3.1.1. Rebounding bubbles

At $s^{\ast }\lesssim 2.0$ , a rising bubble near the acrylic and polyurethane sponge walls shows periodic or non-periodic rebounding motions (figure 3 a,b). The periodic bouncing motions are nicely fitted with a sinusoidal function (selected examples are plotted in figure 5 a) and the variations of corresponding parameters (amplitude and wavelength) are plotted in figure 5(b,c). The cases of non-periodic bouncing were not plotted. In addition to the visual confirmation, we used the value of $r/r_{eq}$ to determine whether the bubble actually collides with the wall or not; that is, $r/r_{eq}<1.0$ for all the cases shown in figure 5(b,c), and the same criterion has been used by Takemura & Magnaudet (Reference Takemura and Magnaudet2003).

Figure 5.

(a) Measured trace (symbols) of a bubble (selected from figure 2) and corresponding curve fit (lines): ○, ——, acrylic at $s^{\ast }=0.26$ ; ♦, – ⋅ – ⋅ –, polyurethane sponge at 0.26; ▵, - - - -, PTFE at 1.53; ▪, $\cdots \cdots$ , roughened PTFE at 2.04. Variations of the (b) normalized amplitude ( $h^{\ast }$ ) and (c) wavelength ( ${\it\lambda}$ ) with $s^{\ast }$ are plotted for acrylic (○), sponge (♦), PTFE (▵) and roughened PTFE (▪) walls. In (b) and (c), dashed lines denote $h^{\ast }$ and ${\it\lambda}$ of a freely rising bubble (i.e. without the wall) at the same $\mathit{Re}$ and $\mathit{We}$ . Here, superscript $\ast$ denotes normalization by the equivalent bubble radius ( $r_{eq}$ ).

For the acrylic wall (having a conventional no-slip boundary condition at the surface), the rising bubble bounces off the wall repeatedly at $s^{\ast }\leqslant 2.04$ (figure 3 a). Among them, the example ( $s^{\ast }=0.26$ ) of periodic bouncing motion ( $0.26\leqslant s^{\ast }\leqslant 1.53$ ) is shown to be nicely fitted with a sinusoidal function in figure 5(a). Except in the close vicinity of the wall (i.e. $s^{\ast }=0.26$ ), the non-dimensional amplitude ( $h^{\ast }=h/r_{eq}$ ) of the periodic bouncing is maintained roughly the same at $s^{\ast }\leqslant 1.53$ (figure 5 b). On the other hand, the wavelength ( ${\it\lambda}$ ) tends to increase with increasing $s^{\ast }$ (figure 5 c). It is noted that the bubble still bounces on the acrylic wall at $s^{\ast }=2.04$ , but its trajectory oscillates non-periodically.

Figure 3(b) shows the case of arising bubble near a polyurethane sponge wall. Similar to the case of the acrylic wall, when $s^{\ast }\leqslant 0.51$ the rising bubble bounces repeatedly off the wall with an approximately constant $h^{\ast }$ and ${\it\lambda}$ which are smaller than those for the acrylic wall at the same $s^{\ast }$ value (figure 5 b,c). Functionally, the effective boundary condition on a porous wall, such as the present polyurethane sponge, felt by the fluid flow over it, has been modelled as a slip velocity in a tangential direction (Beavers & Joseph Reference Beavers and Joseph1967; Hahn, Je & Choi Reference Hahn, Je and Choi2002; Gangloff, Hwang & Advani Reference Gangloff, Hwang and Advani2014), and has been also validated through a direct simulation using a continuum approach (Breugem, Boersma & Uittenbogaard Reference Breugem, Boersma and Uittenbogaard2006) and experimental measurement (Manes et al. Reference Manes, Poggi and Ridolfi2011). On the other hand, according to Gangloff et al. (Reference Gangloff, Hwang and Advani2014), the tangential slip boundary condition is only valid when the bubble is separated sufficiently from the wall, because seepage (i.e. non-zero wall-normal velocity) into the wall may happen as the bubble approaches the wall. Since the surrounding water is static and the fluid flow over the porous wall is solely induced by the bubble motion in the present set-up, the effective boundary condition on the sponge wall is hypothesized as the combination of tangential slip (when the bubble is away from the wall) and non-zero wall-normal velocity (when the bubble is approaching the wall). Therefore, at the same $s^{\ast }$ value, the smaller amplitude and wavelength of bouncing motion on the polyurethane sponge wall than on the acrylic wall is thought to be caused by the interaction between the non-zero water velocity (induced at the wall) and the bubble wake.

