3.1. Overview of toroidal bubble–interface interactions

As the behaviours of a toroidal bubble after passing through the liquid–liquid interface remain elusive, we first describe general features regarding the evolutions of the toroidal bubble and interface. The sequential images of the toroidal bubble and interface are displayed in figure 3(a), and the corresponding ring radius and propagation speed are shown in figure 3(b,c). Although one specific case is presented in figure 3(a), the behaviours demonstrated in the figure are generally observed in the other cases in which the bubble penetrates the interface. These data were tracked until the lower liquid surrounding the bubble became corrugated due to instability because the corrugated interface made the accurate correction of refraction impossible.

Figure 3.

(a) Sequential images of the toroidal bubble rising through an interface between two immiscible liquids for $Fr=0.9$, $\varLambda = 73.2$ and $At = 0.067$. (b,c) Temporal changes in the normalised ring radius and propagation speed. See supplementary movie 1 available at https://doi.org/10.1017/jfm.2023.457 for panel (a).

When the toroidal bubble rises far below the horizontal interface ($t^* < -1$ in figure 3), the bubble hardly interacts with the interface, and it gradually expands the ring radius while reducing the propagation speed. This expansion is common for a toroidal bubble rising vertically in a single uniform fluid (Pedley Reference Pedley1968; Cheng, Lou & Lim Reference Cheng, Lou and Lim2013; Vasel-Be-Hagh et al. Reference Vasel-Be-Hagh, Carriveau and Ting2015a). As the bubble approaches the interface, the interface starts to deform into an ellipsoidal cap (figure 3a, $t^*=-2.8$). Simultaneously, the radial expansion accelerates, and the propagation speed declines drastically (figure 3b,c, $t ^*\approx 0$). These trends of the ring radius and the propagation speed are also observed as a single-phase vortex ring perpendicularly approaches a flat surface or a horizontal density-stratified interface (Walker et al. Reference Walker, Smith, Cerra and Doligalski1987; Olsthoorn & Dalziel Reference Olsthoorn and Dalziel2017; Xu et al. Reference Xu, He, Kulkarni and Wang2017; Yeo et al. Reference Yeo, Koh, Long and New2020).

As the toroidal bubble crosses the initial position of the interface, it moves upwards along with some of the lower liquid instead of passing the interface alone. The lower liquid forms an ellipsoid that envelops the toroidal bubble, preventing direct contact between the bubble and the upper liquid (figure 3a, $t^* =2.8$). Although the horizontal interface at a decent distance from the bubble is lowered by up to 0.3 cm at this point as the lower liquid moves upwards, the dynamics of the bubble are hardly affected by the change in the interface level. After penetration, the ring radius and the volume of the lower liquid surrounding the bubble begin to decrease. The reduced volume of the surrounding liquid results in the vertical liquid column being formed below the bubble. This falling liquid column creates secondary vortices below the initial location of the interface, which rotate in the direction opposite to the flow around the toroidal bubble (figure 3a, $t^* \geq 5.5$). The radially contracting bubble accelerates and rises with a greater speed (figure 3b,c, $t^* >6$). In the later stage, the ring radius and the volume of the lower liquid enveloping the bubble become quite small, with most of the lower liquid lost to the falling liquid column (figure 3a, $t^* =11.0$). The toroidal bubble then becomes unstable, losing its axisymmetric form (figure 3a, $t^*=16.5$); we discuss the origin of this instability in § 3.2. The unstable bubble eventually breaks up into tiny bubbles, primarily in the form of a spherical cap (figure 3a, $t^* =19.3$). The remaining lower liquid near the tiny bubbles loses its momentum and begins to fall down.

The lower liquid encapsulating the toroidal bubble fits into an ellipse when observed from the side, as highlighted in figure 4(a), and it is axisymmetric with respect to the longitudinal $z$-axis (figure 4b). Therefore, we assume that the three-dimensional shape is ellipsoidal, and the combination of the bubble and the lower liquid encapsulating the bubble is termed an ellipsoidal body hereafter. The lower liquid encapsulating the bubble is a salient feature in the penetration of the toroidal bubble. Although non-toroidal bubbles, such as spherical or spheroidal, also lift some of the lower liquid under the same circumstance, only a thin film of the lower liquid surrounds the bubble, and the rest is mostly below the bubble in the form of a liquid column (Reiter & Schwerdtfeger Reference Reiter and Schwerdtfeger1992; Bonhomme et al. Reference Bonhomme, Magnaudet, Duval and Piar2012; Natsui et al. Reference Natsui, Nashimoto, Nakajima, Kikuchi and Suzuki2018).

Figure 4.

(a) Side view of entrained lower liquid. The blue area denotes the entrained lower fluid consisting of an ellipsoidal body with a toroidal bubble. (b) Top view of entrained lower fluid.

When the toroidal bubble is below the initial horizontal interface, the vortical flow of the lower liquid is accompanied by the toroidal bubble, as can be seen from the velocity field (in the laboratory reference frame) measured by PIV (figure 5a); in the figure, the bubble is positioned at the reference position. The velocity field in the reference frame moving with the centre of the bubble can be acquired by superimposing the propagation speed of the bubble at this instant on the velocity field in the laboratory reference frame (figure 5b). In this moving reference frame, the ellipsoidal boundary, which consists of streamlines enclosing the toroidal bubble, is constructed (red dashed line in figure 5c). This boundary can be regarded as the boundary of the vortex ring within which the lower liquid is entrained by the vortical flow (Dabiri & Gharib Reference Dabiri and Gharib2004). As the toroidal bubble passes through the liquid–liquid interface, the generated ellipsoidal body depicted in figure 4(a) is similar to the ellipsoidal boundary of the flow structure in figure 5(c). That is, the entrained volume of the lower liquid inside the ellipsoidal body originates from the vortical flow formed around the toroidal bubble below the horizontal interface. Although reliable velocity fields could not be obtained due to refraction through the interfaces and the data are not presented here, the vortical flow around the toroidal bubble inside the ellipsoidal body is also observed after the bubble penetrates the original interface.

Figure 5.

