Interfacial phenomena of the interaction between a liquid–liquid interface and rising bubble

Abstract

We experimentally investigate the interfacial phenomena involved in the occasions when a rising bubble interacts with the interface between two immiscible liquids, by varying the bubble size (3.0–8.0 mm; i.e., covering the straight, zigzagging, and rocking trajectories) and the viscosity ratio ($\Lambda =1.38$ and $12.8$) of liquids (silicone oil and water–glycerin mixture as the upper and lower liquids, respectively). Our major focus is to understand how the dynamics of a rising bubble and subsequent deformation of the interface are determined, based on the spatiotemporal variation of interface and velocity fields simultaneously measured by using the two-phase particle image velocimetry and laser-induced fluorescence, respectively. When the viscosity ratio is small ($\Lambda =1.38$), even the bubble of the same size, under the path instability, passes through the interface quite differently depending on the colliding angle. This is because the flow dragged by the bubble and wake vortices change according to the bubble position in the oscillating rise path. As the viscosity ratio increases to 12.8, the effect of path instability becomes negligible owing to the enhanced viscous dissipation, so that only the bubble in a rocking motion can escape the liquid interface instantly. By estimating the momentum flux, we find that the bubble-induced momentum transferred to the interface dissipates fast as the viscosity ratio increases. In addition, the maximum height (deformation) of the liquid interface is closely related to the escape dynamics of the bubble. We think the present results will enhance our understanding of how the complex interaction between moving fluid interfaces is determined, and further the effective way of controlling it.

Graphical abstract

1 Introduction

When a solid or fluid particle (bubble or droplet) moves across the interface between multiphase fluids, complex interfacial phenomena occur successively such as the deformation of interfaces, localized rupture, and jet ejection depending on the properties of particle and fluids (Greene et al. 1988; Aristoff and Bush 2009; Bonhomme et al. 2012; Pierson and Magnaudet 2018a, 2018b; Kim and Park 2019a; Choi et al. 2019; Jarvis et al. 2019; Itawi et al. 2020). Fascinated by the physics behind this, many studies have been conducted so far, and we have learned useful insights of multiphase flow, benefited by the areas of liquid extraction, iron processing, and so on (Poggi et al. 1969; Takahashi et al. 1979; Kobayashi 1993; Tanno et al. 2017). Among different configurations of interfacial disturbance, in the present study, we are to focus on the interaction between a rising bubble and the interface of immiscible liquids. Owing to the highly deformable nature of bubble surface and relatively smaller inertia (compared to other fluid particles), it has been considered to involve interesting flow features.

If the inertia of the rising bubble is sufficiently large (bubble Reynolds number ${{\rm Re}}_{b}={U}_{b}{d}_{{\rm eq}}/\gtrsim {\varvec{O}}({10}^{2})$, where the rise velocity (${U}_{b}$) and equivalent diameter (${d}_{{\rm eq}}$) are used as a characteristic scale and ν is the kinematic viscosity of liquid), the liquid–liquid interface starts to deform before the physical contact with the bubble, driven by the high pressure (potential flow effect) in front of the bubble (Manica et al. 2016). During the deformation of the interface, a liquid film is formed between the bubble and interface, which is gradually thinned by the drainage. Depending on the bubble size and liquid conditions, the rising bubble may pass through the liquid interface instantly, escape the interface in a delay or stay under it. Greene et al. (1988), setting up the balance equation with buoyancy and interfacial tension forces, found the minimum volume of the bubble, required to cross the interface and entrain the lower liquid behind the bubble is ${(3.9{\sigma }_{12}/g({\rho }_{2}-{\rho }_{b}))}^{1.5}$ and ${(7.8{\sigma }_{12}/g(3{\rho }_{2}-{\rho }_{1}-2{\rho }_{b}))}^{1.5}$, respectively. Here, ${\rho }_{1}$, ${\rho }_{2}$ and ${\rho }_{b}$ are the density of lower, upper liquid and bubble, respectively, ${\sigma }_{12}$ is interfacial tension, and $g$ is the gravitational acceleration. They assumed that the bubble rising vertically only has a cylindrical symmetry about its vertical axis, and thus, the buoyancy and surface tension are dominant. Dietrich et al. (2008) suggested that the passage time taken by the bubble divided by the characteristic time-scale of the bubble ($={d}_{{\rm eq}}/{U}_{b}$) is proportional to the ratio of its Reynolds number in lower to upper liquids, to the power of 0.34. While they measured the liquid-phase velocity fields around the bubble with the particle image velocimetry, detailed analysis was not provided about the bubble–interface interaction. On the other hand, a regime map describing how the bubble and liquid–liquid interface interact was given by Bonhomme et al. (2012), in terms of Archimedes ($Ar=\rho {(g{d}_{{\rm eq}}^{3})}^{0.5}/\mu \sim {\varvec{O}}({10}^{0}-{10}^{4})$) and Bond ($Bo=\rho g{d}_{{\rm eq}}^{2}/\sigma \sim {\varvec{O}}({10}^{0}-{10}^{2})$) numbers, and viscosity ratio ($\Lambda ={\mu }_{2}/{\mu }_{1}\sim O({10}^{-2}-{10}^{2})$). Here, $\sigma$ is the surface tension of the liquid. It was shown that small bubbles, of which both Bo and Ar are ${\varvec{O}}({10}^{0})$, are not able to pass through the interface and the liquid film drainage is decelerated at higher Bo and $\Lambda$. Natsui et al. (2018) investigated the lower liquid column dragged by the bubble while varying the density, viscosity, and surfactant concentration of the lower liquid. They suggested that dimensionless column lifetime ($\uptau ={\mathrm{tU}}_{b}/{d}_{{\rm eq}})$ is scaled as $\uptau \sim {La}^{1/2}{Bo}^{1/3}$. Here, the Laplace number is defined as $La={\sigma }_{12}\Delta \rho {d}_{{\rm eq}}/{\overline{\mu }}_{12}^{2}$, where the mixture viscosity ${\overline{\mu }}_{12}$ is $2{\mu }_{1}{\mu }_{2}/({\mu }_{1}{+\mu }_{2})$ and ${\sigma }_{12}$ is the interfacial tension. When the surfactant concentration is higher in the lower liquid, the film lifetime became shorter and rupture occurred near the bubble apex. On the other hand, Feng et al. (2016) investigated the dynamics of a bouncing bubble on the interface that does not rupture.