In this regime of $s^{\ast }\;({\lesssim}2.0)$ , considering that the reference bubble trajectory is a two-dimensional zigzag, it is interesting to see that the existence of a vertical wall (with no-slip and porous boundary conditions) does not drastically change the bubble motion, such as the transition to a three-dimensional helical motion; instead, only the amplitude and wavelength of lateral oscillation are affected while elastic rebounding motion occurs. Further discussion on the effects of wall boundary conditions on the bubble motion, in terms of energy components, will be provided in § 3.5.

3.1.2. Sliding bubbles

When the wall has hydrophobicity (figure 3 c), interestingly, the rising bubble slides on the wall, at small $s^{\ast }\;({\leqslant}0.51)$ , while being slanted slightly. Previously, de Vries (Reference de Vries2001) and Zaruba et al. (Reference Zaruba, Lucasa, Prasser and Höhne2007) observed that a spherical rising bubble ( $d_{eq}<1~\text{mm}$ ), smaller than the current one, also slides on the wall after a few bounces with gradually decreasing amplitudes. They attributed the cause of a transition from bouncing to sliding to the weakened inertia compared to the surface tension (de Vries Reference de Vries2001) or the balance between the wall-normal lift and drag forces (Zaruba et al. Reference Zaruba, Lucasa, Prasser and Höhne2007). Although the present bubble is much larger ( ${\sim}3.9~\text{mm}$ ) and bouncing amplitude tends to be maintained approximately constant (i.e. the inertia is not lessened significantly), we were able to make the bubble slide by changing the wall boundary condition, i.e. possibly based on a different mechanism from those previously suggested, which will be discussed below. As $s^{\ast }$ increases, on the other hand, transitions to periodic bouncing ( $1.02\leqslant s^{\ast }\leqslant 2.04$ ) and then to irregular oscillations ( $s^{\ast }\geqslant 2.55$ ) appear in turn. For periodic bounces, $h^{\ast }$ is roughly similar, but ${\it\lambda}$ increases with $s^{\ast }$ (figure 5 b,c).

Figure 3(d) shows the case when the hydrophobicity of the wall is enhanced, i.e. the roughened PTFE wall. When $s^{\ast }$ is small ( ${\leqslant}0.51$ ), the rising bubble also slides on the wall, but the bubble deforms more (see the insets in figure 3 c,d, at $s^{\ast }=0.26$ ; a larger perimeter of the bubble is in contact with the roughened PTFE). Due to the enhanced hydrophobicity, unlike the regular PTFE wall, the transition from bouncing to sliding is observed at a fixed $s^{\ast }~(1.02\leqslant s^{\ast }\leqslant 2.04)$ . Even if the transition to sliding is not captured in the present FoV, the bouncing amplitude is gradually reduced after each collision even at larger $s^{\ast }\;({\geqslant}2.55)$ , and the bubble will eventually slide on the wall outside of the present FoV. For the cases of periodic rebounding on the roughened PTFE wall ( $2.04\leqslant s^{\ast }\leqslant 2.55$ ), on the other hand, $h^{\ast }$ and ${\it\lambda}$ is smaller and larger, respectively, than those on the regular PTFE wall (figure 5 b,c).

As mentioned above, next we discuss the cause of bubble sliding. As the bubble approaches closer to the wall, the liquid layer between the bubble and the wall thins (figure 6). Below the critical thickness of this liquid layer, a three-phase (gas–liquid–solid) contact (TPC) is formed, the dynamics of which is an important phenomenon used in (i) adhesion, spreading of droplets and bubbles on solid substrate (Wasan & Nikolov Reference Wasan and Nikolov2003; Stancik & Fuller Reference Stancik and Fuller2004) and (ii) bubble–particle flotation separation processes (Leja Reference Leja1982; Nguyen & Schulze Reference Nguyen and Schulze2004; Malysa, Kraswska & Krzan Reference Malysa, Kraswska and Krzan2005; Krasowska & Malysa Reference Krasowska and Malysa2007). The temporal rate of change in the thickness ( $h_{l}$ ) of the liquid layer between the (no-slip) soild wall and gas bubble (free surface) has been suggested by Sheludko (Reference Sheludko1967), as