(a) Instantaneous velocity field around the toroidal bubble below the initial interface in the laboratory reference frame, (b) instantaneous velocity field in the reference frame moving with the toroidal bubble and (c) instantaneous streamlines from the velocity field in panel (b). The blue solid lines and greyed-out areas in panels (a–c) denote the toroidal bubble and the shadow of the bubble, respectively, and the red dashed line in panel (c) denotes the boundary of the toroidal bubble in the frame moving with the bubble. Here, $Fr = 2.5$, $\varLambda = 9.7$ and $At = 0.067$.

3.2. Toroidal bubble dynamics and entrained volume

The passage of a toroidal bubble through the liquid–liquid interface exhibits a common pattern, as reported in § 3.1. However, the behaviours of the toroidal bubble, such as ring radius and propagation speed, and the entrained volume of the lower liquid depend on the conditions of the system. In this section, we examine the effects of the Froude number $Fr$ by varying the circulation of the vortical flow around the bubble. The lower and upper liquids are fixed as tap water and V2 oil, respectively: $\varLambda = 73.2$ and $At = 0.067$.

Temporal changes in the normalised ring radius $R/R_{ref}$ and propagation speed $U/U_{ref}$ are depicted in figure 6 for six values of $Fr$. The corresponding values of $R_{{ref}}$, $U_{{ref}}$, $V_{{b,ref}}$ and $\varGamma _{{ref}}$ are listed in table 2 for each case of $Fr$. The trends in the ring radius and propagation speed are similar among the six cases, demonstrating that the trends illustrated by figure 3(b,c) can be generalised. Although $R/R_{ref}$ and $U/U_{ref}$ are hardly affected by $Fr$ before crossing the interface, distinct differences appear after passing the interface. Lower values of $Fr$ produce steeper reductions in $R/R_{ref}$ and $U/U_{ref}$ after $t^* = 0$, followed by steeper growth in $U/U_{ref}$ after approximately $t^* = 6$. This steeper growth in $U/U_{ref}$ appears to begin when the bubble rises sufficiently high to form the liquid column connected with the ellipsoidal body. The relative magnitude of vortex strength is small for lower $Fr$, and the inertial effects of the bubble on the penetration into the upper liquid are accordingly weak, causing a more dramatic reduction in $R/R_{ref}$ and $U/U_{ref}$ as the bubble crosses the interface.

Figure 6.

Temporal changes in (a) normalised ring radius $R/R_{ref}$ and (b) propagation speed $U/U_{ref}$ for $Fr = 0.9\unicode{x2013}2.5$ ($\varLambda = 73.2$ and $At = 0.067$). In each case, data are presented while the bubble remains stable, and a smooth bubble interface can be clearly observed.

Table 2.

Reference values of the toroidal bubble for each case of $Fr$ with $\varLambda =73.2$ and $At=0.067$.

Another main interest is to evaluate how much of the lower liquid the toroidal bubble can entrain and carry upwards across the interface. As shown in figure 5(c), the ellipsoidal boundary of the toroidal bubble in the reference frame fixed with the bubble is caused by the circulatory flow near the bubble. Several studies of the vortex ring have defined the entrained volume as the volume of the fluid inside the ellipsoidal boundary (Dabiri & Gharib Reference Dabiri and Gharib2004). Similarly, the ellipsoidal body in the upper fluid is formed by the circulatory flow near the toroidal bubble. In this sense, the volume inside the ellipsoidal body, excluding the liquid column volume, is only considered for the entrained volume. Consequently, the volume of the ellipsoidal body subtracted by the bubble volume is defined as the entrained volume, $V_{{e}}$, in this study.

In our experiments, toroidal bubbles rose stably to approximately 30 cm above the reference position. The hydrostatic pressure, which considers the densities of the upper and lower liquids, changes along the vertical direction from the reference position. With the adiabatic assumption, the volume change over this vertical distance by the hydrostatic pressure difference is estimated to be approximately 2 %. As the bubble volume is much smaller than the entrained volume, this 2 % change in the bubble volume is negligible in estimating the entrained volume. Thus, the bubble volume is assumed to be constant throughout its penetration phase: $V_{{b}}=V_{{b,ref}}$. The entrained volume is measured from the instant at which the lowermost part of the bubble passes through the initial position of the interface until the bubble and the enveloping lower liquid become unstable and asymmetric. When the entrained volume is not in a completely ellipsoidal form and the liquid column has not yet formed (see figure 3a, $t^*=2.8$), only the volume above the initial position of the interface is considered.

The temporal variations in the entrained volume $V_{{e}}$, made dimensionless by $R_{{ref}}^3$, are depicted in figure 7(a) for six values of $Fr$ (in table 2). The entrained volume drops quickly after the bubble crosses the interface. A greater decrease in $V_{{e}} / R_{{ref}}^3$ is observed for lower values of $Fr$. Comparing figures 6(a) and 7(a), the tendency of the entrained volume is quite similar to that of the ring radius $R$, suggesting the possibility of predicting the entrained volume based on the ring radius.

Figure 7.

(a) Temporal changes in normalised entrained volume $V_{e}/R_{ref}^3$ and (b) normalised volume of the ellipsoidal body, $V_{{s}}/R^3_{{ref}}$, versus normalised ring radius cubed, $(R/R_{{ref}})^3$, for $Fr = 0.9\unicode{x2013}2.5$ ($\varLambda = 73.2$ and $At=0.067$). The black solid line is from (3.1).

The correlation between the ring radius and entrained volume of the toroidal bubble becomes apparent when the bubble is sufficiently far above the initial position of the interface (i.e. $z \geq R_{{ref}}$) to allow the lower liquid encapsulating the bubble to become almost ellipsoidal. The normalised volume of the ellipsoidal body, $V_{{s}}/R^3_{{ref}}$, is plotted versus $(R/R_{{ref}})^3$ for the six values of $Fr$ ($\varLambda = 73.2$ and $At=0.067$) in figure 7(b). Regardless of $Fr$, these parameters collectively exhibit a linear relationship with a non-zero $y$-intercept. As $R$ approaches zero, the bubble becomes spherical, having the non-zero volume of the ellipsoidal body. Thus, a non-zero $y$-intercept is reasonable. Using the least-squares method, the following empirical relation with an R-squared value of 0.97 is acquired:

(3.1)\begin{equation} {V_{{s}}} = {V_{{e}} + V_{{b,ref}}} = {5.8R^3} + 0.6 {R^3_{{ref}}}. \end{equation}

Even for other values of $\varLambda$ and $At$ considered in this study, the data for $V_{{s}}/R^3_{{ref}}$ and $(R/R_{{ref}})^3$ fit well into (3.1), suggesting the universal similarity of the ellipsoidal body with respect to the ring radius.

