This paper investigates the dynamics of an encapsulated bubble within a spherical liquid cell that is surrounded by an infinite elastic solid, aiming to enhance our understanding of bubble oscillations, which is crucial for targeted therapeutic release. The Carreau–Yasuda model is used for the surrounding liquid, and a nonlinear neo–Hookean hyperelastic model is used for the shell, replacing a simpler Newtonian liquid and linear shell models. This increased complexity is necessary to accurately capture bubble oscillations in a parameter range where both the non-Newtonian properties of liquid and the nonlinear behavior of the shell are critical. Resonance occurs when the acoustic field's driving frequency matches the natural frequency, thus, amplifying oscillations. The properties of the shell and elastic solid can dampen or amplify these oscillations, depending on their magnitudes and resonance frequency, making it essential to optimize these properties for balanced control and responsiveness in bubble oscillations. The parametric range for the bubble surface area and the wall liquid shear stress is determined for safe biomedical application. The maximum bubble surface area is 4000 m2 and the maximum wall shear stress is 3000 Pa for the parameters given in this paper. The study also highlights that the damping effect of the power-law index varies with ultrasonic drive frequency, pressure amplitude, Carreau–Yasuda properties, and cavity size, which is not observed for Newtonian fluids.
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