the shell, the material of which has a nonlinear physical law linking stresses and strains, in the form of a linear combination of a quadratic function and a power function with fractional exponent was realized. The fluid is considered in the framework of Newtonian viscous fluid of constant density. The asymptotic analysis of the formulated problem by the method of multiscale expansions and linearization of the equations of dynamics for a thin annular layer of viscous fluid taking into account its inertia was done. As a result, the original problem was reduced to a system of two nonlinear evolution equations. A new difference scheme for the evolution equations system is proposed, and the evolution of nonlinear longitudinal deformation waves was numerically investigated within the framework of the constructed mathematical model. Computational experiments allowed us to establish that solitary deformation waves in the channel walls are supersonic solitons, as well as to estimate the influence of the fluid inertia and the surrounding elastic medium on the nonlinear wave process.

Keywords:

mathematical modeling, solitons, nonlinear deformation waves, cylindrical shell, viscous fluid, annular channel, combined fractional-quadratic nonlinearity, Winkler elastic medium

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