# Modeling of Deformation Waves in Physically Non-linear Shell, Containing Viscous Incompressible Liquid

### Mathematics. Physics. Mechanics

### Аuthors

^{1}

^{*},

^{2}

^{**},

^{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: blinkovua@info.sgu.ru

**e-mail: evilgraywolf@gmail.com

***e-mail: mogilevich@sgu.ru

### Abstract

The coupled hydroelasticity problem formulation: the dynamic equations for the both geometrically and physically nonlinear elastic shell, the equations of dynamics of the viscous incompressible liquid being inside a cylindrical shell, and its boundary conditions has been considered. The equation for non-linear deformation waves in shell has been derived by the use of methods of solution for the mentioned hydroelasticity problem. The used asymptotic transition to the classical equation of hydrodynamic lubrication theory becomes possible when the radius of the shell midsurface is significantly smaller than the deformation wave length. The obtained equation has been solved as the generalized and modified Korteweg-de Vries equation including the term corresponding to the liquid impact inside the shell. In the absence of the liquid the equation has a known exact solution which can be considered as an initial condition for the numerical solution of the new equation.The numerical solution of the Cauchy problem for the new equation considering the effect of the liquid media to the shell dynamics is shown. The formulation 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 the condition for the compatibility of the system and automatically secures the fulfillment of the integral conservation laws in the domains composed form the basic finite volumes.

The presence of liquid inside the shell results a substantial change of longitudinal deformation waves propagation. If there is no liquid in the shell, a solitary wave (soliton) moves retaining its original shape and velocity. The presence of the liquid in the shell with the Poisson ratio less than one-half results the exponential increase of the wave amplitude under and to the absence of wave oscillations at the forefront due to the energy dissipation. In the absence of energy dissipation the oscillations would occur at the leading edge of the wave. Thus, it can be stated that the liquid contributes to a constant extra “feeding” energy (the original source of excitation), providing for the amplitude growth.

Consequently, the use of these models allows one a widening of experimental data analysis possibilities for various principally non-linear systems: fuel supply, cooling, blood and lymph stream pulsating waves etc.

### Keywords:

cylindrical shell, non-linear waves, hydroelasticity, viscous incompressible liquid, soliton### References

- Zemlyanukhin A. I., Mogilevich L. I.
*Izv. vuzov. Prikladnaya nelineinaya dinamika,*1995, vol. 3, no. 1, pp. 52-58. - Zemlyanukhin A. I., Mogilevich L. I.
*Nelineinye volny v tsilindricheskikh obolochkakh: solitony, simmetrii, evolyutsiya*(Nonlinear waves in cylindrical shells: solitons, symmetry, evolution), Saratov, Sarat. gos. tekhn. un-t, 1999, 132 p. - Loitsyanskii L. G.
*Mekhanika zhidkosti i gaza*(Mechanics of liquid and gas), Moscow, Drofa, 2003, 840 p. - Kauzerer K.
*Nelineinaya mekhanika*(Nonlinear mechanics), Moscow, Inostrannaya literatura, 1961, 240 p. - Vol'mir A. S.
*Obolochki v potoke zhidkosti i gaza: zadachi gidrouprugosti*(Shells in the flow of liquid and gas: problems of hydroelasticity), Moscow, Nauka, 1979, 320 p. - Blinkov Yu. A., Mozzhilkin V. V.
*Programmirovanie*, 2006, vol.32, no. 2, pp. 71-74. - Gerdt V. P., Blinkov Yu. A., Mozzhilkin V. V.
*Gröbner bases and generation of difference schemes for partial differential equations. Symmetry, Integrability and Geometry. Methods and Applications*, 2006, vol. 2, 26 p. http://www.emis.de/journals/SIGMA/2006/Paper051/index.html. - Gerdt V. P., Blinkov Yu. A.
*Involution and difference schemes for the Navier-Stokes equations. Computer Algebra in Scientific Computing,*Lecture Notes in Computer Science, Berlin, Springer Heidelberg, 2009, vol. 5743 , pp. 94–105. - SciPy. http://www.scipy.org/

Download