Numerical investigation of the collapse of a static bubble at the free surface in the presence of neighbors

The present study numerically investigates the bubble-bursting phenomenon at the free surface under the influence of symmetric and asymmetric patterns of neighboring bubbles. To conduct the simulation, a static pierced bubble at the interface of liquid and gas has been used and neighboring unpierced bubbles are added, if necessary. In the presence of symmetric neighbors, fluidic stages of the bursting process are established as cavity collapse, jet formation, and consequential fragmentation of the jet into drops. A thorough comparison of bubble bursting with symmetric neighbors (BBSN) and similar bursting of a solitary bubble (BSB) has been shown. Bubble bursting under the influence of an asymmetric pattern is compared with BBSN. Bubble with absent neighbors (BWAN) placement turns into the formation of the bent jet which is lying in the plane of the angular bisector of the absent neighbors. Bending of the jet nullifies as time progresses and turns into a vertical one like the jet dynamics observed in BBSN. A simple pictorial understanding has been presented to describe the reason behind the jet bend in BWAN based on a study of velocity vectors and the gap between the collapsing waves. The effect of a number of absent neighbors is also addressed extensively. As the number of absent neighbors increases, eventually, the jet inclination angle first increases and then decreases to match with the observation of BBSN and BSB. Finally, physical insight behind the rich fluidic phenomenon of the bubble bursting under the influence of different surface tensions and bubble sizes in the same cluster is also addressed.

Figure 1

Illustration of static-shaped bubble in the 2D view: (a)–(c) solitary bubble and (d) multibubble at the free surface; the presence of neighbors.

Figure 2

Computational domain of bubble bursting at the free surface, shown in a 2D plane to depict the placement of center and neighbor bubbles; boundary outflow to top surface and the rest of the outer boundary faces are no-slip, no-penetration type. An enlarged view of octree grid discretization is shown in the inset.

Figure 3

(a) Grid sensitivity study at various levels of refinement and (b) validation of the present study with Duchemin et al. [13].

Figure 4

Bursting of the solitary bubble; temporal evolution of cavity, jet formation, and subsequent breakup into drops. Equivalent radius $R=900\mu \mathrm{m}$, Bond number $\mathrm{Bo}=0.1725$, Laplace number $\mathrm{La}=4.14\times {10}^4$, and Morton number $\mathrm{Mo}=1\times {10}^{-7}$.

Figure 5

Sequences of bursting phenomenon of a bubble surrounded by similar static neighbors. Simulations are performed at conditions reported similar in Fig. 4.

Figure 6

Formation of flower-shaped bubble patterns in the vicinity of collapsing bubble for BBSN; an equivalent radius $R=900\mu \mathrm{m}$, Bond number $\mathrm{Bo}=0.1725$, Laplace number $\mathrm{La}=4.14\times {10}^4$, and Morton number $\mathrm{Mo}=1\times {10}^{-7}$; $t^{*}=t\sqrt{\frac{\sigma }{\mathrm{\Delta }\mu D}}$.

Figure 7

Comparison of stages during bubble bursting for solitary and symmetric neighbors. (a) Evolution of phase contours. (b) Temporal propagation of jet height. (c) Information related to pinch-off of droplets. Inset: observations of Liger-Belair and Jeandet [15].

Figure 8

Comparison of information related to pinched-off droplets, generated due to the bursting of solitary and symmetric neighbors.

Figure 9

Entrapment of tiny bubble after merging of ripples at bottom-most point of the collapsing bubble. Insets: Enlarged view of the phase contour in the vicinity of the entrapped bubble. The present case is shown for BBSN, and a similar case is also observed in BSB.

Figure 10

Numerical simulation of an asymmetric pattern (BWAN) of equisized multibubbles to capture bubble cavity, jet formation, and subsequent jet breakup into drops; an equivalent radius $R=900\mu \mathrm{m}$, Bond number $\mathrm{Bo}=0.1725$, Laplace number $\mathrm{La}=4.14\times {10}^4$, and Morton number $\mathrm{Mo}=1\times {10}^{-7}$.

Figure 11

Qualitative comparison of the jet bent after collapsing bubble in the presence of asymmetric neighbors with experimental work of Liger-Belair et al. [17]. It is to be noted that numerical simulation is for an equivalent bubble diameter of 1.8 mm and experimental work is reported for bubbles in the range of 1 mm; $t^{*}=t\sqrt{\frac{\sigma }{\mathrm{\Delta }\mu D}}$.