(3.1) $$\begin{eqnarray}\frac{\text{d}(1/h_{l}^{2})}{\text{d}t}=\frac{16}{3{\it\mu}R_{F}^{2}}{\rm\Delta}P,\end{eqnarray}$$

where ${\it\mu}$ is the viscosity of the liquid, $R_{F}$ is the radius of the liquid film, and ${\rm\Delta}P\;(\simeq 2{\it\sigma}/r_{eq})$ is the difference between the pressures in the thin layer and in the bulk region. According to Princen (Reference Princen, Matijevic and Eirich1969), $R_{F}^{2}\simeq (r_{eq}/{\rm\pi}{\it\sigma})F$ , where $r_{eq}$ is the radius of the bubble, ${\it\sigma}$ is the surface tension, and $F$ is the force causing the thinning of the liquid layer (figure 6). As mentioned above, for the formation of TPC and subsequent bubble adhesion to the wall, the intervening liquid layer between the bubble and the wall should reach a critical thickness of a rupture (Leja Reference Leja1982; Nguyen & Schulze Reference Nguyen and Schulze2004). Although it may vary depending on the conditions such as the surface roughness of the wall and liquid viscosity, the critical thickness has been reported as roughly $\mathit{O}({\lesssim}1~{\rm\mu}\text{m})$ (Schulze, Stockelhuber & Wenger Reference Schulze, Stockelhuber and Wenger2001; Krasowska, Zawala & Malysa Reference Krasowska, Zawala and Malysa2009). Once the rupture occurs, the TPC nucleus (i.e. hole in the liquid) is formed and expands until the wetting perimeter of the bubble becomes long enough to make the bubble attach to the wall (Leja Reference Leja1982; Nguyen & Schulze Reference Nguyen and Schulze2004).

Figure 6.

Conceptual schematic to describe the thinning liquid layer between the rising bubble and the wall. Here the driving force ( $F$ ) mainly originates from the circulation by the vortices in the bubble wake (drawn not to scale).

As shown by many studies, on the other hand, a counter-rotating vortex pair is formed in the wake behind a laterally oscillating bubble while rising (de Vries Reference de Vries2001; Ellingsen & Risso Reference Ellingsen and Risso2001; Mougin & Magnaudet Reference Mougin and Magnaudet2002, Reference Mougin and Magnaudet2006; Magnaudet & Mougin Reference Magnaudet and Mougin2007; Yang & Prosperetti Reference Yang and Prosperetti2007; Cano-Lozano et al. Reference Cano-Lozano, Bohorquez and Martínez-Bazán2013). Clearly, the same vortical structure would not exist under the interaction with the wall. As explained above, known as the ground effect, the rebounds and diffusion of a single vortex near the no-slip or slip surface (Barker & Crow Reference Barker and Crow1977; Peace & Riley Reference Peace and Riley1983) and the stabilization of wake vortices behind a sphere in the vicinity of the wall (Tsutsui Reference Tsutsui2008; Lee Reference Lee2012) have been well reported. However, it is reasonable to assume that the force that contributes most to induce the thinning of liquid layer is the wake-induced lift (i.e. wall-normal) force ( $F$ ), which is the main cause of the oscillating (zigzagging in the present study) bubble path (Mougin & Magnaudet Reference Mougin and Magnaudet2006). Since the liquid velocity fields are not available, to calculate the wake-induced force (Mougin & Magnaudet Reference Mougin and Magnaudet2006), we roughly estimate $F$ based on the circulation ( ${\it\Gamma}\simeq 2{\rm\pi}v_{i}r_{eq}$ ) in the wake, where $v_{i}$ is the lateral component of induced liquid velocity in the wake and the equivalent bubble radius ( $r_{eq}$ ) is used as a characteristic length (figure 6). Then, the wall-normal force $F$ is estimated as an inviscid lift force (by the Kutta–Joukowski theorem), $F\simeq {\it\rho}_{l}u{\it\Gamma}r_{eq}=2{\rm\pi}{\it\rho}_{l}ur_{eq}^{2}v_{i}$ , where $u$ is the bubble’s rise velocity. Using these relations, (3.1) is integrated to yield

