3.1.1 Dimensionless cavity depth

The cavity depth $Z_{c}$ is defined as the depth of the base of the bubble from the free, undisturbed, liquid surface (figure 3 a). A theoretical expression of $Z_{c}$ can easily be obtained when neglecting bubble deformation. To leading order,

where $h_{c}$ is the height of the top of the bubble from the free surface (see figure 3 a). Assuming symmetry at the point of inflection of the bubble surface at the rim gives,

where $\unicode[STIX]{x1D702}=h_{c}-h_{r}$ is the height of the bubble cap above the rim, with $h_{r}$ being the height of the rim from the free surface. Figure 3(b) shows $h_{c}/h_{r}=h_{c}/\unicode[STIX]{x1D702}$ , plotted against $Bo$ in the range $10^{-1}<Bo<3$ . The measured values are close to $h_{c}/h_{r}=~h_{c}/\unicode[STIX]{x1D702}=2$ , hence the validity of the assumptions leading to (3.3) is supported by experiments. The relatively large experimental error is due to the very small values of $h_{r}$ and $h_{c}$ .

Figure 3.

(a) Schematic of the static bubble shape; (b) $h_{c}/h_{r}$ as a function of $Bo$ . ▵, Water; ▹, GW48 ( $20\,^{\circ }\text{C}$ ); ▴, GW48 ( $30\,^{\circ }\text{C}$ ); *, GW68; ♢, GW72; ◃, ethanol; $+$ , 2-propanol; —, $h_{c}/h_{r}=2$ .

For a spherical bubble, from geometry,

where $R_{r}$ is the radius of the rim (figure 3 a). From the force balance, $F_{B}=F_{\unicode[STIX]{x1D70E}}$ , where the buoyancy force, $F_{B}=\unicode[STIX]{x1D70C}g(4/3)\unicode[STIX]{x03C0}R^{3}(1-\unicode[STIX]{x1D702}^{2}(3-\unicode[STIX]{x1D702}/R)/4R^{2})$ and the surface tension force, $F_{\unicode[STIX]{x1D70E}}=(2\unicode[STIX]{x1D70E}/R)\unicode[STIX]{x03C0}R_{r}^{2}$ , we get

When $Bo\leqslant 1$ , $\unicode[STIX]{x1D702}/R\leqslant 0.4$ , so the term $\unicode[STIX]{x1D702}^{2}(3-\unicode[STIX]{x1D702}/R)/4R^{2}\leqslant 0.1$ , which can be neglected to first order, resulting in,

which when substituted in (3.4) gives,

Using (3.7) in (3.3), we get from (3.2),

Figure 4 shows the square of the experimental $(Z_{ce}/R)^{2}$ and the square of the theoretical dimensionless cavity depths $(Z_{c}/R)^{2}$ , calculated from (3.8), as a function of $Bo$ . Note that we plot $(Z_{c}/R)^{2}$ rather than $Z_{c}/R$ because the Weber number also has the square of velocity. It is seen that in the range $0.1<Bo<1$ , the experimental $(Z_{ce}/R)$ can be fitted by $(Z_{ce}/R)^{2}\sim Bo^{-1/2}$ , which is the same approximate dependence of $We$ on $Bo$ seen in figure 2. In figure 2, when $Bo$ is large, $Bo>1$ , the jet velocity starts to decrease and at $Bo=2.25$ , $R=4.08~\text{mm}$ in water, $We$ deviates considerably from the $Bo^{-1/2}$ correlation. The experimental $(Z_{ce}/R)^{2}$ in figure 4 shows a similar deviation from the approximate power law $Bo^{-1/2}$ . The theoretical $(Z_{c}/R)^{2}$ has a steeper fall off with $Bo$ than the experimental $(Z_{ce}/R)^{2}$ because when $Bo>1$ , the limit of validity of (3.8) is approached. When $Bo\rightarrow 0$ the theoretical cavity depth (3.8) tends to the asymptotic limit of $2R$ and is practically independent of $Bo$ when $Bo<0.1$ because for $Bo=0.1$ , $Z_{c}/R=1.93$ and for $Bo=0.01$ , $Z_{c}/R=1.99$ , which is only a 3 % variation. We can therefore assume that when $Bo<0.1$ , $Z_{c}/R$ is nearly invariant. On the other hand, between $Bo=0.1$ ( $Z_{c}/R=1.93$ ) and $Bo=1$ ( $Z_{c}/R=1.15$ ) gravity has a large effect on the cavity depth.