When the bubble passes through the liquid interface, as expected, the liquid film drainage and the interface rupture are important phenomena. Debrégeas et al. (1998) showed that the film thickness decreases exponentially, which is explained by the plug flow thinning mechanism. Zawala et al. (2011) claimed that the film drainage time depends on the degree of liquid interface curvature and bubble deformation; when the interface curvature increases and the bubble deforms more, bubble–interface contact time decreases preventing the film from rupturing. On the other hand, Lhuissier and Villermaux (2012) claimed that the plug flow or Poiseuille flow-limited thinning mechanism is not appropriate in explaining the drainage of tap water and suggested the viscous drainage mechanism. They showed that the rupture occurs near the meniscus and explained why the range of puncture time is widely scattered. When a stationary air bubble is suspended between surfactant-free liquid and free surface, Nguyen et al. (2013) showed that the main drainage mechanism changes depending on the size ($R$) of a cap bubble. When $R$ is much smaller than the capillary length of liquid, $\lambda ={(\sigma /(g\Delta \rho ))}^{0.5}$, the capillary film drainage is dominant, or the gravitational force will drive the film drainage for $R\gg \lambda$. For the gravitational drainage, the rupture usually occurs at the bubble apex, driven by the force of ${F}_{g}=(4\pi /3){R}^{3}g\Delta \rho$ (it is ${F}_{s}=2\sigma A/R$, where $A$ is the area of drained film, for the case of capillary drainage). The time required for gravitational and capillary drainage is scaled as ${t}_{g}\sim \mu /(gR\Delta \rho )$ and ${t}_{s}\sim \mu R/\sigma$, respectively, indicating that the bubble size oppositely affects the drainage mechanism.

As we have summarized above, there have been many studies on the interaction between a rising bubble and liquid–liquid interface, but mostly they focused on the effect of bubble size and liquid properties, and did not pay much attention to the bubble dynamics (i.e., the effect of path instability) and resulting interfacial phenomena. Therefore, in the present study, we experimentally investigate how a single bubble, rising in a different trajectory, interacts with the interface of two immiscible liquids (silicone oil and water–glycerin mixture for upper and lower liquids, respectively). While carefully controlling the condition of a single bubble (straight path: ${d}_{{\rm eq}}=3.02-3.14$ mm; oscillating path: ${d}_{{\rm eq}}=4.27-4.45$ mm; rocking motion: ${d}_{{\rm eq}}\simeq 8.0$ mm), we simultaneously conduct the two-phase particle image velocimetry (PIV) (Kim et al. 2016; Kim and Park 2019b; Lee and Park 2020) and laser-induced fluorescence (LIF), to measure the two-phase velocity fields (including the interface) and penetration of lower liquid, respectively. To achieve accurate measurements, we match the refractive index of two liquids. By combining the flow (concentration) fields and interface shape, we will try to understand the relation between the bubble dynamics and disruption of the liquid–liquid interface.