This similarity suggests that the ellipsoidal boundary of the toroidal bubble in the reference frame fixed with the bubble (red dashed line in figure 5c) before penetration retains its shape throughout the penetration process. For general vortex rings in a single phase, the shape of the ellipsoidal boundary is dependent on the flow structure of the vortex ring (Norbury Reference Norbury1973). However, in the present study, due to the unavailability of velocity fields after penetration, it is difficult to reliably analyse the flow structure near the bubble after penetration. Hence, the explanation of the correlation between $V_{{s}}/R^3_{{ref}}$ and $(R/R_{{ref}})^3$ is limited to be conjectural, based on the observation that the boundary shape of the toroidal bubble before penetration is similar to the boundary shape of the ellipsoidal body after penetration (figures 4a and 5c).

How long the ellipsoidal body rises stably is also examined in a qualitative manner. When the toroidal bubble shrinks and the entrained volume decreases (figure 7), the lower liquid falling from the ellipsoidal body forms a thick liquid column (figure 8a). Up to this stage, the liquid falling through the column obstructs the piercing of the upper liquid into the ellipsoidal body on the lower side of the ellipsoidal body. As the bubble travels farther, the reduction in the entrained volume becomes slower, and the flux of the lower liquid falling through the column decreases, resulting in a thinner column (figure 8b). The slender column becomes corrugated due to instability, just as in the case of a single-phase vortex ring crossing a density-stratified interface (Advaith et al. Reference Advaith, Manu, Tinaikar, Chetia and Basu2017), and the bottom surface of the ellipsoidal body becomes unstable. The upper liquid around the ellipsoidal body then pierces the ellipsoidal body through its bottom interface (figure 8c).

Figure 8.

Piercing process of the upper liquid into the lower liquid encapsulating the toroidal bubble. See supplementary movie 2.

The piercing of the upper liquid can occur when the reduction in the ring radius becomes slower, and it commences near the bottom of the ellipsoidal body in all experimental cases, because the circulation of the toroidal bubble induces an inward flow of the entrained liquid near the bottom surface; see figure 5(a). Soon after the piercing, the ellipsoidal body becomes asymmetric, accelerating the instability of the bubble. We confirmed that this kind of instability was not observed for a toroidal bubble rising in a single liquid in the range of $Fr$ used in our experiments. In addition, Pedley (Reference Pedley1968) theoretically predicted that the height to which a toroidal bubble with a volume of $21\,{\rm cm}^3$ could rise stably was over 150 cm when it was immersed in a single liquid. In contrast, $h_{{stable}}$ is mainly below 30 cm in the present study, suggesting that the piercing of the upper liquid is the primary cause of the instability in our cases.

After the piercing, the bubble breaks up at an instant. Although the entrained liquid still has a little momentum, this height at which the piercing occurs appears to be the clear limit of the stable transport height of the toroidal bubble as the bubble breaks shortly. Hence, the height of the bubble centre at which the piercing begins, $h_{{stable}}$, is regarded as a characteristic travel distance to represent the effectiveness of fluid transport across the interface. This distance is measured from the original position of the interface (figure 9a). The travel distance $h_{{stable}}$ is correlated with the Froude number. For high values of $Fr$, the ellipsoidal body possesses large inertia, which makes it more stable and thus delays the piercing of the upper liquid. Indeed, $h_{{stable}}/R_{{ref}}$ tends to increase linearly with $Fr$ (figure 9b).

Figure 9.

(a) Definition of characteristic travel distance $h_{{stable}}$ and (b) travel distance normalised by reference ring radius, $h_{{stable}}/R_{{ref}}$, versus $Fr$ ($\varLambda = 73.2$ and $At = 0.067$).

To evaluate how effectively the entrained volume $V_{{e}}$ is transported until the centre of the toroidal bubble reaches $z = h_{{stable}}$ for a given volume of the toroidal bubble, figure 10 plots $V_{{e}}$ normalised by the reference bubble volume $V_{{b,ref}}$ with respect to the vertical position of the bubble centre, $z$, normalised by $h_{{stable}}$ (see figure 2b for the definition of the $z$-coordinate). With decreasing $Fr$, $V_{{e}}/V_{{b,ref}}$ becomes greater around $z/h_{{stable}} = 0.1\unicode{x2013}0.2$, which seems to conflict with the results shown in figure 7. However, this result is reasonable when the definition of the Froude number (2.2a) is considered. In our experiments, lower values of $Fr$ tend to produce bubbles with a greater reference ring radius $R_{ref}$ (see table 2), which leads to a large entrained volume $V_{{e}}$ immediately above the initial position of the interface. Remarkably, just after crossing the interface, the bubble carries a massive amount of the lower liquid, approximately 15–50 times its volume $V_{{b,ref}}$ up to $z/h_{{stable}} = 0.1\unicode{x2013}0.2$, even without including the volume of the liquid column (figure 10).

Figure 10.

Entrained volume normalised by bubble volume, $V_{{e}}/V_{{b,ref}}$ with respect to dimensionless height $z/h_{{stable}}$ of the toroidal bubble for $Fr = 0.9\unicode{x2013}2.5$ ($\varLambda = 73.2$ and $At = 0.067$).