(3.2) $$\begin{eqnarray}\frac{1}{h_{l}^{2}}\simeq \frac{16}{3}\frac{{\it\sigma}^{2}}{{\it\mu}{\it\rho}_{l}uv_{i}r_{eq}^{4}}t,\end{eqnarray}$$

where $t$ is the duration of bubble–wall collision. Let us first estimate $h_{l}$ for a bouncing bubble on the acrylic wall ( $s^{\ast }=0.26$ ). From the current measurements, the contact duration ( $t$ ) is about 10 ms and the bubble’s rise velocity is $u\simeq 0.27~\text{m}~\text{s}^{-1}$ (see figure 9 a below). For a zigzagging bubble, the maximum induced liquid velocity in the wake is $v_{i}\simeq (0.05{-}0.1)u$ (Ellingsen & Risso Reference Ellingsen and Risso2001). The wake structure would be affected by the interaction with the wall, but we think the order of magnitude estimation based on this value would be valid. As a result, the liquid layer thickness corresponding to bubble–wall collision is estimated to be $h_{l}\simeq 15{-}20~{\rm\mu}\text{m}$ , which is one order of magnitude larger than the critical thickness for rupture. In the case of an acrylic wall, therefore, the bubble–wall collision time is too short to form a TPC, and thus the bubble just bounces off the wall, due to the high pressure in the liquid layer (lubrication theory) (Takemura & Magnaudet Reference Takemura and Magnaudet2003; Sugiyama & Takemura Reference Sugiyama and Takemura2010). Similarly, many previous studies have shown evidence that the most critical parameter in TPC formation is the bubble–wall contact duration, which should be long enough to ensure the expansion of the TPC for stable bubble attachment.

On the other hand, for the roughened PTFE wall, the increase of the bubble–wall collision time ( ${\gtrsim}100~\text{ms}$ ) is clearly shown (figures 3 d, 5 a) and the estimated liquid layer thickness is $h_{l}\lesssim 3~{\rm\mu}\text{m}$ , which is of similar order to the reported critical thickness. It is well known that air pockets are trapped on the rough hydrophobic surface when immersed underwater (Rothstein Reference Rothstein2010; Park, Sun & Kim Reference Park, Sun and Kim2014, to name only a few), and this air layer would encourage the formation of a TPC through the lengthening of the contact time (Krasowska & Malysa Reference Krasowska and Malysa2007; Krasowska et al. Reference Krasowska, Zawala and Malysa2009). Therefore, the sliding motion and deformed bubble shape indicate that the main mechanism of a large bubble sliding on a hydrophobic wall at small $s^{\ast }$ is related to the fast formation of a TPC. Also, it is interesting to see that the submicron-sized air pockets, possibly existing on the regular (not perfectly smooth) Teflon^®, are enough to promote bubble sliding (figure 3 c).

In the context of having a surface slip boundary condition, as stated above, the porous (polyurethane sponge) wall is similar to the current hydrophobic walls on which the effective slip is generated on the interface between water and trapped air pockets. Most numerical approaches have used the same slip boundary condition to model both the porous (Beavers & Joseph Reference Beavers and Joseph1967; Hahn et al. Reference Hahn, Je and Choi2002; Bruneau & Mortazavi Reference Bruneau and Mortazavi2008) and hydrophobic (Min & Kim Reference Min and Kim2004; Martell, Perot & Rothstein Reference Martell, Perot and Rothstein2009; Park, Parn & Kim Reference Park, Parn and Kim2013) walls. However, as shown in figure 3(b), the rising bubble does not slide on the sponge wall. This is because the fully immersed porous wall does not have air pockets physically. Therefore, even with the same slip boundary condition, functionally, the behaviour of a rising bubble shows a different trend.

3.5.1. Typical trends in the variations of energy components for bouncing bubbles

Let us first describe the typical trends of energy components for bouncing bubbles, which are found to be highly correlated with each other. Due to the deforming nature of a high- $\mathit{Re}$ and high- $\mathit{We}$ bubble, the aspect ratio ( ${\it\chi}$ ) of the present bubble fluctuates greatly during its rise (the aspect ratio of the reference bubble is measured to vary within $2.5\pm 0.6$ ) but also shows a trend specific to the bouncing motion; that is, as the bubble approaches the wall, ${\it\chi}$ decreases and then increases as the bubble moves away from the wall after the collision (subfigure iii in figures 12 and 13). As expected from (3.5), the surface energy ( $E_{s}$ ) also follows this trend, especially across a bubble–wall collision, i.e. as the bubble approaches the wall, $E_{s}$ decreases with ${\it\chi}$ (subfigure iv in figures 12 and 13). Although it is clearly observed that $E_{s}$ has a local minimum during the collision, its fluctuating nature makes it difficult to discern any characteristic trend relevant to the bubble motion. After trying many parameters to explain the role of $E_{s}$ in the bubble motion, we find that the ratio of $E_{s}$ to $E_{t}$ (where $E_{t}$ is the sum of $E_{s}$ and $E_{k}$ ) shows a good correlation with the bubble motion. As shown in subfigure (ii) in figures 12 and 13, $E_{s}/E_{t}$ varies periodically according to the rebounding trajectory of the bubble, and $E_{s}$ has a larger contribution to $E_{t}$ than $E_{ku}$ and $E_{kv}$ , indicating that $E_{s}$ is a dominant source for maintaining the present rebounding bubble motion.