Figure 4.

Square of the dimensionless cavity depth of the bubble as a function of the Bond number. ▵, Water; ▹, GW48 ( $20\,^{\circ }\text{C}$ ); ▴, GW48 ( $30\,^{\circ }\text{C}$ ); *, GW68; ♢, GW72; ◃, ethanol; $+$ , 2-propanol; – –,  $1.32Bo^{-1/2}$ ; –.–,  $(Z_{c}/R)^{2}=4~(1-2/3~Bo)$ ; —,  $(Z_{cd}/R)^{2}=4(\sqrt{1-2/3Bo}-0.17~Bo^{0.8})^{2}$ .

The bubble deformation is negligible up to $Bo\approx 0.1$ and remains small up to $Bo\approx 3/2$ , with the deformation varying from approximately 4 % to 15 % of $R$ as $Bo$ increases from $0.1$ to $1$ . Although these deformations are relatively small, this seems to affect the cavity depth sufficiently when $Bo<1$ to make the theoretical depths (3.8) deviate noticeably from the experiments, as can be seen in figure 4 when $Bo<1$ . $Z_{c}/R$ given by (3.8) can be corrected for these small deformations by assuming an oblate spheroid shape of the bubble. The expression for such a corrected cavity depth is $Z_{cd}/R\approx 2((R/R_{m})^{2}-1+\sqrt{1-(2/3)~Bo})$ , where $R_{m}$ is the measured horizontal radius at the equator, approximated by $(R/R_{m})^{2}~\approx 1-0.17~Bo^{0.8}$ , to get,

As shown by the continuous line in figure 4, we obtain a better match of $(Z_{cd}/R)^{2}$ versus $Bo$ obtained from (3.9) with the experimental variation of $(Z_{ce}/R)^{2}$ versus $Bo$ , compared to that obtained by (3.8).

3.1.2 Cavity depth model for gravity effects

Gravity effects can be best demonstrated with data from one fluid alone rather than data from different viscosity fluids as in figure 2. Figure 5 shows the experimental jet velocities for water plotted in terms of $We$ versus $Bo$ , along with the data from BSB, the drop velocities measured by Spiel (Reference Spiel1995), the jet velocity measured from the images of Kientzler et al. (Reference Kientzler, Arons, Blanchard and Woodcock1954) and the jet velocity data of Ghabache et al. (Reference Ghabache, Antkowiak, Josserand and Séon2014). The continuous line is

while the dashed line is (3.1). The following three regimes can be identified in figure 5.

1. (i) $Bo<0.1$ : at these low $Bo$ numbers the theoretical cavity depth varies only by 3 % from $Bo=0$ to 0.1 (see figure 4) and according to the cavity depth model (3.10), $We$ should also asymptote to a constant value, as shown by the solid line in figure 5. Our data and those of BSB tend to asymptote toward such a constant value of $We$ rather than to increasing $We$ with decreasing $Bo$ , following a $Bo^{-1/2}$ law, given by Ghabache et al. The deviation of our data from the cavity depth model (3.10), seen as a slight increase in $We$ with decreasing $Bo$ , for $Bo<0.1$ is due to capillary wave damping. As we discuss in § 3.2.1, (figure 9), if the Weber number is corrected for this damping, it is practically a constant for $Bo<0.1$ .