It is worth comparing the performance of liquid transport between the toroidal bubble and non-toroidal bubbles. Non-toroidal bubbles rising in a single liquid have various shapes depending on their size and liquid properties such as viscosity and surface tension coefficient, and their paths are also diverse, in addition to vertical straight motion without oscillation, because of instability (Magnaudet & Mougin Reference Magnaudet and Mougin2007; Tripathi et al. Reference Tripathi, Sahu and Govindarajan2015; Sharaf et al. Reference Sharaf, Premlata, Tripathi, Karri and Sahu2017). When a non-toroidal bubble passes through a liquid–liquid interface, the colliding angle between the bubble and the interface also plays a big role (Choi & Park Reference Choi and Park2021). Although the trajectory and colliding angle are regarded as critical parameters for liquid transport, only a few cases of non-toroidal bubbles are chosen for comparison with the toroidal bubble. To compare the toroidal bubble and non-toroidal bubbles of the same volume, an equivalent diameter $D_{{eq}} = (6V_{{b,ref}}/{\rm \pi} )^{1/3}$ is useful. According to Bonhomme et al. (Reference Bonhomme, Magnaudet, Duval and Piar2012), the total volume of the lower liquid transported by a vertically ascending spherical bubble (including the volume of the liquid column below the bubble) is three times the bubble volume when the vertical position of the bubble is around $z=D_{{eq}}$. The volume of the transported liquid drops quickly afterwards and reaches zero at $z=4D_{{eq}}$ (figures 4 and 5 in Bonhomme et al. Reference Bonhomme, Magnaudet, Duval and Piar2012). The volume of the transported lower liquid by a spherical cap bubble rising vertically (including the liquid column) reaches up to seven times the bubble volume at $z=4D_{{eq}}$ (figures 8 and 9 in Bonhomme et al. Reference Bonhomme, Magnaudet, Duval and Piar2012). Furthermore, Choi & Park (Reference Choi and Park2021) reported that bubbles in zigzagging paths transported the lower liquid for a distance much shorter than bubbles in rocking motion; bubbles in rocking motion can rise with the lower liquid approximately by 5$D_{{eq}}$ from the position of the undisturbed interface.

By comparison, the maximum entrained volume of the toroidal bubble (excluding the liquid column) is roughly 15–50 times the volume of the toroidal bubble at $z = 0.6$–$1.0D_{{eq}}$. The value of $V_{{e}} / V_{{b,ref}}$ decreases as the bubble continues to rise towards $z = h_{{stable}}$ (figure 10). Here, the travel distances $h_{{stable}}$ for the six $Fr$ cases correspond to 5.6–9.6$D_{{eq}}$. Despite the long travel distance, the toroidal bubble can still transport approximately five times the volume of the lower liquid. Although the bubble and liquid conditions are not identical, our results suggest that toroidal bubbles have superior transport performance compared with bubbles of other shapes. Interestingly, while there are huge differences in $V_{{e}} / V_{{b,ref}}$ between the six $Fr$ cases at $z/h_{{stable}} <0.5$, their values tend to collapse onto each other when $z/h_{{stable}} >0.5$. Furthermore, the converged values of $V_{{e}} / V_{{b,ref}}$ at $z/h_{{stable}} = 1$ are in quite a narrow band between 4.3 and 6.0. This unique feature – convergence of the volume ratio $V_{{e}} / V_{{b,ref}}$ – is related to the balance of linear momentum of the ellipsoidal body, which is analysed in the following section.

3.3. Prediction of temporal change in ring radius

The relationship between the entrained volume and the ring radius, as shown in (3.1), indicates that the transport capability of a toroidal bubble can be evaluated from the temporal change in the ring radius of the bubble. In this section, we theoretically predict the temporal variation in the ring radius. The ring radius before passing the liquid–liquid interface is discussed first. When the bubble is far below the interface, the deformation of the interface is not observable; that is, the bubble has little effect on the interface. At this time, the behaviour of the toroidal bubble can be explained by referring to a toroidal bubble rising in a single liquid. For a single liquid, the theoretical ring radius $R$ is given by

(3.2)\begin{equation} R = \left[ R^2_{{a}} + \frac{F_{{B}} (t-t_{{a}}) }{{\rm \pi} \rho_{{l}} \varGamma} \right] ^{{1}/{2}}, \end{equation}

assuming that the circulation $\varGamma$ remains constant (Turner Reference Turner1957). Here, $t_{{a}}$ is an arbitrary time, $R_{{a}}$ is the vortex ring radius at $t=t_{{a}}$ and $\rho _{{l}}$ is the density of the surrounding liquid. Additionally, $F_{{B}}$ is the buoyancy force acting on the toroidal bubble, which is assumed to remain constant: $F_{{B}} = (\rho _{{l}} - \rho _{{air}}) gV_{{b}}$, where $\rho _{{air}} = 1.2\,{\rm kg}\,{\rm m}^{-3}$. According to (3.2), the ring radius increases over time, which agrees with our observation at $t^* < 0$ in figure 6(a). Dividing both sides by $R_{{ref}}$ and substituting (2.3) gives the dimensionless form of (3.2):

(3.3)\begin{align} \frac{R}{R_{{ref}}} &= \left[ 1 + \frac{ \rho_{{l}}-\rho_{{air}}}{ \rho_{{l}}} \frac{gV_{{b,ref}}}{{\rm \pi}\varGamma^2_{{ref}}} (t^* - t^* _{{ref}}) \right] ^{{1}/{2}} \nonumber\\ &= \left[ 1 + \frac{1}{Fr^2} \frac{ \rho_{{l}}-\rho_{{air}}}{ \rho_{{l}}} \frac{V_{{b,ref}}}{{\rm \pi} R^3_{{ref}}} (t^* - t^* _{{ref}}) \right] ^{{1}/{2}}, \end{align}

where $t^* _{{ref}}$ is the dimensionless time at $t = t_{{ref}}$. Here, $t_{{a}}$, $R_{{a}}$, $\varGamma$ and $V_{b}$ are replaced by $t_{{ref}}$, $R_{{ref}}$, $\varGamma _{{ref}}$ and $V_{b,ref}$, respectively.

In figure 6(a), $R/R_{ref}$ is hardly affected by $Fr$ before crossing the interface. The independence of $R/R_{ref}$ by $Fr$ can be derived as follows. Walters & Davidson (Reference Walters and Davidson1963) theoretically estimated the circulation of the toroidal bubble, $\varGamma$, as

(3.4)\begin{equation} \varGamma = 3g^{{1}/{2}}V_{{b}}^{{1}/{2}}, \end{equation}

and showed that their experimental data were in reasonable agreement with (3.4) although this relation was derived from the instant when a spherical bubble deforms significantly and turns toroidal. In (3.3), the air density is negligible compared to the liquid density, and $V_{{b}}=V_{{b,ref}}$. Combining (3.4) with (3.3) yields

(3.5)\begin{equation} \frac{R}{R_{{ref}}} = \left[ 1 + \frac{1}{9 {\rm \pi}} (t^* - t^* _{{ref}}) \right] ^{{1}/{2}}, \end{equation}

which is independent of $Fr$. The dash–dotted line in figure 11 is from (3.5), which shows the theoretical prediction of the ring radius is in good agreement with the experimental data.

