On the forced oscillations of a small gas bubble in a spherical liquid-filled flask

Abstract

A spherically-symmetric problem is considered in which a small gas bubble at the centre of a spherical flask filled with a compressible liquid is excited by small radial displacements of the flask wall. The bubble may be compressed, expanded and made to undergo periodic radial oscillations. Two asymptotic solutions have been found for the low-Mach-number stage. The first one is an asymptotic solution for the field far from the bubble, and it corresponds to the linear wave equation. The second one is an asymptotic solution for the field near the bubble, which corresponds to the Rayleigh–Plesset equation for an incompressible fluid. For the analytical solution of the low-Mach-number regime, matching of these asymptotic solutions is done, yielding a generalization of the Rayleigh–Plesset equation. This generalization takes into account liquid compressibility and includes ordinary differential equations (one of which is similar to the well-known Herring equation) and a difference equation with both lagging and leading time. These asymptotic solutions are used as boundary conditions for bubble implosion using numerical codes which are based on partial differential conservation equations. Both inverse and direct problems are considered in this study. The inverse problem is when the bubble radial motion is given and the evolution of the flask wall pressure and velocity is to be calculated. The inverse solution is important if one is to achieve superhigh gas temperatures using non-periodic forcing (Nigmatulin et al. 1996). In contrast, the direct problem is when the evolution of the flask wall pressure or velocity is given, and one wants to calculate the evolution of the bubble radius. Linear and nonlinear periodic bubble oscillations are analysed analytically. Nonlinear resonant and near-resonant periodic solutions for the bubble non-harmonic oscillations, which are excited by harmonic pressure oscillations on the flask wall, are obtained. The applicability of this approach bubble oscillations in experiments on single-bubble sonoluminescence is discussed.

Liquid compressibility effects during the collapse of a single cavitating bubble

“…Analysis of the liquid dynamics during the bubble collapse in scarce. 16,17 The works of Prosperetti and Lezzi,18,19 theoretically quantifying the order of accuracy of various RP-type equations as a function of the Mach number, are relevant examples.…”

Regular and chaotic dynamics of a spherical bubble

“…We will assume that the antinode of the acoustic wave is located at the centre of the flask and that the closest nodes are on its boundary. Such a simplified formulation of the problem is often used when simulating the dynamics of a single bubble [20][21][22][23][24]. The acoustic pressure distribution in the liquid, which is a solution of the linear wave equation for a weakly compressible liquid, can be represented in the form sinkr p(r, t) = Po + Pa kr where r is the translational coordinate, t is the time, P0 is the atmospheric pressure, Pa = -Pa sino~t is the acoustic pressure at the centre of the flask with amplitude Pa, k = co/c is the wave number, c is the velocity of sound in the liquid and ~ is the angular frequency of the acoustic wave.…”

“…In the case of a bounded volume of liquid, a mathematical model is used which describes the radial oscillations of the bubble surface in a spherical flask filled with a weakly compressible liquid when the flask walls serve as the source of the perturbation of the oscillations in the liquid [22][23][24]. It includes an ordinary differential equation which is analogous to the Herring-Flinn-Gilmore equation and an equation with a delay which relates the pressure on the flask walls to the change in the bubble radius.…”

Dynamics of laser-induced cavitation bubbles

“…The ''liquid drop'' external radius RðtÞ is defined by a generalized Rayleigh-Plesset equation [18] 1 À…”