The history effect in bubble growth and dissolution. Part 1. Theory

Abstract

The term 'history effect' refers to the contribution of any past mass transfer events between a gas bubble and its liquid surroundings towards the current diffusion-driven growth or dissolution dynamics of that same bubble. The history effect arises from the (non-instantaneous) development of the dissolved gas concentration boundary layer in the liquid in response to changes in the concentration at the bubble interface caused, for instance, by variations of the ambient pressure in time. Essentially, the history effect amounts to the acknowledgement that at any given time the mass flux across the bubble is conditioned by the preceding time-history of the concentration at the bubble boundary. Considering the canonical problem of an isolated spherical bubble at rest, we show that the contribution of the history effect in the current interfacial concentration gradient is fully contained within a memory integral of the interface concentration. Retaining this integral term, we formulate a governing differential equation for the bubble dynamics, analogous to the well-known Epstein-Plesset solution. Our equation does not make use of the quasi-static radius approximation. An analytical solution is presented for the case of multiple step-like jumps in pressure. The nature and relevance of the history effect is then assessed through illustrative examples. Finally, we investigate the role of the history effect in rectified diffusion for a bubble that pulsates under harmonic pressure forcing in the non-inertial, isothermal regime.

History effects on the gas exchange between a bubble and a liquid

“…This is the standard approximation made in a majority of papers on the present subject, most recently in Ref. [9]. Some further considerations on these issues will be found in the next section.…”

“…Tao [14,15] tried to improve on this approximation but his method of solution leads to an infinite series that must be truncated, introducing errors that are difficult to quantify. A better approach was developed in a recent paper [9] by the introduction of a modified time variable. The radius of a dissolving bubble decreases very slowly for a significant fraction of its lifetime; it is only near the end of the process that it varies rapidly.…”

“…In this case, the boundary condition at the moving bubble surface can be treated as a domain perturbation as shown in one of the cases treated in Ref. [9]. When R ∼ R, however, the two terms of the convective time derivative are of a comparable order and, therefore, the ratio of ∂ρ g /∂t to diffusion is also of the order of the Péclet number.…”

“…History effects have also been studied in a very recent paper [9] focusing, however, on situations different from those considered here, namely the response of the bubble when the ambient pressure undergoes jump discontinuities and rectified diffusion in the limit of small-amplitude oscillations. The method of investigation is similar, except for the use of a modified time variable (a point which we briefly consider in the appendix).…”

Diffusion of dissolved CO2 in water propagating from a cylindrical bubble in a horizontal Hele-Shaw cell

“…Apart from the effect of advection, the (positive) error is mainly attributed to the extension of the quasistationary approximation to the interfacial concentrations of the species, since these are treated as constant in the obtention of the expression for the concentration gradient [28]. In other words, the error comes from not taking into account the so-called history effect [30,31]. The history effect can be summarized as the contribution of the preceding time history of the interfacial concentration on the current rate of growth or dissolution.…”