Interestingly, $E_{s}/E_{t}$ shows the opposite trend to $E_{s}$ during the bubble–wall collisions. As the bubble moves away from the wall, $E_{s}/E_{t}$ decreases ( $E_{s}$ increases) while ${\it\chi}$ increases, and is kept to a moderate level ( ${\sim}0.5$ ); however, it increases significantly as the bubble approaches the wall while ${\it\chi}$ decreases. After the collision, $E_{s}/E_{t}$ decreases back and this oscillating behaviour is repeated along the bubble’s rise.

On the other hand, $E_{ku}/E_{t}$ (the contribution of the vertical component of bubble kinetic energy to $E_{t}$ ) shows the opposite trend to that of $E_{s}/E_{t}$ (subfigure ii in figures 12 and 13). That is, $E_{ku}/E_{t}$ decreases as the bubble is attracted toward the wall and recovers after the collision, in step with $E_{s}/E_{t}$ . This sharp decrease of rise velocity, i.e. $E_{ku}$ , just before the bubble–wall collision has been also reported by Moctezuma et al. (Reference Moctezuma, Lima-Ochoterena and Zenit2005) and Figueroa-Espinoza et al. (Reference Figueroa-Espinoza, Zenit and Legendre2008). The wall-normal component of $E_{k}~(E_{kv})$ contributes less to $E_{t}$ than the others, but shows a small peak just before the collision.

When the instant of bubble–wall collision is examined closely (see the dashed-line boxes in figures 12 and 13), we can discuss the transport between energy components and the opposite trend of $E_{s}/E_{t}$ to $E_{s}$ . Closer to the collision, both $E_{s}$ and $E_{ku}$ are reduced while $E_{kv}$ and energy dissipation ( $E_{D}$ ) increase (subfigures i, ii and vi in figures 12 and 13). Then, the increase of $E_{s}/E_{t}$ before the collision indicates that the loss of $E_{ku}$ (to $E_{D}$ , for example) is larger than the release of $E_{s}$ . Although it is difficult to quantitatively characterize the transport between energy components, it is thought that some of the deficits of $E_{s}$ and $E_{ku}$ are converted to $E_{kv}$ before the collision and also some are dissipated. Across the collision, $E_{ku}/E_{t}$ and $E_{kv}/E_{t}$ become very small and $E_{s}$ contributes most to $E_{t}$ , indicating that most of $E_{k}(=E_{ku}+E_{kv})$ has dissipated. Just after the impact, most of $E_{kv}$ and $E_{ku}$ is recovered. As the bubble moves away from the wall after the collision, both $E_{s}$ and $E_{ku}$ increase while $E_{D}$ and $E_{kv}$ are reduced. Based on the reduction of $E_{s}/E_{t}$ at this stage, we can say that the recovery of $E_{ku}$ is larger than the increased $E_{s}$ (i.e. the enlarged bubble surface area). When the bubble is shifted furthest from the wall (when $E_{D}$ is smallest), therefore, the deformation-related force (energy) is small while the hydrodynamic forces (wake-induced forces, for example) prevail again, which eventually attracts the bubble toward the wall for rebounding motion. Therefore, it is understood that the accumulation of available (or excess) kinetic energy in the shape deformation (i.e. $E_{s}$ ) and then the transport back to $E_{kv}$ across the collision is responsible for the rebounding motion with a large amplitude, even considering the large energy dissipation during the process (Zaruba et al. Reference Zaruba, Lucasa, Prasser and Höhne2007).

