Conditions for the sliding-bouncing transition for the interaction of a bubble with an inclined wall

In this study we analyze the interaction of a single rising bubble with an inclined wall. We conduct experiments considering different liquids and bubble sizes, to cover a wide range of Reynolds and Weber numbers, with wall angles from nearly horizontal to nearly vertical. For all cases, the bubble initially collides with the wall; after the initial interaction, in accord with previous studies, the bubble either steadily slides on the wall or ascends, colliding repeatedly with it. Considering a force balance for the bubble motion on the wall, we propose a set of conditions for the transition from sliding to bouncing that is validated with the present and previous data.

Figure 1

A bubble rises in a viscous liquid at terminal conditions $V_{\text{term}}$ and ${\chi }_{\text{term}}$; it then collides with a wall, inclined at an angle $\theta$. After a transient phase, the bubble reaches a new time-average steady velocity $V_w$.

Figure 2

(a) Map of terminal Reynolds ${\text{Re}}_{\text{term}}$ and Weber ${\text{We}}_{\text{term}}$ numbers showing all the experiments conducted in this investigation and (b) terminal Reynolds number ${\text{Re}}_{\text{term}}$ as a function of wall inclination angle $\theta$. The symbols are according to Table 1. In all cases, the closed and open symbols show the experiments in which sliding or bouncing was observed, respectively.

Figure 3

Bubble motion for two typical behaviors: (a) impact and sliding for $\theta ={50}^{\circ }$ and (b) impact and bouncing for $\theta ={60}^{\circ }$. In both cases $D_{\text{eq}}=1.6$ mm, ${\text{Re}}_{\text{term}}=500$, and ${\text{We}}_{\text{term}}=2.3$, corresponding to experiment E7 from Table 1. The image is composed by superposing bubble positions at different instants, $\mathrm{\Delta }t=5$ ms. Note that the image was rotated to make the wall appear horizontal, in both cases. (c) Repeated bouncing behavior for $\theta ={60}^{\circ }$; the experimental conditions are the same as in (b) but further along the plate.

Figure 4

Evolution of the normalized bubble velocity as a function of normalized time $t{D}_{\text{eq}}/V_{\text{term}}$ for the experiments shown as closed and open symbols. The top, middle, and bottom plots show the velocity magnitude $\left|V\right|/V_{\text{term}}$, the perpendicular velocity $V_{\perp }/V_{\text{term}}$, and the parallel velocity $V_{\parallel }/V_{\text{term}}$, respectively.

Figure 5

Map of wall Reynolds ${\text{Re}}_w$ and Weber ${\text{We}}_w$ numbers showing all the experiments conducted in this investigation. The symbols are according to Table 1. In all cases, the closed and open symbols show the experiments in which sliding or bouncing was observed, respectively. The black squares show the data from Tsao and Koch [9]; the asterisks and crosses are results from Takemura and Magnaudet [3] and De Vries et al. [4], respectively.

Figure 6

Images for all bubbles in the sliding motion, at an angle just below the transition. The symbols are according to Table 1. (a) E1, ${\text{Re}}_w=106, {\text{We}}_w=1.07$, and $\theta ={75}^{\circ }$; (b) E10, ${\text{Re}}_w=311, {\text{We}}_w=0.71$, and $\theta ={50}^{\circ }$; (c) E9, ${\text{Re}}_w=522, {\text{We}}_w=1.57$, and $\theta ={40}^{\circ }$; and (d) E11, ${\text{Re}}_w=627, {\text{We}}_w=1.63$, and $\theta ={40}^{\circ }$. Images are shown on the same scale.

Figure 7

Conditions for transition: (a) ${\text{We}}_{\text{wall}}<1.2$ and wall Reynolds ${\text{Re}}_{\text{wall}}$ as a function of $cot\theta$ and (b) ${\text{We}}_{\text{wall}}>1.5$ and wall Weber ${\text{We}}_{\text{wall}}$ as a function of inclination angle $\theta$. The symbols are according to Table 1. In all cases, the closed and open symbols show the experiments in which sliding or bouncing was observed, respectively. The black squares are the data from Tsao and Koch [9]; the asterisks and crosses are results from Takemura and Magnaudet [3] and De Vries et al. [4], respectively. The dashed line in (a) corresponds to Eq. (4). The vertical dash-dotted line in (b) corresponds to $\theta =43.7^{\circ }$.

Figure 8

Visualization of the bubble wake for a sliding bubble, slightly below the transition angle. The image was taken at steady sliding $t{D}_{\text{eq}}/V_{\text{term}}=54$ for E10 in Table 1 for $\theta ={50}^{\circ }$. The colors show the value of the vorticity, normalized by $D_{\text{eq}}/V_{\text{term}}$.