Nonlinear Waves in three Coaxial Elastic Cylinderic Shells with Viscous Incompressible Fluid between them

Аuthors

Blinkov Y. A.^1*, Kovaleva I. A.^2**, Kuznetsova E. L.^3***, Mogilevich L. I.^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
3. Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: blinkovua@gmail.com
**e-mail: irinakovaleva1406@gmail.com
***e-mail: vida_ku@mail.ru
****e-mail: mogilevichli@gmail.com

Abstract

The equations which describe the strain waves by means of asymptotic methods for solving the hydro-elastic problem that includes the dynamic equations of three coaxial geometrically nonlinear elastic shells are obtained. Energy dissipation and equations for an incompressible viscous fluid between cylindrical shells with appropriate boundary conditions are taken into account. We obtained equations, which generalize the well-known Korteweg-de Vries equations by introducing the term describing the liquid impact between the shells. The radius of the medial surface of the shell is significantly smaller than the wavelength of deformation, and therefore the asymptotic transition to the classical equation of hydrodynamic lubrication theory is made in the equations of viscous incompressible fluid.
This paper describes the numerical solution to the Cauchy problem for the resulting system of new equation with the influence of the fluid. This approach to the construction of the difference scheme is based on the construction of the predetermined system of differential equations, derived from the integral approximation of conservation laws and the integral relations, connecting the unknown functions and their derivatives. As a result, the difference scheme is defined as a condition for the compatibility for the system and the resulting difference scheme automatically ensures the fulfillment of the integral conservation laws in the areas, made of the basic finite volumes.
The presence of fluid between the co-axial shells gives rise to deformation waves not only in the outer shell but also in the inner ones, where the initial deformation moment is equal to zero. Hence, the deformation wave of stable amplitude and velocity takes place. This fact is in accordance with the solitary wave solution, which cannot be described analytically. The construction under consideration can be characterized as a five layered packet with liquid as a filler.
Consequently, the use of these models allows for a widening of experimental data analysis possibilities in the course of the investigating various systems — fuel supply, cooling for aerospace engineering and etc., which dynamics is principally non-linear.

Keywords:

non-linear waves, coaxial cylinder shells, incompressible liquid, soliton

References

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