2. (ii) $Bo>1$ : in this range there is no doubt a clear deviation of our $We$ data from the $Bo^{-1/2}$ trend. This is because the cavity depth decreases more rapidly with $Bo$ than $Bo^{-1/2}$ and there is also larger bubble deformation for $Bo>1$ .

3. (iii) Intermediate range $0.1<Bo<1$ : in this range, the data of Ghabache et al. are well fitted by $Bo^{-1/2}$ whereas the present results and those of BSB deviate noticeably from this power law and closely follow the cavity depth model (3.10). For $0.06<Bo<0.15$ Spiel’s results follow $We\sim Bo^{-1/2}$ but deviate from this power law when $Bo$ increases. Spiel measured the first drop velocities and not the jet tip velocities when the jet emerges from the free surface; the values of the drop velocities are different from the unbroken jet tip velocities. Figure 6 shows the measured jet velocities at increasing heights as time increases, culminating in the first drop velocity due to jet fragmentation at some height. We see that at moderate $Bo$ ( $0.2\leqslant Bo\leqslant 0.63$ ) there is a 30 %–60 % reduction in drop velocity compared to the jet velocity at the free surface. This reduction in drop velocities increases with increasing $Bo$ since jets from larger $Bo$ bubbles fragment farther away from the free surface (figure 15). This decrease in drop velocities at moderate $Bo$ is the reason why Spiel’s data are lower than the solid curve from the cavity model in figure 5.

Figure 5.

Variation of jet Weber number with Bond number in water. ▵, Present experiments; $\star$ , BSB; ♢, drop velocities of Spiel (Reference Spiel1995);  $\blacksquare$ , Kientzler et al. (Reference Kientzler, Arons, Blanchard and Woodcock1954); —,  $We=62.5~(Z_{cd}/R)^{2}$ ; – –,  $We=55~Bo^{-1/2}$ ; $\times$ , jet velocities of Ghabache et al. (Reference Ghabache, Antkowiak, Josserand and Séon2014) for water.

Figure 6.

Variation of the jet $We$ with the dimensionless time for bubbles of different $Bo$ in water. ♢,  $Bo=0.2$ ; ▫,  $Bo=0.3$ ; ○,  $Bo=0.49$ ; ▵,  $Bo=0.63$ . Unfragmented jet velocities are shown by hollow symbols while the velocities of the first drop after jet fragmentation are shown by filled symbols. The cubic polynomial fits, show the progression of velocities with time of each jet.

Based on the above considerations, it can be concluded that the present experimental data for water and the BSB data, when considered over the whole range of $Bo$ , show better agreement with (3.10) than with (3.1). This agreement implies that the jet velocity scales as $U_{j}\sim (Z_{cd}/R)~U_{c}$ , the gravity effects in jet velocity comes purely from the gravity effects in the cavity depth. This scaling can also be demonstrated by measuring the jet velocity from the vertical retraction of cavity, as shown in appendix A. It is shown in appendix A that the time taken by the jet to travel the distance of cavity depth is independent of gravity effects, implying that the gravity effects in jet velocity comes only from the gravity dependence of static cavity depths. As we show later in § 3.2, this conclusion is further strengthened when the experimental $We$ , compensated for capillary wave damping is plotted verses $Bo$ (see figure 9).

It could be argued that the dynamic cavity depth when the singular collapse commences (figure 1 m) is of importance and not the static depth just after the surface film disintegration (figure 1 c). An estimate of the change in cavity depth during the time of bubble collapse $t_{bc}\approx 0.3~t_{c}$ (Krishnan & Puthenveettil Reference Krishnan and Puthenveettil2015), where the capillary time $t_{c}=R\sqrt{\unicode[STIX]{x1D70C}R/\unicode[STIX]{x1D70E}}$ , is obtained by evaluating the upward bubble displacement $\unicode[STIX]{x0394}z=gt_{bc}^{2}/2$ to get,

which is negligible when $Bo<1$ . Any decrease in cavity depth would hence have to be due to capillary forces caused by the curvature of the bubble base; our experiments show that this is small for small bubble sizes.