3.5.2. Variation in energy components with wall configuration

We now look into the variation of energy components specific to each wall type. Figure 12(a) shows the result for the rebounding bubble on the acrylic wall at $s^{\ast }=0.26$ . This case fits well the general picture explained above. That is, $E_{k}$ is mostly dominated by the vertical component ( $E_{ku}$ ) except just before the collision when $E_{kv}$ takes up to 40 % of $E_{k}$ (figure 12 a i), which is similar to the suggestion by Figueroa-Espinoza et al. (Reference Figueroa-Espinoza, Zenit and Legendre2008) that the bubble may experience an enhanced bouncing (i.e. larger amplitude) due to the transfer from vertical to horizontal (i.e. wall-normal) momentum, despite the obvious loss of energy during collisions. Note that in the previous section, we approximately described the transport between $E_{ku}$ , $E_{kv}$ , $E_{s}$ and $E_{D}$ . The contribution of $E_{s}$ to $E_{t}$ becomes almost 90 % at the instant of collision but that of $E_{ku}$ recovers to be comparable to that of $E_{s}$ when the bubble is at the maximum distance from the wall (figure 12 a ii). The variations of $E_{s}/E_{k}$ and $E_{k}/E_{t}$ are well correlated with the change in ${\it\chi}$ (in figure 12 a iii) along the bubble trajectory. As the bubble approaches the wall, ${\it\chi}$ decreases while $E_{s}/E_{t}$ and $E_{ku}/E_{t}$ increase and decrease, respectively. The trend is reversed after the collision. $E_{t}$ shows a local minimum during the bubble–wall collision (figure 12 a v), while the wall effect (i.e.  $E_{D}$ ) shows a sharp peak at the instant of collision (figure 12 a vi).

For a porous wall at $s^{\ast }=0.26$ (figure 12 b), general trends of energy components are similar to those for the acrylic wall. However, the rates of change of each energy component are not significant, compared to those on the no-slip wall; the contribution of $E_{kv}$ to $E_{t}~(E_{kv}/E_{t})$ and the reduction of $E_{ku}$ just before the collision become smaller than those on the acrylic wall (figure 12 b i), which indirectly indicates the existence of non-negligible wall-normal velocity (i.e. seepage effect) into the wall (Gangloff et al. Reference Gangloff, Hwang and Advani2014). And this will result in the reduced repulsive force that makes the bouncing amplitude become smaller than others (see figure 5 a). Since the bubble’s momentum is spent less on the bouncing (and also on the bubble’s rolling movement during the collision, see figure 8), estimated $E_{D}$ is smaller than other cases and thus the large $E_{t}$ is achieved (figure 12 b v,vi).

Figures 13(a) and 13(b) show the rebounding and bouncing–sliding transition events on the regular and roughened PTFE walls, respectively. For both walls, the trends of each energy component with bubble trajectory are similar, but the reduction of $E_{ku}$ is measured to be larger than on the acrylic and sponge walls. On the other hand, the role of hydrophobicity is identified more clearly for the roughened PTFE wall. For example, after the first two collisions with a broad peak in $E_{D}$ (note that $E_{D}$ had a relatively sharp peak for the acrylic and sponge walls), the kinetic energy (and also the bouncing amplitude) is reduced gradually and the bubble starts to slide when the available inertia (or $E_{kv}$ ) does not preclude the formation of a TPC (figure 13 b i,ii). When the bubble slides, $E_{s}$ contributes to $E_{t}$ more than 90 % and $E_{D}$ remains very large. As shown in figure 3(d), the rate of shape deformation is very large (figure 13 b iii) for the roughened PTFE wall. As a result, it is understood that the hydrophobicity increases $E_{D}$ on a bouncing bubble, compared to that of the rebounding bubble on the acrylic wall, i.e. at $1.02\leqslant s^{\ast }\leqslant 2.04$ for regular PTFE and at $2.04\leqslant s^{\ast }\leqslant 2.55$ for roughened PTFE (figure 14).

Figure 14.

Variations of $E_{D}/E_{to}$ with $s^{\ast }$ for its (a) average and (b) maximum peak: ○, acrylic; ♦, polyurethane sponge, ▵, PTFE and ▪, roughened PTFE walls.

3.5.3. Wall effect on the energy dissipation

Shown in subfigure (v) in figures 12(a,b) and 13(a,b) are the variations of $E_{t}$ along $x^{\ast }$ . It is calculated that $E_{t}$ is maintained approximately constant during the rise but decreases locally at the collision and thus the wall effect ( $E_{w}$ ) on the energy (i.e. loss, $E_{D}$ ) shows a peak at that time (subfigure vi in figures 12 a,b and 13 a,b). There may be various sources of the energy loss during the bubble’s rise near the wall, such as a viscous dissipation (especially in a thin liquid film between the bubble and the wall), damping due to surface deformation and the diffusion of fluid energy into the secondary vortices generated by the interaction between the bubble and the wall. Contrary to the unsteady liquid droplet motion in air (Nam et al. Reference Nam, Kim and Shin2013), the viscous dissipation due to the motion inside the air bubble is assumed to be relatively small. In the present case, on the other hand, the wall effect ( $E_{w}$ ) is thought to be dominant, because $E_{D}$ decreases remarkably with increasing $s^{\ast }$ . Note also that a sharp change in the bubble orientation (due to the tumbling motion) accompanies the bubble–wall collision (figure 8). As shown in (3.6), detailed velocity field measurements around the bubble and the wall are required to identify the exact causes. However, the current estimation of $E_{D}$ according to various wall configurations would also provide useful insight.