Figure 11.

Comparison of normalised ring radius $R/R_{ref}$ between experimental measurements (markers) and numerical solutions of (3.10) (solid lines) for $Fr = 0.9\unicode{x2013}2.5$ ($\varLambda = 73.2$ and $At = 0.067$). The black dash–dotted line is from (3.5), and the solid lines are from (3.10).

However, after passing the interface, the buoyancy force of the toroidal bubble is not the only force critical for the bubble dynamics. To take other forces into account without considering the complexity outside the ellipsoidal body, the boundary of the ellipsoidal body (figure 4a) is selected as a control surface. By doing so, mass flux across the control surface appears only through the liquid column below the ellipsoidal body. As the control volume moves upwards at a propagation speed $U$, the $z$-momentum equation from the Reynolds transport theorem is given as

(3.6)\begin{equation} \sum {F_z} - \int_{{CV}}{a_z} \,\mathrm{d}m = \frac{\mathrm{d}}{\mathrm{d} t}\int_{{CV}}u_{z,{rel}}\,\mathrm{d}m+\int_{{CS}}u_{z,{rel}}\,\mathrm{d}\dot{m}_{out}, \end{equation}

where $a_z (= \mathrm {d}U/\mathrm {d}t)$ is the $z$-directional acceleration of the control volume with respect to the ground reference frame, and $u_{z,{rel}}$ is the $z$-directional velocity of flow relative to the reference frame moving with the control volume (White Reference White2015). CV and CS represent the control volume and control surface, respectively. Here, $\mathrm {d}m$ is the differential mass of the fluid inside the control volume and $\mathrm {d}\dot {m}_{out}$ is the differential mass flow rate out of the control surface. The first and second terms on the left-hand side are the sum of the forces applied on the ellipsoidal body and the inertial force accounting for the accelerating control volume, respectively, and the first and second terms on the right-hand side are the momentum change within the control volume and the momentum flux through the control surface, respectively.

Some assumptions should be made because the exact velocity field inside the control volume cannot be obtained. We assume that the first term on the right-hand side of (3.6) is minor because the flow is circulatory around the ring core of the toroidal bubble and thus $u_{z,{rel}}$ inside the control volume is mostly cancelled out for volume integral. Although the downward flow near the lower region of the ellipsoidal body exists towards the liquid column, the temporal variation of the downward flow velocity is minimal. Because the momentum flux through the control surface occurs only through the liquid column descending from the ellipsoidal body, the second term on the right-hand side can be rewritten as $(\dot {m}u_{z,{rel}})_{{out}}$, where $\dot {m}$ at the outlet means the outward mass flow rate from the ellipsoidal body into the liquid column. As the momentum of the liquid column is relatively small compared with that of the ellipsoidal body, the liquid column is assumed to be nearly still in the ground reference frame, and the relative $z$-velocity $u_{z,{rel}}$ at the outlet is supposed to be equal to $-U$. That is, the second term on the right-hand side of (3.6) is dominant over the first term. Equation (3.6) is then simplified as

(3.7)\begin{equation} \sum {F_z} = m \frac{\mathrm{d}U}{\mathrm{d}t}- \dot{m}_{out}U = \frac{\mathrm{d}}{\mathrm{d}t} (mU), \end{equation}

where $\dot {m}_{out} = -\mathrm {d}m/\mathrm {d}t$; $m$ is the total mass inside the control volume.

For $z$-directional forces on the left-hand side in (3.7), the body force acting on the control volume and the surface force acting on the control surface should be considered. Neglecting air density, the gravitational body force acting on the ellipsoidal body in the downward direction is $\rho _{{l}} gV_{{e}}$, and the buoyancy force due to the hydrostatic pressure distribution on the surface of the ellipsoidal body is $\rho _{{u}}g(V_{{e}}+V_{{b,ref}})$ in the upward direction. To evaluate the drag from the pressure and shear stress distributed on the surface of the ellipsoidal body, we refer to Gan, Dawson & Nickels (Reference Gan, Dawson and Nickels2012) in which the drag force over the ellipsoidal boundary of the turbulent and single-phase vortex ring is obtained. The drag force acting on the ellipsoidal boundary in the downward direction is given by

(3.8)\begin{equation} F_{{D}} = C''_{{D}} \times \frac{1}{2} \rho _{{u}} U^2 \times {\rm \pi}R^2 k_4, \end{equation}

where $C''_{{D}}$ is the drag coefficient and $k_4$ is the geometrical coefficient related to the shape of the ellipsoidal body. According to their estimation, $C''_{{D}}k_4$ is approximately 0.042 for their cases. The drag by the shear stress on the ellipsoidal boundary is an order of magnitude smaller than the pressure drag for turbulent vortex rings. Because of similarity between the ellipsoidal boundary of the vortex ring noted by Gan et al. (Reference Gan, Dawson and Nickels2012) and the surface of the ellipsoidal body in our study, we adopt $C''_{{D}}k_4 = 0.042$ and assume that the viscous drag is negligible; the effect of viscous drag is discussed again in § 3.4 with other experimental data using the three upper liquids with different viscosities.

The liquid column below the ellipsoidal body has an additional effect on the pressure for the lower surface of the ellipsoidal body (Ji, Yang & Feng Reference Ji, Yang and Feng2021). The Laplace pressure due to the curvature of the liquid column, which acts on the lower surface, is $\sigma _{{I}}/R_{{c}}$, where $R_{{c}}$ is the curvature radius of the liquid column surface. Consequently, the force in the upward direction due to the Laplace pressure in the liquid column is approximately $(\sigma _{{I}}/R_{{c}}){\rm \pi} R^2_{{c}}$. Assuming that $R_{{c}}$ is of the same order of magnitude as $R$, this force can scale as ${\rm \pi} \sigma _{{I}}R$, which is only 1/400 of the gravitational force acting on the ellipsoidal body. Therefore, the force by the Laplace pressure is negligible. In addition, the interfacial tension force acting on the region connecting the ellipsoidal body and the liquid column scales as $2 {\rm \pi}R \sigma _{{I}}$ and is negligible, considering that its scale is only approximately 1/200 of the gravitational force acting on the ellipsoidal body.

