The history effect on bubble growth and dissolution. Part 2. Experiments and simulations of a spherical bubble attached to a horizontal flat plate

Abstract

The accurate description of the growth or dissolution dynamics of a soluble gas bubble in a super-or undersaturated solution requires taking into account a number of physical effects that contribute to the instantaneous mass transfer rate. One of these effects is the so-called history effect. It refers to the contribution of the local concentration boundary layer around the bubble that has developed from past mass transfer events between the bubble and liquid surroundings. In Part 1 of this work (Peñas-López et al. 2016b), a theoretical treatment of this effect was given for a spherical, isolated bubble. Here, Part 2 provides an experimental and numerical study of the history effect regarding a spherical bubble attached to a horizontal flat plate and in the presence of gravity. The simulation technique developed in this paper is based on a streamfunction-vorticity formulation that may be applied to other flows where bubbles or drops exchange mass in the presence of a gravity field. Using this numerical tool, simulations are performed for the same conditions used in the experiments, in which the bubble is exposed to subsequent growth and dissolution stages, using step-wise variations in the ambient pressure. Besides proving the relevance of the history effect, the simulations highlight the importance that boundary-induced advection has to accurately describe bubble growth and shrinkage, i.e. the bubble radius evolution. In addition, natural convection has a significant influence that shows up in the velocity field even at short times, though, given the supersaturation conditions studied here, the bubble evolution is expected to be mainly diffusive.

Ultrasound-enhanced mass transfer during single-bubble diffusive growth

“…1(b), the ambient bubble radius R 0 (defined as the mean sphere-equivalent radius about which the bubble oscillates) grows by diffusion until it approaches the resonant size. The early growth dynamics are well predicted by the classical Epstein-Plesset theory for diffusive growth [27] despite some deviations which can be attributed to perturbations in the initial condition of the concentration field [28]. Taking into account the presence of the substrate, the asymptotic solution reads…”

Collective dissolution of microbubbles

“…namely, the atmospheric pressure far from the bubble (P ∞ ), the hydrodynamic pressure induced by the fluid motion (P h ), and the capillary pressure (P γ ). The latter is retained here, in contrast to most analytical derivations on bubble dissolution and growth [25,29], effectively restricting their results to bubble radii greater than 1-10 µm, and therefore excluding the final stages of the bubble's life. In the following, we assume that hydrodynamic pressure is negligible; when the fluid motion results from the bubble collapse, this effectively assumes that the Capillary number, Ca = ηṘ/γ, is always small, i.e.Ṙ γ/η ≈ 70 m.s −1 for water under normal conditions.…”

“…This shows that the potential part u pot does not contribute to the force that is solely given in terms of the viscous contribution and can be computed directly using Stimson's result as in Eq. (29).…”

“…For most gases under ambient conditions, diffusion of the dissolved gas is much faster than the evolution in the bubble radius due to the molar density contrast between the concentration of gas in the bubble and in the liquid [26]. As a result, transient and convective effects are essentially negligible except initially or when fluctuations in the background pressure are taken into account [27][28][29]. This is in stark contrast with the heat-driven dynamics of vapour bubbles for which both time-scales are comparable and inertial effects can be significant [30], or with gas bubbles with comparable concentrations in both phases [31].…”

Ultrasound-enhanced mass transfer during single-bubble diffusive growth

“…perturbations in the initial condition of the concentration field [27]. Taking into account the presence of the substrate, the asymptotic solution reads [28]…”