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

Abstract

Rectified diffusion is a bubble growth phenomenon that occurs in acoustic fields. Despite the existence of a well-established spherically symmetric mathematical model, theoretical results have been unsuccessful in reproducing the bubble growth even in the case of a single spherical bubble in the bulk. In the latter case, the influence of surfactants and acoustic microstreaming have been speculated as the explanation for this disagreement. In this article, an exact solution for the leading-order concentration of gas in the liquid is determined. Using this exact solution, the well-established mathematical model is reduced to a system of two ordinary differential equations for the spherical bubble radius and the mass of gas in the bubble. This simplified model predicts the rapid bubble growth observed in experiments of a single spherical bubble in the bulk. The new results show that the bubble growth is asymptotically larger than at later times when the mass flux is limited by the slow diffusion of gas in a much larger region of the liquid surrounding the bubble. The new results also show that this bubble growth is relatively insensitive to a reduction in surface tension.

Cavitation in a periodontal pocket by an ultrasonic dental scaler: A numerical investigation

“…5 a. This discrepancy could be due to the inaccuracy of the present model and/or the pre-existed air bubbles that grow due to rectified diffusion [9] , [23] .…”

A tractable mathematical model for rectified diffusion

“…The following is a modified version of the initial-stage analysis described by Smith & Wang (2022). The initial boundary value problem (3.45)–(3.47) is nonlinear, but the bubble radius in (3.45) only depends on time.…”

“…Fyrillas & Szeri (1994) describe the theoretical difficulty as hinging on the determination of the net flux of dissolved gas at the bubble interface; in particular, the influence of convection associated with bubble volume oscillations. Based on the Lagrangian change of variables developed by Plesset & Zwick (1952), Smith & Wang (2022) recently found an exact solution for the gas flux at the bubble interface including the influence of convection, but only for the initial rapid bubble growth over hundreds of cycles of oscillations. Smith & Wang (2022) performed matched asymptotic expansions in space.…”

“…Based on the Lagrangian change of variables developed by Plesset & Zwick (1952), Smith & Wang (2022) recently found an exact solution for the gas flux at the bubble interface including the influence of convection, but only for the initial rapid bubble growth over hundreds of cycles of oscillations. Smith & Wang (2022) performed matched asymptotic expansions in space. In this article, our approach is to extend the result in Smith & Wang (2022) to allow for the simulation of millions of cycles of oscillation.…”

“…Smith & Wang (2022) performed matched asymptotic expansions in space. In this article, our approach is to extend the result in Smith & Wang (2022) to allow for the simulation of millions of cycles of oscillation. We will establish the model below using both matched asymptotic expansions in space and multi-scales in time.…”