Substitution of the aforementioned forces into (3.7) leads to

(3.9)\begin{equation} (\rho_{{u}} -\rho_{{l}})gV_{{e}} + \rho_{{u}} gV_{{b,ref}} - \frac{1}{2} {\rm \pi}C''_{{D}}k_4 \rho _{{u}} U^2 R^2 = \frac{\mathrm{d}}{\mathrm{d} t} (mU). \end{equation}

According to (3.1), $m (= \rho _{{l}} V_{{e}})$ is a function of $R$; the mass of the toroidal bubble inside the ellipsoidal body is negligible. Also, from the assumption that the bubble volume remains constant, $V_{b} (=2 {\rm \pi}^2 R a^2)=V_{{b,ref}}$ and $a = (V_{{b,ref}}/2{\rm \pi} ^2 R)^{1/2}$. Hence, (3.9) has two variables ($U$ and $R$), which can be reduced to one if we express $U$ in terms of $R$. If the passage through the interface does not significantly affect the circulation, $\varGamma$ can be replaced by $\varGamma _{{ref}}$, and $U$ can be expressed as a function of only $R$ in (2.1). Combining (2.1), (3.1) and (3.9) yields the equation for $R$,

(3.10)\begin{align} &(\rho_{{u}} -\rho_{{l}}) g({5.8R^3 + 0.6R^3_{{ref}} -V_{{b,ref}}}) + \rho_{{u}}gV_{{b,ref}} \nonumber\\ &\qquad- \frac{1}{2}{\rm \pi} C''_{{D}}k_4 \rho _{{u}} { \left[ \frac{\varGamma_{{ref}}}{4{\rm \pi}} \left\{ \frac{1}{2}\mathrm{ln} \left(\frac{128 {\rm \pi}^2 R^3}{V_{{b,ref}}}\right) - \frac{1}{4} \right\} \right]}^2 \nonumber\\ &\quad = \frac{\mathrm{d}}{\mathrm{d} t} \left[ \rho_{{l}} ({5.8R^3 + 0.6R^3_{{ref}} -V_{{b,ref}}}) \frac{\varGamma_{{ref}}}{4{\rm \pi} R} \left\{ \frac{1}{2}\mathrm{ln} \left( \frac{128 {\rm \pi}^2 R^3}{V_{{b,ref}}}\right) - \frac{1}{4} \right\}\right]. \end{align}

Discretisation using the forward Euler method for $R$ with a time step of $\Delta t = 0.003$ s is employed to solve (3.10) numerically. Because (3.10) is derived under the condition in which the ellipsoidal body is fully formed after penetration, the numerical simulation uses the value of $R$ at $z=R_{{ref}}$, which is high enough to have the complete ellipsoidal body, as the initial condition.

Admittedly, the theoretical prediction from (3.10) has several limitations. As the ring radius decreases, the inward radial velocity makes the Kutta–Joukowski lift, accelerating the core upwards (Lundgren & Mansour Reference Lundgren and Mansour1991). Therefore, (2.1) that represents the relation between $U$ and $R$ needs to be modified after passing through the interface, especially for low $Fr$ where the bubble undergoes significant speed change. In addition, although we use a single drag coefficient $C''_{{D}}$ which was originally acquired for turbulent and single-phase vortex rings, the presence of the three phases in this study makes the detailed flow structure of the toroidal bubble differ from that of the turbulent vortex rings, which also causes errors in the theoretical prediction. Despite these limitations, the numerical solutions of (3.10) successfully capture the temporal change in the ring radius from the experiments (figure 11). This theoretical result explains that the downward force acting on the ellipsoidal body eventually leads to its contraction.

For a solid sphere passing through a horizontal liquid–liquid interface, Pierson & Magnaudet (Reference Pierson and Magnaudet2018) theoretically estimated the amount of liquid that adheres to the sphere and moves together, using vertical force balance acting on the sphere with buoyancy and surface tension forces. Even for a non-toroidal bubble coated with oil, the volume fraction of the bubble and oil is determined where the resultant force of vertical forces approaches zero (Ji et al. Reference Ji, Yang and Feng2021). Similarly, we examine the vertical forces acting on the ellipsoidal body to find the relationship between the vertical forces and the entrained volume. Figure 12 plots the left-hand side of (3.10) (i.e. $\sum F_{z}$) with respect to the vertical position $z/h_{{stable}}$ of the bubble. Here, $\sum F_{z}$ is normalised by the buoyancy force acting on the bubble when it is immersed entirely in the lower liquid, $F_{{B}}$ in (3.2): $F^*=\sum F_{z}/F_{{B}}$. In figure 12, the vertical net force $F^*$ is very close to zero when the piercing of the upper liquid into the ellipsoidal body occurs at $z/h_{{stable}}=1$ (figure 8). In other words, because $F^*$ close to zero causes slow change in the ring radius as inferred from (3.10), the liquid column becomes thinner, resulting in the piercing of the upper liquid.

Figure 12.

Normalised force in $z$-direction with respect to dimensionless height $z/h_{{stable}}$ of the toroidal bubble for $Fr=0.9\unicode{x2013}2.5$ ($\varLambda = 73.2$ and $At = 0.067$).

As the piercing of the upper liquids occurs when the vertical forces acting on the ellipsoidal body balance each other, we can also predict the entrained volume at $z/h_{{stable}}=1$ from the force balance in (3.9). From our observation, the magnitude of pressure drag is minimal at $z/h_{{stable}}=1$ for all cases, while gravitational and buoyancy forces are dominant. Neglecting the pressure drag, the following approximation is valid at $z/h_{{stable}}=1$:

(3.11a)\begin{gather} (\rho_{{u}} - \rho_{{l}})gV_{{e,f}} + \rho_{{u}} gV_{{b,ref}} \approx 0, \end{gather}

(3.11b)\begin{gather} \frac{V_{{e,f}}}{V_{{b,ref}}} \approx \frac{\rho_{{u}} }{\rho_{{l}} - \rho_{{u}}}, \end{gather}

where $V_{{e,f}}$ is the entrained volume at $z/h_{{stable}} = 1$. For our cases, (3.11) yields $V_{{e,f}}/V_{{b,ref}} \approx 6.7$, which is very close to the experimental values of $V_{{e}}/V_{{b,ref}}$ at $z/h_{{stable}} = 1$ in figure 10. In short, the final value of the entrained volume is simply determined by the densities of the liquids and the bubble volume.

