Mathematical modeling of vibrations of walls with an elastic support, forming two plane channels with a common wall, filled with viscous liquids


Аuthors

Popov V. S.1*, Popova A. A.1**, Orlova A. A.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: vic_p@bk.ru
**e-mail: anay_p@bk.ru
***e-mail: kostylevaaa@mail.ru
****e-mail: mari.popova.2004@internet.ru

Abstract

This paper presents a mathematical model for studying the dynamics of interaction between parallel rigid walls of two narrow channels. That channels sharing a common wall and filled with pulsating incompressible liquids with different physical properties. The case considered is one in which the upper wall of the first channel is stationary, while its bottom wall is elastically supported and serves as the upper wall of the second channel, the bottom wall of which also has an elastic support. The walls on the elastic supports can move in the direction normal to their surfaces. Liquid motion in the first channel is excited by its interaction with the vibrating channel walls. Liquid motion in the second channel occurs due to a static pressure drop and a harmonically pulsating pressure specified at its ends. The pressure at the ends of the first channel is assumed to be constant and equal to the initial static pressure in the channels under consideration. When developing the model, the viscosity of the liquids and differences in their physical properties were taken into account, and steady-state oscillations of the channel walls were studied. To describe a complex oscillatory system of heterogeneous bodies, the Lagrange approach for the equations of motion of rigid bodies and the Euler approach for the equations of fluid dynamics were used. These approaches were reconciled when writing the boundary conditions at the contact surface between rigid bodies and viscous liquids. As a result, a mathematical model was formulated as a nonlinear boundary value problem of mathematical physics. It including: ordinary differential equations for the dynamics of two channels walls under consideration, partial differential equations for the dynamics of viscous liquids, supplemented by boundary conditions at the channel ends and at the contact surface between the liquids and the walls, as well as expressions for the forces acting from the fluids on the channel walls. An asymptotic analysis of the mathematical model was done. For this purpose, dimensionless variables was proposed and small parameters of the oscillatory system under consideration were identified. Liquid dynamics were considered within the framework of hydrodynamic lubrication and a perturbation method was applied, which allowed for the linearization of the model and the determination of the hydroelastic responses of the channel walls in the form of amplitude and phase frequency characteristics. At the final stage, a numerical study of the obtained hydroelastic responses was conducted, allowing us to determine the characteristics of hydroelastic oscillations in the system under study. The presence of two resonant frequencies of wall oscillations was demonstrated, as well as the existence of an antiresonance for each wall. The effect of varying the channel gap size and fluid viscosity on the suppression of channel wall oscillations was assessed.

Keywords:

mathematical modeling; vibrations; viscous fluid; rigid wall with an elastic support; hydroelasticity; computational experiment.

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