Mathematical model of movement of a pulsing layer of viscous liquid in the channel with an elastic wall

Abstract

The problem of hydroelasticity of the plate forming at wall of the slot-hole channel with a pulsing layer of viscous incompressible liquid at the set harmonious law of a pulsation of pressure at its end face in flat statement is put and analytically solved. The set regional task represents nonlinear related system of the equations of Navier-Stokes for a layer of viscous incompressible liquid and the equation of dynamics of a plate (beam strip). Conditions of sticking of liquid act as regional conditions to impenetrable walls of the channel, a condition of the free expiration of liquid at end faces of the channel and a condition of a hinged supporting of a plate wall of the channel. The complex of dimensionless variables of a considered task is created and small parameters of a task are allocated. As small parameters we have chosen the relative thickness of a layer of liquid and relative amplitude of a deflection of a plate. Considering asymptotic decomposition in the allocated small parameters of a task we have carried out its linearization by a method of indignations. The solution of the linearized task is obtained by a method of the set forms for a mode of the established harmonic oscillations. Thus, proceeding from boundary conditions for a channel plate wall, the form of its deflection is set in the form of ranks on trigonometrical functions from longitudinal coordinate. The law of a deflection of an elastic wall of the channel and distribution of hydrodynamic parameters are found in liquid. We have obtained frequency dependent functions of distribution of amplitudes of a deflection and dynamic pressure along the channel and frequency dependent functions of distribution of phase shift of a deflection of a wall and pressure in the channel of rather initial indignation at an end face. On the basis of calculations it is shown that resonant fluctuations of an elastic wall of the channel, pressure excited by insignificant pulsations at its end face, can cause essential changes of dynamic pressure and be the main reason of vibration cavitation in liquid.

Hydroelastic Oscillations of a Circular Plate, Resting on Winkler Foundation

“…By substituting (14) in (10) and expanding in the series in eigenfunctions of the Sturm-Loiville problem, we get: …”

“…The investigation of a beam hydroelastic oscillations in a viscous liquid flow with an application to piezoelectric elements, aimed at obtaining energy from the flow, is made in [13]. The problem of bending hydroelastic oscillations of the plate forming the narrow channel wall under the impact of viscous liquid pulsating layer is solved in [14]. The forced hydroelastic oscillations of a three-layered circular plate, interacting with viscous incompressible liquid layer under channel foundation vibration, are investigated in [15].…”

The Study of Wave Propagation in a Shell with Soft Nonlinearity and with a Viscous Liquid Inside

“…In those papers, the case of interaction of shells and a viscous incompressible fluid is not considered. References [12][13][14] consider the case of interaction of shells and a viscous incompressible fluid without taking into account wave phenomena; neither were local terms of inertia influence investigated. Various methods are used to solve related and unrelated problems.…”

Mathematical modeling of three-layer beam hydroelastic oscillations

“…Reference [10] investigates hydroelastic oscillations of the beam in a viscous liquid flow for the case of piezo-electric elements with the aim of obtaining energy from the flow. The problem of the bending hydroelastic oscillations of the plate under the impact of the viscous liquid pulsating layer was studied in [11]. The study of the hydroelastic oscillations of the plate, resting on Winkler foundation was made in [12][13][14].…”