Figures 14(a) and 14(b) show the variations of normalized $E_{D}$ with $s^{\ast }$ , for its average and maximum peak, respectively. The maximum peak occurs during the bubble–wall collision. As shown, the maximum peak and average values of normalized $E_{D}$ show the same trend with $s^{\ast }$ and wall boundary condition, indicating that the trend of $E_{D}$ explained here is not limited to local events during the collision. As $s^{\ast }$ increases, in general, $E_{D}$ tends to decrease, indicating the reduced wall effect on the energy dissipation. According to Figueroa-Espinoza et al. (Reference Figueroa-Espinoza, Zenit and Legendre2008), the wall-induced vorticity decreases as $(s^{\ast })^{3}$ , which might be one of the explanations for the trend in figure 14.

As can be expected from the different bubble rises depending on the wall boundary conditions at $s^{\ast }\lesssim 2.0$ , the estimated $E_{D}$ also shows a discernible deviation at smaller $s^{\ast }$ values. As shown in figure 14, for example, normalized $E_{D}$ on the porous wall is smaller than that on the acrylic wall. Less energy loss seems to be related to the slip boundary condition at the porous wall. That is, the interaction between the bubble wake and the porous wall may produce a weaker secondary vortex than that for the no-slip wall, due to the apparent surface slip. Also, we showed that the slip condition requires less angular momentum to be used for the bubble during the collision (figure 8). Therefore, the wall-induced energy loss on the porous wall is smaller than for the other walls, and this agrees well with the fact that the bubble rises faster over the sponge wall than the other cases (figure 11).

An interesting trend of $E_{D}$ is found on hydrophobic walls. As shown in figure 14, the energy dissipation on the hydrophobic surfaces is larger than that on the no-slip surface at $s^{\ast }\lesssim 2.0$ for regular PTFE and at $s^{\ast }\lesssim 2.5$ for roughened PTFE walls. In these ranges of $s^{\ast }$ , the bubble actually touches the wall for sliding or bouncing. As explained in § 3.1.2, the three-phase-contact (TPC) formation encouraged by the air pockets trapped on the hydrophobic surface will retard the rebounding motion of the bubble, which is evidenced by the elongated bubble–wall contact time (figure 3 d) and larger change rate in the bubble orientation during collision (figure 8). Therefore, even with the same (functionally) slip boundary condition as the porous wall, the existence of air pockets on the surface induces strong energy dissipation when colliding. When the bubble does not touch the hydrophobic surface ( $s^{\ast }>2.0$ for regular PTFE and $s^{\ast }>2.5$ for roughened PTFE walls), conversely, $E_{D}$ becomes smaller than that on the no-slip surface (figure 14). Now the bubble does not touch the wall, and thus the direct contact with the trapped air pockets has little effect on $E_{D}$ while the slip boundary condition is in place. With the same reason for small $E_{D}$ on the porous wall, the hydrophobic surfaces impose smaller $E_{D}$ on the rising bubbles than that imposed by the no-slip wall.

As $s^{\ast }$ increases ( ${\gtrsim}3.0$ ), on the other hand, the difference in $E_{D}$ with wall boundary condition is reduced, and the dominating variable affecting $E_{D}$ seems to be $s^{\ast }$ . In previous sections, we showed that the bubble motion and variation of mean rise velocity are also classified with a similar criterion for $s^{\ast }$ .

