Radiative decay of the nonlinear oscillations of an adiabatic spherical bubble at small Mach number

Abstract

A theoretical study is carried out for bubble oscillation in a compressible liquid with significant acoustic radiation based on the Keller–Miksis equation using a multi-scaled perturbation method. The leading-order analytical solution of the bubble radius history is obtained to the Keller–Miksis equation in a closed form including both compressible and surface tension effects. Some important formulae are derived including: the average energy loss rate of the bubble system for each cycle of oscillation, an explicit formula for the dependence of the oscillation frequency on the energy, and an implicit formula for the amplitude envelope of the bubble radius as a function of the energy. Our theory shows that the frequency of oscillation does not change on the inertial time scale at leading order, the energy loss rate on the long compressible time scale being proportional to the Mach number. These asymptotic predictions have excellent agreement with experimental results and the numerical solutions of the Keller–Miksis equation over very long times. A parametric analysis is undertaken using the above formula for the energy of the bubble system, frequency of oscillation and minimum/maximum bubble radii in terms of the dimensionless initial pressure of the bubble gases (or, equivalently, the dimensionless equilibrium radius), Weber number and polytropic index of the bubble gas.

A Review of Models for Bubble Clusters in Cavitating Flows

“…Keller and Miksis [115], Gilmore [81] and Tomita [193] have proposed different corrected versions of the Rayleigh-Plesset equation that accounts for compressibility effects showing that the amplitude of the rebound is significantly attenuated during the bubble collapse processes. The relevance of compressibility effects and the accuracy of the various modified equations is discussed by Prosperetti and Lezzi [165,130], and an interesting assymptotic analysis on the radiative decay of non-linear bubble oscillations at small Mach numbers has been recently presented by Smith & Wang [186]. For strong bubble collapses mechanisms such as the damping originated by emission of shock waves during the collapse are not captured by the corrections mentioned above and one has to resort to the numerical solution the full Navier-Stokes in the liquid [71,122,76].…”

A theoretical model for the growth of spherical bubbles by rectified diffusion

“…This new system of equations is important for the following reasons. These ordinary differential equations are straightforward to integrate numerically, whereas the well-established mathematical model has proved to be intractable. The spatial extent of the diffusion boundary layer in the liquid adjacent to the bubble interface is sufficiently small (), that the model applies to bubbles even in a dense cloud. This model, together with our recent research on strongly nonlinear analysis (Smith & Wang 2017, 2018, 2021), is the basis of our objective to simulate bubble growth over many millions of cycles of oscillation. …”

“…(3) This model, together with our recent research on strongly nonlinear analysis (Smith & Wang 2017, 2018, 2021, is the basis of our objective to simulate bubble growth over many millions of cycles of oscillation.…”

The pitfalls of investigating rotational flows with the Euler equations

“…Kuzmak (1959) introduced the strongly nonlinear analysis of ordinary differential equations. His technique has recently been applied to the Rayleigh–Plesset equation in order to model the viscous decay of oscillating spherical bubbles (Smith & Wang 2017) and to the Keller–Miksis equation in order to describe the radiative decay of oscillating spherical bubbles (Smith & Wang 2018). Kuzmak's method has also been applied to the three-dimensional Navier–Stokes equations to successfully predict the viscous decay of an oscillating drop (Smith 2010).…”