Figure 7.

Variation of the dimensionless jet velocity with the dimensionless bubble radius for low $Bo$ . The velocities and radii are normalised by the viscous capillary scales: ▵, water; ▴, GW48 ( $30\,^{\circ }\text{C}$ ); ▹, GW48 ( $20\,^{\circ }\text{C}$ ); ▫, GW55; *, GW68; ♢, GW72; ○, jet velocities of Duchemin et al. (Reference Duchemin, Popinet, Josserand and Zaleski2002) for water; $\times$ , jet velocities of Ghabache et al. (Reference Ghabache, Antkowiak, Josserand and Séon2014) for water; —,  $16(R/R_{\unicode[STIX]{x1D707}})^{-1/2}$ ; –.–, the viscous cutoff at $R/R_{\unicode[STIX]{x1D707}}=Oh^{-2}\simeq 730$ ; – –, the vertical dashed lines demarcate the $Bo<0.1$ data on the left with the $Bo>0.1$ data on the right for water (W), GW48 and GW55. Viscous–capillary scaling is seen for $Bo<0.1$ part of each data set.

The proposed scaling law for gravitational effects on jet velocity, namely $We\sim (Z_{cd}/R)^{2}$ , where $Z_{cd}/R$ is given by (3.9), implies that the jet $We$ becomes practically independent of $Bo$ at $Bo<0.1$ . Hence at these low Bond numbers the viscous–capillary scaling of Duchemin et al. (Reference Duchemin, Popinet, Josserand and Zaleski2002) should be appropriate. As shown in figure 7, the data closely follow the relation

i.e. $Ca=16~Oh$ for $Bo<0.1$ where the capillary number $Ca=\unicode[STIX]{x1D707}U_{j}/\unicode[STIX]{x1D70E}$ . This power-law variation of jet velocity gives $U_{j}=16~U_{c}$ or $We=(U_{j}/U_{c})^{2}\simeq ~250$ as seen in figure 5 for $Bo<0.1$ . The data of Ghabache et al. (Reference Ghabache, Antkowiak, Josserand and Séon2014) however show a $We\sim Bo^{-1/2}$ scaling for their whole range of $Bo$ , $0.007<Bo<1$ . The relation (3.12) implies that $U_{j}\sim 1/\sqrt{R}$ (Duchemin et al. Reference Duchemin, Popinet, Josserand and Zaleski2002). Curiously their data deviate from the $R^{-1/2}$ scaling when $R/R_{\unicode[STIX]{x1D707}}<5\times 10^{3}$ , which is not observed in the present experiments. The data of Ghabache et al. (Reference Ghabache, Antkowiak, Josserand and Séon2014) show a $We\sim Bo^{-1/2}$ scaling for their whole range of $Bo$ , $0.007<Bo<1$ .

3.2.1 Capillary wave damping

Figure 8 shows the damping of capillary waves on the collapsing cavity surface with increase in $Oh$ . Figure 8(a,b) is a bubble collapse sequence of a moderate $Bo$ water bubble ( $Bo=0.63$ ) with $Oh=0.0026$ . Here, in the wave train preceding the kink, two wavelengths $\unicode[STIX]{x1D706}_{1}\approx 0.36R$ and $\unicode[STIX]{x1D706}_{2}\approx 0.17R$ can be clearly identified, with faster, shorter waves being practically damped. The wave train group velocity is $C_{g}=3/2\sqrt{\unicode[STIX]{x1D70E}~k/\unicode[STIX]{x1D70C}}\simeq 3/2\sqrt{2\unicode[STIX]{x03C0}\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70C}\unicode[STIX]{x1D706}_{1}}\approx 6U_{c}$ which corresponds to the measured speed of the kink. Figure 8(c,d) is a collapsing sequence of a smaller water bubble ( $Bo=3\times 10^{-2}$ ) with a relatively larger $Oh=0.0055$ which shows only wavelength $\unicode[STIX]{x1D706}_{1}$ clearly; the shortest wave ( $\unicode[STIX]{x1D706}_{2}$ ) is nearly damped. Figure 8(e,f) shows the capillary waves in GW48 ( $30\,^{\circ }\text{C}$ ) bubble with $Oh=0.0139$ in which the longer wave ( $\unicode[STIX]{x1D706}_{1}$ ) alone moves ahead of the kink, with the amplitude noticeably decreased. As can be seen in figure 8(g,h), a further increase in $Oh$ from $0.0139$ to $0.0225$ results in complete damping of the capillary waves on the cavity surface.

