Поширення хвиль у пружному шарі, що контактує з шаром в'язкої стисливої рідини

Abstract

У рамках тривимірних лінеаризованих рівнянь Нав'є-Стокса для рідкого середовища та тривимірних лінійних рівнянь класичної теорії пружності для твердого тіла наведено постановку задачі про поширення гармонічних хвиль у пружному шарі, який контактує з шаром в'язкої стисливої рідини. Проведено чисельне дослідження, побудовано дисперсійні криві та встановлено залежності фазових швидкостей і коефіцієнтів згасання мод від в'язкості рідини, товщини шару рідкого середовища та товщини твердого шару для широкого спектра частот.

The problem about the propagation of acoustic waves in elastic layer which interacts with a layer of viscous compressible fluid is considered. The study is carried out basing on the three-dimensional linearized Navier-Stokes equations for viscous fluid and linear equations of the classical theory of elasticity for the solid layer. Statement of the problem and approach that are based on the use of general solutions of linearized Navier-Stokes equations for viscous fluid and linear equations for elastic body is applied. The dispersion equation is obtained, which describes the propagation of harmonic waves in the hydroelastic system within the wide range of frequencies. Numerically the characteristic equation is solved and dispersion curves are constructed for the wide spectrum of frequencies in the case of a thick elastic layer. Dependencies of phase velocities of normal modes of Lamb from the thickness of elastic layer in the case of absence of interaction with fluid are given. For the hydroelastic waveguide dependencies of phase velocities and attenuation coefficients of modes from the thickness of layer of viscous compressible fluid are presented. Dependencies of relative changes of phase velocities of normal modes from the thickness of fluid layer are also given. An influence of the viscosity of liquid medium, thickness of layers of elastic body and fluid on the phase velocities and attenuation coefficients is analyzed. For hydroelastic system with increasing of the thickness layer of viscous fluid the velocity of zero antisymmetric mode tends to the Stoneley wave velocity and velocity of zero symmetric mode tends to the Rayleigh wave velocity was shown. With increasing of thickness of liquid layer the velocity of the first antisymmetric mode goes to the wave velocity, the value of which is less than the propagation velocity of sound in liquid. Phase velocities of all other higher modes tends to the propagation velocity of sound in liquid. It was showed that in the case of thick elastic layer for all modes there are layers of fluid of certain thickness and there are certain frequencies for which the influence of viscosity of the fluid on the phase velocities and attenuation coefficients is minimal. It was also found that for certain of modes, exist as certain frequencies and certain ranges of frequencies wherein the influence of fluid viscosity on the phase velocities and attenuation coefficients of these modes is significant. The developed approach and the results obtained allow for wave processes to establish the limits applicability of the model of an ideal fluid.