Particle-bubble interaction inside a Hele-Shaw cell

Hydrodynamic interactions between air bubbles and particles have wide applications in multiphase separation and reaction processes. In the present work, we explore the fundamental mechanism of such complex processes by studying the collision of a single bubble with a fixed solid particle inside a Hele-Shaw cell. Physical experiments show that an air bubble either splits or slides around the particle depending on the initial transverse distance between the bubble and particle centroids. An air bubble splits into two daughter bubbles at small transverse distances, and slides around the particle at large distances. In order to predict the critical transverse distance that separates these two behaviors, we also develop a theoretical model by estimating the rate of the bubble volume transfer from one side of the particle to the other based on Darcy's law, which is in good agreement with experiments.

Figure 1

(a) Schematic of the experimental setup with key parameters and the coordinate system. (b) The image sequence from three representative experiments demonstrating the interaction of the bubble and a fixed particle: $\left(i\right)$ equal splitting, $\left(ii\right)$ unequal splitting, and $\left(iii\right)$ sliding bubbles. The bubble diameter $\left(2R\right)$ in $\left(i\right)$–$\left(iii\right)$ corresponds to 1.187, 1.183, and 1.184 cm, while the values of the transverse distance between the particle and bubble centroids, $d$, are 0.03, 0.16, and 0.51 mm, respectively.

Figure 2

(a) The evolving boundaries and centroid positions of the bubbles corresponding to cases $\left(i\right)$–$\left(iii\right)$ shown in Fig. 1. (b) The in-plane boundary curvature $\left(R{\kappa }_{\exp }\right)$ of case $\left(iii\right)$ at three different frames. Here $s$ denotes the coordinate defined along the bubble boundary.

Figure 3

The normalized area difference, $\left|\right(A_r-A_l)/A|$, is plotted as a function of the dimensionless time, $t/(R/U)$ for all different values of $d$. The dark blue lines correspond to the sliding bubbles, while the light gray lines are the splitting bubbles.

Figure 4

(a) Deformed bubble shapes at two different instants during the bubble-particle interaction process. (b) Modeled bubble area shift with a range of initial $\mathrm{\Delta }A/A$ values, compared with experiment data shown in Fig. 3. Parameters $\overline{\alpha }=0.8$ and $\overline{\beta }=0.75$ are used in the model [Eq. (4)]. The critical time $t_c^{*}=0.75$ is assumed in this figure. Corresponding to the final time $t_N^{*}=2$ and 2.5, the predicted transition curves that separate the sliding and splitting cases are shown as dashed curves.

Figure 5

(a) The critical transition $d_c^{*}$ predicted by our model for different $t_c^{*}$ and $t_N^{*}$ values. (b) The $d\text{−}R$ phase diagram that exhibits the transition between sliding and splitting bubbles. The blue circles correspond to sliding bubbles, while the gray upside-down triangles indicate splitting bubbles. The predicted minimum and maximum values of this transition are $d_c^{*}=0.027$ and $d_c^{*}=0.151$ for $R=5$ mm (solid lines) and $d_c^{*}=0.035$ and $d_c^{*}=0.173$ for $R=7.5$ mm (dashed lines). The three representative cases $\left(i\right)$–$\left(iii\right)$ are labeled with red edges.

Figure 6

The BEM simulation results for (a) $p^{\prime }/(\sigma /R)$ and (b) $p/(\sigma /R)$ distribution around a deforming bubble.

Figure 7

The velocity field derived from the numerically computed pressure field using Eq. (3).

Figure 8

(a) The curvature difference $\left|\mathrm{\Delta }\kappa \right|$ as a function of $|1/\sqrt{A_l}-1/\sqrt{A_r}|$ based on measurement (blue dots). The black line has a slope of 1.2. (b) The measured vertical distance $\left|\mathrm{\Delta }y\right|$ as a function of $|\sqrt{A_r}-\sqrt{A_l}|$ (blue dots). The black line has a slope of 1.5. In both plots, $x$ and $y$ axes are normalized by the length scale $R$.