3.5.4. Coefficient of restitution

In previous sections, we discussed that the interplay between the energy storage (i.e. transport between surface and kinetic energies) and the energy dissipation determines the detailed motion of a rising bubble near a wall. The same conclusion has been drawn from studies of the dynamics of normal bouncing of solid (Joseph et al. Reference Joseph, Zenit, Hunt and Rosenwinkel2001; Gondret, Lance & Petit Reference Gondret, Lance and Petit2002) and liquid (Legendre, Daniel & Guiraud Reference Legendre, Daniel and Guiraud2005) particles on a horizontal wall. These studies adopted the coefficient of restitution ( ${\it\epsilon}$ ), defined as ${\it\epsilon}=|v_{departure}/v_{approach}|$ , as an indirect measure of energy dissipation resulting from a collision. Here, $v_{departure}$ and $v_{approach}$ denote the wall-normal velocity of the particle, after and before the collision, respectively. For example, ${\it\epsilon}$ of a solid particle scales well as ${\it\epsilon}\sim f(St)$ , where the impact Stokes number (ratio of the particle inertia to the viscous forces) is defined as $\mathit{St}={\it\rho}_{p}d_{eq}v_{approach}/(9{\it\mu})$ based on the particle density ${\it\rho}_{p}$ (Joseph et al. Reference Joseph, Zenit, Hunt and Rosenwinkel2001; Gondret et al. Reference Gondret, Lance and Petit2002). As $\mathit{St}$ increases, ${\it\epsilon}$ increases sharply. On the other hand, for the normal collision of a liquid droplet, Legendre et al. (Reference Legendre, Daniel and Guiraud2005) suggested a correlation of ${\it\epsilon}\sim f(\mathit{St}^{\ast })$ , where the modified Stokes number includes the effect of added mass such that $\mathit{St}^{\ast }=({\it\rho}_{p}+c_{m}{\it\rho})d_{eq}v_{approach}/(9{\it\mu})$ . Here, $c_{m}$ is the added mass coefficient (3.4). Unlike the solid particle, the normal bouncing of a liquid droplet was highly dissipative and thus the added-mass effect should be considered.

Zenit & Legendre (Reference Zenit and Legendre2009) tried to show that the above correlation also works for the normal bouncing of a gas bubble. From a wide range of experiments ( $\mathit{Re}=4{-}540;\mathit{We}=0.02{-}3.52$ ), they found the relationship $-\log \,{\it\epsilon}\sim (\mathit{Ca}/\mathit{St}^{\ast })^{1/2}$ , where $\mathit{Ca}={\it\mu}U/{\it\sigma}$ is the capillary number.

Although the bubble considered herein collides with the vertical wall (i.e. lateral bouncing) (figure 3), it would be interesting to study the bubble–wall collision events in terms of ${\it\epsilon}$ . Figure 15 shows the variation of ${\it\epsilon}$ with modified Stokes number, $\mathit{St}^{\ast }$ , for all rebounding ( ${\it\epsilon}>0$ ) and bouncing–sliding transition ( ${\it\epsilon}=0$ ) events measured for all types of wall. As shown, even the lateral bouncing of the large deformable bubble is well presented in terms of $\mathit{St}^{\ast }$ ; that is, $\log {\it\epsilon}\sim \mathit{St}^{\ast }$ . Here, $\mathit{St}^{\ast }$ is calculated for each event, based on the maximum wall-normal velocity of the bubble approaching the wall. The best fit is given by ${\it\epsilon}=\exp (5\mathit{St}^{\ast })/30$ . Compared to the acrylic wall, ${\it\epsilon}$ (and the corresponding $\mathit{St}^{\ast }$ ) on the sponge wall are larger, indicating that the dominance of inertia over the viscous effect is more pronounced (i.e. less energy loss). On the other hand, ${\it\epsilon}$ on the hydrophobic walls is smaller than on the acrylic wall and the corresponding $\mathit{St}^{\ast }$ is small to represent the increased viscous effect (i.e. more energy loss). Clearly, this trend is closely associated with the analysis of transports between energy components in previous sections.

Figure 15.

Variation of the coefficient of restitution ( ${\it\epsilon}$ ) with the modified impact Stokes number ( $\mathit{St}^{\ast }$ ) for the rebounding and bouncing–sliding transition events: ○, acrylic; ♦, polyurethane sponge; ▵, PTFE and ▪, roughened PTFE walls.

In the present analysis, we find that the correlation between ${\it\epsilon}$ and $\mathit{Ca}/\mathit{St}^{\ast }$ , as suggested by Zenit & Legendre (Reference Zenit and Legendre2009), is not clear as that with $\mathit{St}^{\ast }$ . This may due to the difference between normal and lateral bouncing or the fact that the present bubble ( $d_{eq}\simeq 3.9~\text{mm}$ ) is larger than the capillary length of an air–water system, ${\it\lambda}_{c}=\sqrt{{\it\sigma}/{\it\rho}g}\simeq 2.7~\text{mm}$ . In Zenit & Legendre (Reference Zenit and Legendre2009), the size of the bubble considered was smaller than or equivalent to ${\it\lambda}_{c}$ .