Mathematical model of pulsating viscous liquid layer movement in a flat channel with elastically fixed wall

Abstract

The problem of dynamic interaction of elastically fixed channel wall with pulsating viscous incompressible liquid layer is set up and analytically solved. The problem in a flat setting for the regime of a stationary pulsating liquid movement in the cannel under the suggested harmonic law of pressure pulsating at its butt end is considered. The formulated bound problem represents non-linear connected Navier-Stocks equations system for viscous incompressible liquid layer and the equation of elastically fixed channel wall dynamics. The conditions of liquid adhesion to impenetrable channel walls and the condition of free leakage of liquid at channel butt ends are presented in the paper as the bound ones. The complex of dimensionless variables of the problem under consideration and its small parameters are singled out well. The relative thickness of liquid layer and relative amplitude of channel wall fluctuations are taken as small parameters. The linearization of the problem by means of perturbation method is made in the course 7900 R. R. V. Ageev et al. of asymptotic expansions, according to the small parameters, singed out above. The solution of the linearized problem is made by means of the assigned forms method for adjusted harmonic fluctuations. The law of elastically fixed channel wall shift and hydrodynamic parameters distribution in liquid are defined. Frequency dependent function of dynamic pressure distribution along the channel, frequency dependent function of phase shift pressure in the channel, amplitude frequency characteristics and phase frequency characteristics of elastically fixed channel wall in relation to the initial perturbation in the butt end are received.

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