A Fourth-Order Rayleigh–Plesset Approximation for Nonlinear Bubble Dynamics in Viscoelastic Media

Understanding the dynamics of gas bubbles in viscoelastic media is crucial for applications involving stable cavitation under ultrasound, such as drug delivery, materials processing, and biomedical imaging. The Rayleigh–Plesset equation formulated in terms of bubble volume variation, incorporating viscoelastic effects via the linear Kelvin–Voigt model, is extended here to a fourth-order approximation. This extension should allow a more accurate description of nonlinear bubble dynamics at finite acoustic amplitudes.

The resulting equation is solved numerically under various acoustic conditions, with particular emphasis on driving frequencies near the bubble’s resonance. The fourth-order model represents a more accurate and detailed version of the equation describing bubble oscillations. This increased precision allows for a better understanding of how the medium’s shear elasticity and viscosity influence bubble behavior, especially under high amplitude oscillations or near-resonance driving frequencies. Comparisons with lower-order models will be carried out to show whether this fourth-order approximation significantly enhances the description of nonlinear oscillations. In particular, it may reveal dynamic features that traditional second-order formulations fail to capture.

This work provides a robust framework for analyzing how viscoelastic properties affect bubble dynamics, contributing to improved prediction and control of stable cavitation phenomena in complex media. These findings highlight the importance of using the fourth-order model to fully account for viscoelastic behavior when modeling bubble volume variations under ultrasound excitation.