3.4. Effects of viscosity and density difference

Two dimensionless parameters, the viscosity ratio $\varLambda$ and the Atwood number $At$, are now controlled to explore how the dynamics of a toroidal bubble change with the relative magnitudes of viscosity and density between the two liquids. First, by changing the viscosity of the upper liquid, $\varLambda$ varies from 15.1 to 217.9, while $Fr$ and $At$ remain fixed at 1.7 and 0.067, respectively (three cases in table 3). Comparing the cases of $\varLambda = 15.1$ and 217.9, the overall sequential patterns of the toroidal bubble after crossing the interface are similar at the same $t^*$, although the interface of the liquid column is smoother for $\varLambda = 217.9$ (figure 13).

Table 3.

Reference values of the toroidal bubbles in figures 13 and 14(a–c).

Figure 13.

Sequential images of the toroidal bubble rising through an interface between two immiscible liquids for (a) $\varLambda = 15.1$ and (b) $\varLambda = 217.9$ ($Fr=1.7$ and $At=0.067$). See supplementary movie 3.

Obviously, the ring radius and propagation speed are not affected by the viscosity ratio before $t^*=0$ because only the upper liquid has changed (figure 14a,b). After $t^*=0$, however, the toroidal bubble exhibits faster radial contraction at given $t^*$ for higher $\varLambda$, leading to a slightly greater propagation speed. The solid lines in figure 14(a) are from the numerical solutions of (3.10). Note that (3.10) neglects the drag force from the viscous stress on the ellipsoidal body. Thus, it shows good accuracy in predicting the change in the ring radius for low $\varLambda$, but deviates from the experimental observations for high $\varLambda$. The viscous drag resists the upward motion of the ellipsoidal body, adding a downward force along with the gravitational force. As a result, a stronger viscous drag force would make the ellipsoidal body and bubble shrink more rapidly.

Figure 14.

Temporal changes in (a) ring radius $R/R_{{ref}}$, (b) propagation speed $U/U_{{ref}}$ and (c) entrained volume $V_{{e}}/V_{{b,ref}}$ for $\varLambda = 15.1$, 73.2, 217.9 ($Fr = 1.7$ and $At = 0.067$). (d) Travel distance $h_{{stable}}/R_{{ref}}$ versus $Fr$ ($At=0.067$). In panel (a), the black dash–dotted line is from (3.5), and the solid lines are from (3.10).

The entrained volume also exhibits a slightly faster decrease for the higher $\varLambda$, following the trend of $R/R_{ref}$ (figure 14c). As a consequence, $V_{{e}}/V_{{b,ref}}$ at $z=h_{{stable}}$ becomes smaller for the higher value of $\varLambda$. The values of $V_{{e,f}}/V_{{b,ref}}$ are 6.4 and 3.6 for $\varLambda$ = 15.1 and 217.9, respectively. Although not presented here, $V_{{e}}/V_{{b,ref}}$ tends to decrease with increasing $\varLambda$ for fixed values of $Fr$ other than 1.7 reported in figure 14(c). As discussed in § 3.3, the entrained volume is determined by the vertical forces imposed on the ellipsoidal body, and the entrained volume at $z = h_{{stable}}$ corresponds to a zero net vertical force in (3.11). However, the force acting on the surface of the ellipsoidal body from the viscous stress of the upper liquid was neglected in § 3.3 because of the difficulty in rigorously estimating the viscous stress. As a result, a stronger viscous drag force would make the ellipsoidal body and bubble shrink more rapidly, leading to the reduced $V_{{e,f}}/V_{{b,ref}}$ for higher $\varLambda$. The characteristic travel distance $h_{{stable}}/R_{{ref}}$ is almost unaffected by $\varLambda$ in the range considered in this study (figure 14d); for this plot, other cases with different $Fr$ values are included in addition to the three cases in table 3. This is because, for the higher $\varLambda$, the increase in the propagation speed owing to the faster reduction in the ring radius offsets the negative effect of the stronger viscous force on the travel distance.

Despite an increase in the viscosity ratio by a factor of approximately 15, the changes are relatively minor compared with those in figure 6, in which the effects of $Fr$ are depicted. In other words, changing the viscosity ratio has a limited effect on the interactions between the bubble and the interface compared with the Froude number, at least for the range of viscosities considered in this study. This is because the toroidal bubble does not directly contact the upper liquid owing to the surrounding lower liquid. In contrast, non-toroidal bubbles penetrating through the interface are sensitive to the viscosity ratio since they undergo direct contact with the upper liquid (Greene et al. Reference Greene, Chen and Conlin1991; Bonhomme et al. Reference Bonhomme, Magnaudet, Duval and Piar2012; Magnaudet & Mercier Reference Magnaudet and Mercier2020; Choi & Park Reference Choi and Park2021).

Next, while the upper liquid (V2 oil) remains unchanged, the density of the lower liquid is varied using tap water and an NaCl solution to investigate the effect of the Atwood number between $At = 0.067$ and $0.133$. The viscosity ratios of these two cases differ because the viscosity of the NaCl solution is greater than that of tap water: $\varLambda = 73.2$ for $At = 0.067$ and $\varLambda = 51.7$ for $At = 0.133$ (first two rows in table 4). Nevertheless, this difference is negligible because the bubble–interface interactions are insensitive to such small variations in the viscosity ratio, as explained previously.

Table 4.

Reference values of the toroidal bubbles. The first two rows are compared in figures 15 and 16(a–c), while the first and third are in figure 17(a,b).