The amplitude $\unicode[STIX]{x1D6FC}$ of capillary waves falls off exponentially in the form $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FC}_{0}\text{e}^{-\unicode[STIX]{x1D705}t}$ with the damping rate $\unicode[STIX]{x1D705}=8\unicode[STIX]{x03C0}^{2}\unicode[STIX]{x1D707}/\unicode[STIX]{x1D70C}\unicode[STIX]{x1D706}^{2}$ . In the collapse time $t_{bc}\approx 0.3t_{c}$ (Krishnan & Puthenveettil Reference Krishnan and Puthenveettil2015), the decrease in capillary wave amplitude is then given by

Figure 8.

Progressive damping of capillary waves with increase in $Oh$ , shown in image pairs from (a) to (h). Collapsing sequence, (a,b) water, $Oh=0.0026$ ( $Bo=0.63$ ); (c,d) water, $Oh=0.0055$ ( $Bo=3\times 10^{-2}$ ); (e,f) GW48 ( $30\,^{\circ }\text{C}$ ),  $Oh=0.0139$ ( $Bo=0.17$ ); (g,h) GW55, $Oh=0.0225$ ( $Bo=0.47$ ). In each image pair, images are separated by $741$ , $100$ , $148$ and 375 ms, respectively. The lines in (a,b) 1 mm, (c,d) 0.2 mm, (e,f) 0.5 mm, (g,h) 0.5 mm.

Capillary waves can be considered completely absent when $\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D6FC}_{0}=\text{e}^{-n}$ with $n\approx 4$ . Equation (3.13) with $n=4$ then implies that capillary waves with wavelength less than $\unicode[STIX]{x1D706}/R\approx 0.17$ are absent at a value of $Oh\approx 0.0048$ (see figure 8 a–d). Similarly, capillary waves of wavelength $\unicode[STIX]{x1D706}/R\approx 0.36$ will be absent at $Oh\approx 0.022$ (see figure 8 e–h). We can hence infer that capillary waves are progressively damped as $Oh$ increases and there is an increase in jet velocity up to about $Oh\approx 0.02$ due to this damping.

For given fluid properties, $Oh$ increases with decreasing bubble radius, resulting in a corresponding decrease in $Bo$ . We saw in figure 5 that with decreasing $Bo$ when $Bo<0.1$ , the jet $We$ increases, deviating from the cavity model (3.10) and this is due to increasing capillary wave damping. If we assume that the radius at the bottom of the cavity increases by a factor $R/\unicode[STIX]{x1D706}$ , given by (3.13), due to the presence of capillary waves, then $We$ has to be compensated by $\sqrt{Oh}$ to account for capillary wave damping. As shown in figure 9, $We/Oh^{1/2}$ is practically a constant for the experimental data for $Bo<0.1$ and matches well with the cavity depth model. The $We$ of Spiel is also practically independent of $Bo$ when $Bo<0.06$ in figure 9. Even Ghabache’s data could be considered to be following our cavity depth model in figure 9 for $Bo<0.1$ . Unfortunately, Ghabache et al. (Reference Ghabache, Antkowiak, Josserand and Séon2014) have no measurements at smaller $Bo$ , say at a $Bo$ of $0.003$ for a clearer verification.

