Longitudinal Waves in a Nonlinear Cylindrical Shell Containing a Viscous Liquid

Deformable body mechanics


Аuthors

Ivanov S. V.1*, Mogilevich L. I.2**, Popov V. S.2***

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

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

Abstract

The problem of wave propagation in gas dynamics and elastic shells theory is being studied by linearized equations, propagation velocity being constant and equal to sound propagation velocity in unperturbed medium. Nevertheless, a certain set of occurrences, despite the dependent variables small values, are fully defined by perturbations propagation velocity dependence on dependent variables values and are studied basis on non-linear equations. These studies are performed by the perturbation method. The equations for an elastic body dynamics and liquid for the joint hydroelasticity problem are solved simultaneously, the corresponding boundary conditions impenetrable surfaces are being accounted for herewith. This approach is employed to sudy the non-linear deformation waves of elastic shells, containing viscous incompressible liquid.

The perturbation method for studying deformation waves in physically non-linear elastic cylinder shell, containing viscous incompressible liquid, was developed. The method of two-scale disintegration leads to Korteweg – de Vries generalized modified equation not having the exact solution. The surrounding elastic medium, the structure damping in the longitudinal direction and viscous liquid impact require the numerical solution of the equation.

There are no studies on viscous incompressible liquid impact on the non-linear wave process in elastic shells, the liquid being inside them, with considering its inertia movement local members. The presented article studies consideration viscous incompressible liquid impact on non-linear deformation wave propagation, which requires computer modeling, the liquid being inside the shell. The existing methods of mathematical models qualitative analysis do not fully allow studying deformation wave models in case of the shell filled with viscous incompressible liquid. Transition to initial discrete analogue models represents a far more universal method of models investigation. This publication studies the impact of the structure damping in longitudinal direction, surrounding the elastic medium and viscous incompressible liquid inside the shell on the wave amplitude and velocity. Systematic studying of this model wave movements in physically non-linear elastic shell is performed by the difference scheme in analogy with Crank-Nicholson for heat conduction equation.

Keywords:

non-linear waves, elastic cylinder shells, viscous incompressible liquid

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