A fourth-order approximation Rayleigh-Plesset equation written in volume variation for an adiabatic-gas bubble in an ultrasonic field: Derivation and numerical solution

Abstract

The derivation of a nonlinear ordinary differential equation for modeling the nonlinear oscillations of a gas bubble placed in an ultrasonic field is performed in terms of bubble-volume variations up to the fourth-order approximation. The equation, written within the Rayleigh-Plesset framework, is solved through numerical approximations. Results from simulations are compared to data obtained from the classic second-order approximation equation derived in the 1960–70’s, usually used in this framework, and from the third-order approximation equation derived in the 1990’s. This comparison shows that the fourth-order approximation allows us to observe the nonlinear behavior of the bubble at high finite amplitude, which differs from the other approximations when the nonlinearity of the phenomenon is higher, i.e., when the driving acoustic frequency is close to the bubble resonance.

Keywords

• Nonlinear acoustics
• Bubble dynamics
• Mathematical modeling
• Nonlinear ordinary differential equation
• Numerical solution