Figure 12

Liquid jet comparison between (a) (I) and (II) symmetric and (b) (I) and (II) asymmetric pattern multiple bubbles; an equivalent radius $R=900\mu \mathrm{m}$, Bond number $\mathrm{Bo}=0.1725$, Laplace number $\mathrm{La}=4.14\times {10}^4$, and Morton number $\mathrm{Mo}=1\times {10}^{-7}$.

Figure 13

Temporal evolution of bubble cavity, liquid jet, and consequential jet fragmentation into drops for (a) asymmetric and (b) symmetric pattern of multiple bubbles; an equivalent radius $R=900\mu \mathrm{m}$, Bond number $\mathrm{Bo}=0.1725$, Laplace number $\mathrm{La}=4.14\times {10}^4$, and Morton number $\mathrm{Mo}=1\times {10}^{-7}$.

Figure 14

Comparison of details related to pinched-off droplets, generated due to the bursting of symmetric and asymmetric neighbors: (a) Droplet velocity and (b) corresponding pinch-off time. Parameters are similar to Fig. 13.

Figure 15

Temporal evolution of the gap between collapsing waves in the asymmetric pattern of multiple bubbles. The comparison is made with the corresponding symmetric pattern. Parameters for simulations are the same as Fig. 13.

Figure 16

Temporal sequence of line contours obtained in a symmetric (solid line) and asymmetric (dashed line) pattern of bubbles when observed from the interface.

Figure 17

Thematic representation of proposal for reasons behind jet bending in BWAN.

Figure 18

Demonstration of velocity vectors obtained from numerical simulation during rupture of bubble placed in (a) symmetric pattern, (b) asymmetric pattern, (c),(d) closer view in ABCD block, and (e) quantitative comparison at point $P$.

Figure 19

Transient spatial location of the jet tip at equitime interval of $5\mu \mathrm{s}$ for BWAN (as shown in Fig. 10). BBSN shows trajectory along center line given in the figure.

Figure 20

Jet inclination obtained after the bubble bursting for each pattern as mentioned in the figure at the same time instance $(t=2.2\mathrm{ms})$; an equivalent bubble radius $R=900\mu \mathrm{m}$, Bond number $\mathrm{Bo}=0.1725$, Laplace number $\mathrm{La}=4.14\times {10}^4$, and Morton number $\mathrm{Mo}=1\times {10}^{-7}$.

Figure 21

Top view of jet to depict the plane of jet bending for each pattern in a manner conforming to Fig. 17 at the same time instance $(t=3.0\mathrm{ms})$; an equivalent bubble radius $R=900\mu \mathrm{m}$, Bond number $\mathrm{Bo}=0.1725$, Laplace number $\mathrm{La}=4.14\times {10}^4$, and Morton number $\mathrm{Mo}=1\times {10}^{-7}$.

Figure 22

(a) Comparison of jet inclination at a fixed time for different degrees of absenteeism. (b) Temporal evolution in jet inclination for symmetric and asymmetric cases. Simulation parameters are taken the same as Fig. 20.

Figure 23

Jet profile along with bent angle for $\mathrm{BWA}{\mathrm{N}}_3$; association of (a) similar-sized $(\mathrm{B}{\mathrm{o}}_{\mathrm{collapse}}/\mathrm{B}{\mathrm{o}}_{\mathrm{neighbor}}=1)$ and (b) dissimilar-sized $(\mathrm{B}{\mathrm{o}}_{\mathrm{collapse}}/\mathrm{B}{\mathrm{o}}_{\mathrm{neighbor}}=1.366)$, one-sided neighboring wrap; reported time instance is at 2.2 ms. (c) Temporal history of jet bending for $\mathrm{B}{\mathrm{o}}_r=1$ and 1.366.

Figure 24

The effect of Bo (or Mo) on profile of jet bend: (a) $\mathrm{Bo}=0.17,\mathrm{Mo}=1\times {10}^{-7}$; (b) $\mathrm{Bo}=0.017,\mathrm{Mo}=1\times {10}^{-10}$; (c) $\mathrm{Bo}=0.0017,\mathrm{Mo}=1\times {10}^{-13}$. (d) A temporal history of jet bend at different Bo.