WAVE DYNAMICS OF COATED INCLUSIONS IN A VISCOELASTIC MEDIUM
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WAVE DYNAMICS OF COATED INCLUSIONS IN A VISCOELASTIC MEDIUM D. A. Gubaidullin and Yu. V. Fedorov∗
UDC 532.529:534.2
Abstract: A modified Rayleigh–Lamb equation is obtained, which takes into account the radial vibrations of a liquid drop that is covered with a viscoelastic shell, whose center contains a gas bubble, and which is placed in a viscoelastic medium. For the case of small vibrations of the inclusion, the heat transfer problem for gas, liquid phase, viscoelastic shell, and carrier liquid is solved. A dispersion equation is derived for a bubble medium. The influence of the inclusion shell and the viscoelasticity of the carrier phase on the dynamics of acoustic waves is investigated. Calculation results are compared with experimental data. Keywords: acoustic waves, gas bubbles, liquid interlayer, viscoelastic shell, modified Rayleigh– Lamb equation, interfacial heat transfer, dispersion equation. DOI: 10.1134/S0021894420040033
INTRODUCTION Bubbles covered with a viscoelastic shell are observed in many scientific fields, but mainly used in biomedicine as contrast agents for ultrasound diagnostics [1, 2]. Existing theoretical models are based on various forms of writing equations of vibrations of spherical bubbles with account for elasticity and viscosity of the surface layer. In [3], a modified Rayleigh–Plesset equation is obtained, which accounts for radial oscillations of a gas bubble with a viscoelastic shell of finite thickness in a liquid on the basis of the Kelvin–Voigt rheological equation, and the impact of the shell parameters on the dynamics of acoustic waves is analyzed. In [4], the model from [3] is simplified for the case where the thickness of a viscoelastic bubble shell is close to zero, and the effect of the polymer shell of microbubbles on the attenuation of a pulse perturbation is studied theoretically and experimentally. In [5], based on the results of [3], a mathematical model was obtained that determines the propagation of acoustic waves in a liquid with gas bubbles covered with a viscoelastic shell in the approximation of interacting and interpenetrating continua [6]. However, with the advent of new substances, coated gas bubbles are less commonly used in applied research. Recently, some attention has been paid to the study of emulsions with a phase shift, which have the following feature. Under the action of ultrasound, gas bubbles form inside the liquid droplets. This process is known as acoustic evaporation of droplets [7]. Emulsions with a phase shift are actively used in gas embolotherapy [8–10]. This method is aimed at fighting cancer cells and tumors. Droplets of a special substance are injected into the blood vessels that feed cancer cells, each enclosed in a viscoelastic biological shell. Under the action of
Institute of Mechanics and Engineering—A Separate Structural Subdivision of the Federal Research Center “Kazan Scientific Center of the Russian Academy of Sciences,” Kazan, 420111 Russia; [email protected]; ∗ kopperfi[email protected]. Translated from Prikladnaya Mekhanika i Tekhnicheskay
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