Viscous liquid movement in a flat channel, formed by a vibrating stamp and a simply supported plate

Аuthors

Ageev R. V.^*, Mogilevich L. I.^**, Popov V. S.^***, Popova A. A.^****

Yuri Gagarin State Technical University of Saratov, 77, Politechnicheskaya str., Saratov, 410054, Russia

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

Abstract

The problem of viscous incompressible liquid layer movement in a flat channel with the walls formed by a vibrating stamp and an elastic plate is set up and analytically solved in a two-dimensional problem’s statement. The liquid’s movement is excited by the known pressure difference at the butt ends. The law of the stamp movement as an absolutely rigid body is also considered as a known function. The problem statement results a coupled system of the Navier-Stocks equations, the continuity equation for viscous liquid layer, the plate-stripe dynamics equation, and the boundary conditions. The boundary conditions of liquid adhesion to the channel walls, of free liquid leakage at the butt ends and the ones of simply support of the plate at the butt ends make up the boundary vaue problem’s statement. The complex of dimensionless variables of the problem is formed; the relative liquid layer thickness and the stamp oscillations amplitude are taken as small parameters. The linearization of the problem by means of perturbation method is made. The solution of the linearized problem for established harmonic oscillations regime is carried out. The form of the plate deflection is approximated by the trigonometric series of longitudinal coordinate. The laws of elastic channel wall deflection and liquid pressure in the channel are found.

The solution is carried out by the means of the perturbation method for the established harmonic oscillations. The pressure distribution in the liquid and the deflections of the channel wall are found. The frequency dependent deflection amplitude and the dynamic pressure along the channel are obtained, and the investigation of the hydroelastic oscillations of channel walls is provided. On the basis of the performed calculations it is shown that the keeping of two or three members of the series in the solution is quite enough for practical purposes. Considering each consecutive series’ member leads to the emergence of additional resonance frequency of the plate oscillation.

Keywords:

hydroelasticity, viscous liquid, oscillations, beam, oscillating stamp, perturbation method

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