Figure 9.

Variation of the Weber number compensated for capillary wave damping with $Bo$ . Symbols are the same as in figure 5. – –,  $WeOh^{-1/2}=830Bo^{-1/2}$ ; —,  $WeOh^{-1/2}=900(Z_{cd}/R)^{2}$ .

Figure 10.

Normalised $We$ , uncompensated and compensated for capillary wave damping, $We/(Z_{cd}/R)^{2}\times 10^{3}$ and $We/((Z_{cd}/R)^{2}Oh^{1/2})$ , as functions of $Oh$ . ▵, Water; ▴, GW48 ( $30\,^{\circ }\text{C}$ ); ▹, GW48 ( $20\,^{\circ }\text{C}$ ); ▫, GW55; *, GW68; ♢, GW72; $\times$ , water data of Ghabache et al. (Reference Ghabache, Antkowiak, Josserand and Séon2014); – –,  $We/(Z_{cd}/R)^{2}=70$ ; –.–,  $We/((Z_{cd}/R)^{2}Oh^{1/2})=900$ ; —,  $We/((Z_{cd}/R)^{2}Oh^{1/2})=8.3\times 10^{-9}Oh^{-7.3}-0.17$ ; $\ldots \,$ , the vertical dotted line represents $Oh_{c}=0.037$ .

In figure 10, we now plot $We/(Z_{cd}/R)^{2}$ as a function of Ohnesorge number, $Oh$ . There is a small region of decreasing $We/(Z_{cd}/R)^{2}$ with increasing $Oh$ at $Oh<0.003$ , which is an artefact of the deviation of the theoretical cavity depth (3.9) from the experimental values at large $Bo$ (see figure 4). When $Oh<Oh_{c}\simeq 0.037$ , except in the range of $0.02<Oh<Oh_{c}$ , the data sets of water and GW48 indicate an increase of $We/(Z_{cd}/R)^{2}$ with $Oh$ in figure 10. As mentioned above, this increase of $We/(Z_{cd}/R)^{2}$ with $Oh$ occurs because capillary waves are more and more damped as $Oh$ increases, leading to a smoother cavity and hence a stronger pressure impulse. Hence we plot the compensated $We$ , namely $We/((Z_{cd}/R)^{2}Oh^{1/2})$ , in figure 10 as a function of $Oh$ . This rescaling collapse the present water and GW48 data reasonably well to a nearly constant value of compensated $We$ . The increase in compensated $We$ with decrease in $Oh$ at $Oh<0.003$ (large $Bo$ ) still persists, being an outcome of (3.9) being valid only till $Bo\approx 1$ .

