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 formulation allows 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 and differences between Newtonian and viscoelastic media. To identify the physical conditions under which higher-order nonlinearities become necessary, a decision-tree classification analysis is performed. The results show that the proximity to resonance and the excitation amplitude are the primary determinants of higher-order nonlinear effects, while rheological properties act as modulators, with viscosity exerting a stronger influence than elasticity within the explored ranges. This work provides a physically interpretable criterion for selecting the appropriate model order, improving the prediction and control of nonlinear bubble oscillations under ultrasound excitation in viscoelastic media.