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

• Published in: Results in physics 2021-06, Vol.25, p.104193, Article 104193
• Main Author: Vanhille, Christian
• Format: Article
• Language: English
• Subjects: Bubble dynamics; Mathematical modeling; Nonlinear acoustics; Nonlinear ordinary differential equation; Numerical solution
• ISSN: 2211-3797; 2211-3797
• DOI: 10.1016/j.rinp.2021.104193

•We derive a Rayleigh-Plesset equation of 4th-order approximation in bubble volume.•We solve the initial-value problem for an adiabatic-gas bubble by a numerical model.•Results show the physical importance of the new terms introduced into the equation. 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.