Diffusion of dissolved ${\mathrm{CO}}_2$ in water propagating from a cylindrical bubble in a horizontal Hele-Shaw cell

The dissolution of a gas bubble in a confined geometry is a problem of interest in technological applications such as microfluidics or carbon sequestration, as well as in many natural flows of interest in geophysics. While the dissolution of spherical or sessile bubbles has received considerable attention in the literature, the case of a two-dimensional bubble in a Hele-Shaw cell, which constitutes perhaps the simplest possible confined configuration, has been comparatively less studied. Here, we use planar laser-induced fluorescence to experimentally investigate the diffusion-driven transport of dissolved ${\mathrm{CO}}_2$ that propagates from a cylindrical mm-sized bubble in air-saturated water confined in a horizontal Hele-Shaw cell. We observe that the radial trajectory of an isoconcentration front, $r_f\left(t\right)$, evolves in time as approximately $r_f-R_0\propto \sqrt{t}$, where $R_0$ denotes the initial bubble radius. We then characterize the unsteady ${\mathrm{CO}}_2$ concentration field via two simple analytical models, which are then validated against a numerical simulation. The first model treats the bubble as an instantaneous line source of ${\mathrm{CO}}_2$, whereas the second assumes a constant interfacial concentration. Finally, we provide an analogous Epstein-Plesset equation with the intent of predicting the dissolution rate of a cylindrical bubble.

Figure 1

(a) Plan view and (b) side view of the experimental setup. PLIF reveals a ${\mathrm{CO}}_2$-containing region [dark green region enclosed by $r_f\left(t\right)$] propagating radially outward from the dissolving cylindrical bubble of radius $R\left(t\right)$.

Figure 2

Raw PLIF snapshots of the bubble surrounded by a propagating ${\mathrm{CO}}_2$ front taken at (a) $t=10$ s, (b) $t=1$ min, (c) $t=3$ min, and (d) $t=8$ min after bubble injection. The dotted white circle marks the initial perimeter of the dissolving bubble. The horizontal dark slab left of the bubble is the bubble shadow. These snapshots correspond to the experiment with initial ${\mathrm{CO}}_2$ fraction $X_0=0.55$ (see Fig. 3).

Figure 3

Evolution of the bubble radius with time for six different experiments, each possessing a distinctive initial mole fraction of ${\mathrm{CO}}_2$, $X_0$. Experiments (markers) are compared with simulation (solid lines).

Figure 4

Comparison of (a) a raw image and (b) a processed image. The white dotted circle marks the initial bubble circumference, while the red dotted circle marks the front position at which $\overline{G}=0.9$.

Figure 5

Evolution in time of the radial intensity profile corresponding to the experiment with $X_0=0.39$. The profile is plotted at several log-spaced times after bubble injection. $R_0$ (in the abscissa label) denotes the initial bubble radius.

Figure 6

(a) Front radial trajectories corresponding to $\overline{G}=0.9$ for the six experiments. Upper and lower values of the error bars mark the front radii corresponding to $\overline{G}=0.95$ and 0.85, respectively. (b) Same front trajectories plotted in dimensionless form.

Figure 7

Experimental front trajectories (markers) corresponding to $\overline{G}=0.95$. ILS diffusion model curves given by Eq. (10) are drawn with ${\tau }_s=0.25$ and $u_f=0.02\%$ (upper curve), 0.1% (middle) and 0.5% (lower).

Figure 8

Experimental front trajectories (markers) corresponding to $\overline{G}=0.95$ labeled with the predicted front concentration $u_f\left(c_f\right)$ according to simulation. Simulation, ILS model (10) and CIC model (13) trajectories are also shown.

Figure 9

Concentration profiles according to—simulation and - - - the ILS model. The corrected intensity $\overline{G}$ is plotted assuming that ($•$) $\mathrm{\Delta }\overline{G}\propto \mathrm{\Delta }C$ or ($\circ$) $\mathrm{\Delta }\overline{G}\propto \mathrm{\Delta }\mathrm{pH}$.

Figure 10

pH profiles from - - - the ILS model and—simulation. The corrected intensity $\overline{G}$ is plotted assuming that ($•$) $\mathrm{\Delta }\overline{G}\propto \mathrm{\Delta }C$ or ($\circ$) $\mathrm{\Delta }\overline{G}\propto \mathrm{\Delta }\mathrm{pH}$.

Figure 11

Time history of the concentration gradient at a stationary cylindrical, planar, and spherical interface.

Figure 12

Dissolution dynamics of a cylindrical bubble under ambient conditions in degassed water. The curves correspond to a bubble entirely composed of (a) ${\mathrm{CO}}_2$ gas (high solubility) and (b) air (poor solubility). We show the solutions according to our numerical simulation (cf. Appendix pp1) and quasistationary model with the full interfacial gradient (numerically computed) given by Eqs. (16) and (B11). We also plot the solutions of our simplified quasistationary model in Eq. (17) setting $K=1/2$ (short-time approximation) and $K=0$ (planar interface).

Figure 13

Dissolution dynamics of a ${\mathrm{CO}}_2$-air cylindrical bubble in air-saturated water according to our numerical simulation and our quasistationary model in Eq. (18) upon the assumption of a circular ($K=1/2$) or planar ($K=0$) interface.

Figure 14

Large gap thickness ($L=4$ mm) experiment 3 min after bubble injection. Two snapshots reveal the ${\mathrm{CO}}_2$ boundary layer observed (plan view) when the laser sheet penetrates the cell (a) close to the top glass slide and (c) close to the bottom slide. Panel (b) shows the cross section of the bubble at that time.