For $At = 0.133$, the greater density of the lower liquid causes a stronger downward body force on the ellipsoidal body, resulting in a significant reduction in the ring radius at a given time (figures 15 and 16a). Because of the greater mass in the ellipsoidal body for $At = 0.133$, the upward motion is more severely retarded immediately after the bubble crosses the interface (figure 16b). However, the greater contraction in the ring radius eventually induces a steeper increase in the propagation speed after $t^* = 6$. Remarkably, just a twofold increase in $At$ induces dramatic variations in $R/R_{ref}$ and $U/U_{ref}$, while their variations are minor despite a 15 times increase in $\varLambda$, as shown in figure 14(a,b). Also, in figure 16(a), the numerical solutions of (3.10) (solid lines) are in excellent agreement with the experimental results for both cases.

Figure 15.

Sequential images of the toroidal bubble rising through an interface between two immiscible liquids for (a) $At=0.067, \varLambda = 73.2$ and (b) $At=0.133, \varLambda = 51.7$ ($Fr=1.4$). See supplementary movie 4.

Figure 16.

Temporal changes in (a) ring radius $R/R_{ref}$, (b) propagation speed $U/U_{ref}$ and (c) entrained volume $V_{{e}}/V_{{b,ref}}$ for $At$ = 0.067 and 0.133 ($Fr=1.4$). (d) Travel distance $h_{{stable}}/R_{{ref}}$ versus $Fr$. Here, $\varLambda = 73.2$ for $At=0.067$ and $\varLambda = 51.7$ for $At=0.133$. In panel (a), the black dash–dotted line is from (3.5), and the solid lines are from (3.10).

The higher Atwood number leads to a faster reduction in the entrained volume due to a significant contraction in the ring radius (figure 16c). Consequently, $V_{{e}}/V_{{b,ref}}$ at $z= h_{{stable}}$ becomes smaller for the higher $At$: these values in figure 16(c) are 6.0 and 3.0 for $At$ = 0.067 and 0.133, respectively. Substituting the densities of the liquids used for $At=0.133$ into (3.11) yields $V_{{e,f}}/V_{{b,ref}} \approx 3.2$, which matches well with the experimental value of $V_{{e,f}}/V_{{b,ref}}=3.0$. Moreover, under a similar $Fr$ value, the travel distance $h_{{stable}}/R_{{ref}}$ is noticeably smaller for the higher $At$ because of the heavier lower liquid (figure 16d); additional cases with $Fr$ values other than 1.4 are considered for this plot. The value of $h_{{stable}}/R_{{ref}}$ is determined by $At$ rather than $\varLambda$, as seen from a comparison between figures 14(d) and 16(d).

In short, both the radius change and travel distance are more sensitive to $Fr$ and $At$ rather than $\varLambda$, given the ranges of these dimensionless parameters. Furthermore, in figures 6 and 16, the increase in $At$ and decrease in $Fr$ have similar effects on the variations in $R$ and $U$, as well as $h_{{stable}}$. Thus, it appears reasonable to introduce a dimensionless parameter that integrates the opposite effects of $Fr$ and $At$. Dahm et al. (Reference Dahm, Scheil and Tryggvason1989) reported that, for a single-phase vortex ring impinging a density-stratified interface with a small density variation, the dynamics were primarily determined by $Fr$ divided by the square root of $At$. In some aspects, the interaction between a toroidal bubble and an interface between two immiscible liquids is similar to that of the single-phase vortex ring and the density-stratified interface. Accordingly, it is worth examining whether the effective Froude number $Fr_{{eff}}(=Fr At^{-1/2})$ can better characterise the toroidal bubble interacting with the immiscible interface.

For two cases with very similar $Fr_{{eff}}$ values (5.4 and 5.5), but distinctly different $Fr$ and $At$ values, $R/R_{{ref}}$ and $U/U_{{ref}}$ collapse onto each other at a given $t^*$ (figure 17a,b); these two cases correspond to the first and third rows in table 4. Furthermore, $h_{{stable}}/R_{{ref}}$ is collectively in a linear relation with $Fr_{{eff}}$ in figure 17(c). The feasible range of $Fr$ is limited in our apparatus, and there were not enough experimental cases to make identical $Fr_{{eff}}$ pairs, except for the cases of $Fr_{{eff}} \approx 5.5$ presented in figure 17. Due to experimental difficulties, we are able to validate only a few cases, so additional validation with wider ranges of $Fr$ and $At$ is essential to support the reliability of our argument. Nevertheless, the results in figure 17 indicate that the combined parameter $Fr_{{eff}}$ successfully reflects the effects of density difference between the two liquids.

Figure 17.

Temporal changes in (a) ring radius $R/R_{ref}$ and (b) propagation speed $U/U_{ref}$ for $Fr_{{eff}}= 5.4$ ($Fr=2.0$, $At=0.133$) and $Fr_{{eff}}= 5.5$ ($Fr=1.4$, $At=0.067$). (c) Travel distance $h_{{stable}}/R_{{ref}}$ versus $Fr_{{eff}}$ for the two $At$ values. In panel (a), the black dash–dotted line is from (3.5), and the solid lines are from (3.10).

Finally, using the effective Froude number, we briefly compare the penetration capability of a toroidal bubble with that of the single-phase vortex rings analysed in previous studies. The single-phase vortex ring can only partially penetrate the density-stratified interface and bounces backwards at $Fr_{{eff}}=8.6$ in figure 7 of Dahm et al. (Reference Dahm, Scheil and Tryggvason1989), and it does not penetrate the interface at $Fr_{{eff}}=5.2$ in figure 7 of Stock et al. (Reference Stock, Dahm and Tryggvason2008). Furthermore, the single-phase vortex ring deforms the interface between immiscible liquids and rebounds without complete penetration even at $Fr_{{eff}}= 9.6$ in figure 7 of Song et al. (Reference Song, Choi and Kim2021). By contrast, in the present study, the toroidal bubble clearly penetrates the interface at the small values of $Fr_{{eff}}= 3.5$ (figure 3a: $Fr=0.9, At=0.067$) and $Fr_{{eff}}=3.9$ (figure 15b: $Fr=1.4, At=0.133$). Despite much smaller values of $Fr_{{eff}}$, the toroidal bubble can pass through the interface stably by virtue of the buoyancy force acting on the bubble, which implies that a toroidal bubble is capable of mixing surrounding liquids more easily across a broader range of conditions, compared with a single-phase vortex ring.