Non-linear waves in cylinder shells with viscous incompressible liquid inside, in consideration of energy dissipation

Аuthors

Blinkova A. Y.^1*, Blinkov Y. A.^2**, Mogilevich L. I.^1***

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: anblinkova@yandex.ru
**e-mail: blinkovua@gmail.com
***e-mail: mogilevichli@gmail.com

Abstract

An equation for nonlinear waves propagating in cylindrical shells containing viscous liquids is derived. Both the geometrical and physical nonlinearity for the shell as well as the energy dissipation are taken into account. The viscous liquid model is used. The nonlinear boundary value problem of coupled hydroelasti city is formulated and solved on the basis of the asymptotic approach.
The models of structural damping in the material of the shell and of the viscoelastic material of the shell were considered. It is shown that both of them lead to one and the same equation which generalizes the well-known modified Korteweg-de Vries-Burgers equation. The new term describing the liquid impact inside the shell is introduced. The equations of viscous incompressible liquid are asymptotically reduced to the classical equation of hydrodynamic lubrication theory. This simplification becomes possible when the radius of the shell midsurface is significantly smaller than the deformation wavelength.
In the absence of liquid the equation has a known exact solution, which can be considered as an initial condition for the numerical solution of the equation described above.
This paper describes the numerical solution of the Cauchy problem for the shells dynamic equation considering the effect of the liquid. This approach to the formulation of the finite-difference schema is based on the construction of the predetermined system of differential equations derived from the integral approximation of conservation laws and the integral relations between the unknown functions and their derivatives. As a result, the finite-difference schema is defined as a condition for the compatibility for the system and the resulting difference schema automatically secures the fulfillment of the integral conservation laws in the areas compounded from the basic finite volumes.
The presence of liquid inside the shell leads to a substantial change of the longitudinal deformation waves propagation. In the abscence of the liquid in the shell a solitary wave (kink-antikink) moves retaining its original shape and velocity. The presence of fluid in shells made from the materials with Poisson ratio less than 0.5 results the exponential increasing of the wave amplitude under small time magnitudes and the absence of wave oscillations at the forefront, which is due to the energy dissipation. In the absence of energy dissipation the oscillations at the leading wave front occur. It can be stated that the fluid contributes to a constant extra «feeding» energy from the original excitation source that provides the growth of amplitudes.
Consequently, the use of the proposed models increases the possibilities of experimental data analysis during the investigation of various systems like the fuel supply, the cooling, the blood and lymph stream pulsating waves and etc. which dynamics can be only by nonlinear models described.

Keywords:

non-linear waves, cylindrical shells, energy dissipation

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