Sliding motion of a bubble against an inclined wall from moderate to high bubble Reynolds number

The motion of a bubble sliding over an inclined wall from moderate to high bubble Reynolds number is studied experimentally for a wide range of liquid properties and bubbles sizes, considering wall inclination angles from nearly horizontal to nearly vertical. All experiments are restricted to sliding behavior, below the transition to steady bouncing motion. We study both the shape of the bubble and its drag coefficient. For small angles, the bubble shape is dominated by gravitational effects resulting in a flattened shape against the wall; for large angles, the bubble remains in constant contact with the wall but adopts a shape that is aligned perpendicularly to the wall, closer to that observed for an inertia-dominated free rising bubble. We model this transition of shape considering balances among surface tension, gravitational, and inertial forces; we observe good agreement with experiments. We found that the drag coefficient is strongly influenced by the shape that the bubble adopts as it slides over the wall. By considering the flow in the film and around the bubble, we propose a correlation to predict the drag coefficient for each regime of bubble shape. In the regime dominated by viscous effects, the drag of a single bubble is increased due to the mirror effect with the wall and by the friction in the film formed between the wall; conversely, for the case dominated by inertial effects, the drag coefficient is constant. The behavior for a single bubble is changed: no significant increase due to deformation. In both shape regimes the proposed expression agrees well with the experimental measurements.

Figure 1

Sketch of bubble terminal and wall conditions. Away from the wall, the bubble ascends at $V_{\mathrm{term}}$, with a constant aspect ratio $d_{\mathrm{ma}}/d_{\mathrm{mi}}$, where $d_{\mathrm{ma}}$ and $d_{\mathrm{mi}}$ measure the major and minor axis of the bubble. While sliding on the wall, the bubble moves at a constant speed, $V_w$, with an aspect ratio $d_{\parallel }/d_{\perp }$, where $d_{\parallel }$ and $d_{\perp }$ measure the bubble length and width with respect to the wall.

Figure 2

Aspect ratio, $\chi$, as a function of terminal Weber number, for all the experiments. Each bubble-fluid combination is represented by a different symbol, according to Table 1. The continuous and dashed lines correspond to Eq. (6) plotted without the Morton term and for $\text{Mo}=2.1\times {10}^{-11}$, respectively.

Figure 3

Drag coefficient, $C_D$, as a function of terminal Reynolds number, ${\text{Re}}_{\mathrm{term}}$, for all the experiments in Table 1. The continuous black line shows the prediction of Eq. (9). The other continuous lines show the predictions of Eq. (10) for four cases: blue line, for $\text{Mo}=5.32\times {10}^{-9}$ (E1, E2, E3); cyan line, for $\text{Mo}=1.15\times {10}^{-9}$(E5); green line, for $\text{Mo}=5.03\times {10}^{-11}$(E7, E9); and red line, for $\text{Mo}=2.18\times {10}^{-11}$ (E10, E11). The colors correspond to the nomenclature in Table 1. The dashed black line shows the drag calculated according to Eq. (11).

Figure 4

Images for the sliding bubble shape evolution at different inclination angles in experimental conditions E3 (see Table 1): (a) $V_w=40.1$ mm/s, ${\text{Re}}_w=59$, ${\text{We}}_w=0.04$; (b) $V_w=80.9$ mm/s, ${\text{Re}}_w=119$, ${\text{We}}_w=0.16$; (c) $V_w=115.0$ mm/s, ${\text{Re}}_w=169$, ${\text{We}}_w=0.33$; (d) $V_w=138.8$ mm/s, ${\text{Re}}_w=204$, ${\text{We}}_w=0.48$; (e) $V_w=155.2$ mm/s, ${\text{Re}}_w=228$, ${\text{We}}_w=0.60$. Images are shown with the same scale.

Figure 5

Dependence of wall aspect ratio for gravitational and inertial effects for all the experiments. (a) Wall aspect ratio, ${\chi }_w$, as a function of $\text{Bo}cos\theta$. (b) The wall aspect ratio, ${\chi }_w$, as a function of the wall Weber number, ${\text{We}}_w$. The symbols correspond to Table 1: open and solid symbols represent experiments for ${\chi }_w>1$ and ${\chi }_w<1$, respectively.

Figure 6

Comparison between the experimental value of ${\chi }_w$ and the prediction of Eq. (15), considering $\alpha =0.1$ and $\beta =3/32$, for all experiments in Table 1: open and solid symbols represent experiments for ${\chi }_w>1$ and ${\chi }_w<1$, respectively. The dashed line shows the perfect correlation.

Figure 7

Sliding bubble velocities, $V_w$, as a function of $sin\theta$, where $\theta$ is the inclination of the wall, for all the experimental conditions of Table 1. The black + symbols show the results of Dubois et al. [17] for water ($D_{\mathrm{eq}}=2.0\mathrm{mm}$). The dashed line shows the trend of the experiments by Tsao and Koch [10] ($D_{\mathrm{eq}}=1.6\mathrm{mm}$), also for water.

Figure 8

Wall drag coefficient ${C_D}_w$, as a function of wall Reynolds number, ${\text{Re}}_w$, for all the experimental conditions of Table 1. The solid and open symbols show the experiments for ${\chi }_w>1$ and ${\chi }_w<1$, respectively. The black continuous and dashed lines show the prediction of Eqs. (17) and (20), respectively.

Figure 9

Comparison between the measured wall coefficient and the prediction of Eq. (25). The continuous and dashed lines are the predictions for minimum and maximum values of ${\text{Ca}}_w$, respectively; the black lines show the prediction for $\theta =5^{\circ }$; the red lines show the prediction for $\theta ={74}^{\circ }$, corresponding to the minimum and maximum angles of the entire set. (a) $\text{Bo}<1$, $n=1/2$, experiments E4, E6, E7, E8, and E10; (b) $\text{Bo}=0.58$, $n=1/2$, experiment E1; and (c) $\text{Bo}>1$, $n=1/4$, experiments E5, E9, and E11. In all cases the blue dash-dotted line shows the prediction of Eq. (25) without the film drag ($K_o=0.0$). The magenta lines show the prediction from the model of Dubois et al. [17] for the corresponding values of the capillary numbers.

Figure 10

Comparison between the experimental drag coefficient and the predictions from Eq. (25). The line shows a perfect correlation.

Figure 11

Wall drag coefficient, ${C_D}_w$, as a function of wall Reynolds number, ${\text{Re}}_w$, for all the experiments from Table 1 for which ${\chi }_w<1$. The dash-dotted line shows ${C_D}_w=0.7$. For comparison, the dashed line shows the prediction from Eq. (20). The solid and dashed magenta lines show the predictions from Eq. (26) from Dubois et al. [17], for two different values of the capillary number.