3.2.2 Direct viscous damping of jet

It is seen that when $Oh>Oh_{c}\simeq 0.037$ there is a rapid decrease in jet velocity. This is because when $Oh>Oh_{c}$ the Reynolds number on the bubble scale falls below $10^{2}$ , viscous effects become important on the bubble scale and the jet velocity decreases rapidly with increasing $Oh$ . Note that $Oh_{c}=0.037$ corresponds to $R/R_{\unicode[STIX]{x1D707}}=Oh^{-2}\simeq 730$ in figure 7, below which the jet velocity drops off rapidly in agreement with the numerical simulations of Duchemin et al. (Reference Duchemin, Popinet, Josserand and Zaleski2002). Corresponding to $Oh_{c}$ , we can estimate a critical Bond number $Bo_{c}=\unicode[STIX]{x1D707}^{4}g/(0.037)^{4}\unicode[STIX]{x1D70E}^{3}\unicode[STIX]{x1D70C}$ , which is $1.5\times 10^{-5}$ in water, beyond the range of our experiments. For higher viscosity fluids like GW68 and GW72, $Bo_{c}=0.3$ and $Bo_{c}=1.28$ respectively, which can be seen in figure 2, where a rapid drop of $We$ occurs for $Bo<Bo_{c}$ . In figure 10, we have empirically fitted this viscous regime by $We/((Z_{cd}/R)^{2}Oh^{1/2})=8.3\times 10^{-9}Oh^{-7.3}-0.17$ . This critical value of $Oh$ above which the jet velocity decreases rapidly corresponds to the value proposed by Walls, Henaux & Bird (Reference Walls, Henaux and Bird2015) as a critical value beyond which the jet does not breakup into drops. Figure 11 shows images of jets for different $Oh$ values. It is seen that for conditions of figure 11(d), $Oh=0.05>Oh_{c}$ , for instance, there is no breakup of the jet into drops (figure 16 d), whereas breakup occurs for lower values of $Oh$ (figure 11 a,b). In figure 11(c) there is no breakup either, even though $Oh<Oh_{c}$ (see figure 15(a) for full sequence of jet evolution). This is because the corresponding $Bo$ is large, as discussed below in § 3.2.3. Furthermore, we find that no jet emerges when $Oh=Oh^{\ast }\simeq 0.1$ , a value larger than the value of $Oh^{\ast }=0.052$ proposed by San Lee et al. (Reference San Lee, Weon, Park, Je, Fezzaa and Lee2011); the difference could possibly arise from the small liquid layer depth (of the order of $R$ ) in their experiments.

Figure 11.

Images of jets for different combinations of $Bo$ and $Oh$ to show the effect of gravity and viscosity: (a) and (b) are at approximately the same $Bo$ , but (b) has a much larger $Oh$ , still less than $Oh=0.02$ ; (a) and (c) are at the same $Oh<0.02$ , but (c) has a much larger $Bo$ ; (a) and (d) have approximately the same $Bo$ , but (d) has a much larger $Oh>Oh_{c}=0.037$ . (a) Bubble of $R=2.15$  mm in water, $Bo=0.629$ , $Oh=2.55\times 10^{-3}$ ; (b) $R=2$  mm in GW48 ( $30\,^{\circ }\text{C}$ ),  $Bo=0.646$ , $Oh=10\times 10^{-3}$ ; (c) $R=4.08$  mm in water, $Bo=2.25$ , $Oh=1.85\times 10^{-3}$ ; (d) $R=1.46$  mm in GW72, $Bo=0.39$ , $Oh=4.9\times 10^{-2}$ . The lines in (a,b,d) are 1 mm in length. The grid size in (c) is 1 mm.

Figure 10 clearly indicates the existence of an intermediary regime in the range $0.02<Oh<Oh_{c}$ where the compensated Weber number for GW48 ( $20\,^{\circ }\text{C}$ ), GW55 and GW68 falls below the nearly constant value observed when $Oh<0.02$ . This reduction in jet velocity is most likely due to a viscous effect on the jet scale expressed here by the jet Reynolds number $Re=\unicode[STIX]{x1D70C}U_{j}R/\unicode[STIX]{x1D707}$ that decreases from approximately $10^{3}$ in the case of GW48 to $10^{2}$ for GW55 (see table 1). For the latter experiment, when $Re$ is defined with the jet radius instead the bubble radius, it is well below $10^{2}$ .

3.2.3 Large Bond number effects

At very large $Bo$ numbers, the ascending velocity of the bubble due to the buoyancy force approaches the bubble collapse velocity due to the capillary force. An estimate of $Bo$ for no jet formation can be obtained from (3.11) by using $\unicode[STIX]{x0394}z\approx 0.5~R$ during the collapse time to obtain $Bo\approx 12$ . Walls et al. (Reference Walls, Henaux and Bird2015) indicate that there is still jet formation at $Bo\approx$ 5 (bubble in water) but no breakup into drops. As seen in figure 11(c), our experiments also show no drop formation for a $R=4.08~\text{mm}$ bubble in water at $Bo=2.25$ . Hence we expect that at $Bo\approx 10$ no jet will be formed.