The remaining part of this paper is organized as follows. In Sect. 2, we will explain the experimental methods and the tested conditions of liquid and gas phases, followed by Sect. 3, in which we discuss the interaction between the bubble and liquid interface to examine how the bubble rise path affects the interfacial phenomena with the viscosity ratio of the liquids. Finally, in Sect. 4, we will summarize our findings and provide a future outlook.

2 Experimental setup and methods

2.1 Test chamber with two immiscible liquids and a single rising bubble

The present experiments were conducted in an acrylic tank with a size of 150 × 150 × 400 mm³, as shown in Fig. 1a. Clift et al. (1978) suggested that the tank width needs to be at least 8 times larger than the bubble size to avoid the wall effect, and considering the size of the bubble used in the present study, the tank size is sufficiently large to avoid the interference of the side wall. For the upper and lower liquids, we used the silicone oil and water–glycerin mixture, respectively, to control its viscosity ratio and make the immiscible liquid interface reliably. To conduct the PIV and LIF successfully in the multilayered liquids, it is critical to match the refractive index of the stacked liquids. Since the refractive index of tap water (1.333) is quite different from that (1.399–1.406) of silicone oil, we added the glycerin to water (Wiederseiner et al. 2011), by which it was possible to reduce the difference in the refractive index as low as O(10^–4) without affecting the bubble dynamics substantially. According to Patil and Liburdy (2012), the uncertainty of PIV caused by difference of refractive index as much as 0.0016 was approximately 2%. Unlike adding the sodium chloride solution (Daviero et al. 2001; Bai and Katz 2014), the addition of glycerin changed the mixture viscosity fairly compared to tap water, which did not have a drastic change in the rise path and size of the bubble. As a result of systematic tests, we chose the lower–upper liquids combinations of (case i) 50.7 wt% (wt: weight percentage of glycerin in the mixture) water–glycerin mixture and 10 cp silicone oil, and (case ii) 53.3 wt% water–glycerin mixture and 100 cp silicone oil, which corresponds to the viscosity ratio of $\Lambda ={\mu }_{2}/{\mu }_{1}$= 1.38 and 12.8, respectively (subscript 1 and 2 denotes lower and upper liquid, respectively). The properties of upper and lower liquids are provided in Table 1. The refractive index was measured by the digital refractometer (R09-105–457, OPTi®), whose minimum measurement unit is 0.0001, and the viscosity was measured by the sine wave vibro viscometer (sv-10, A&D). Surface tension and interfacial tension coefficients were obtained using the pendent drop method (SDS-TEZD10013, Smartdrop), and the density was measured using 10 ml pipette (Gilson). All the liquid properties were measured at 20 (± 1) °C and the experiments were conducted at the same temperature, as well.

Fig. 1

a Schematic for the experimental setup to measure the rising bubble through the liquid–liquid interface. b Example images of the tested bubble (d_eq: equivalent bubble size). The insect in a illustrates the definition of colliding angle, $\theta$

Table 1 Properties of the tested liquids

To introduce a single bubble, we located a stainless steel nozzle at the bottom of the tank and used different sizes of inner diameter (1.194 mm and 0.584 mm) to control the rising path as planned such as the straight (${d}_{{\rm eq}}=3.08 (\pm 0.3)$ mm) and oscillating trajectories (${d}_{{\rm eq}}=4.36 (\pm 0.4)$ mm), respectively. The representative bubble images are shown in Fig. 1b. It is noted that the size of the bubble and its rise trajectory is slightly different from previous studies (Clift et al. 1978; Ern et al. 2012; Jeong and Park 2015; Lee and Park 2017; Maeng and Park 2021), which used water as a working fluid. Since we are using a water–glycerin mixture solution for the lower liquid, which is more viscous than water, for the purpose of refractive index matching as we explained above, the bubble size is larger when the path instability starts to occur, compared to using tap water. The gas flow rate through the nozzle was controlled by the syringe pump (Fusion 100 Touch, Revodix Inc.) to be fixed at 0.1 ml/min, which is sufficiently low to avoid successive bubble injection. The equivalent diameter (${d}_{{\rm eq}}$) of the bubble is defined based on the total injection bubble volume (${V}_{b}$), as ${d}_{{\rm eq}}={(6{V}_{b}/\pi )}^{1/3}$. We defined the bubble size based on the initial bubble volume (Bonhomme et al. 2012; Kong et al. 2019) since the two-dimensional image is not sufficient to evaluate the size of a deformable bubble, especially for the case of a rocking motion in the present study. Because the compressibility effect or mass transport (e.g., phase change) is not involved, the bubble size is maintained to be the same during the experiment. On the other hand, to generate a much larger bubble rising with a rocking motion, we used the reversely rotatable cup method (cup diameter is 40 mm), used in common to make a bubble as large as 20 mm (Greene et al. 1988; Bonhomme et al. 2012). We were able to generate the bubble up to ${d}_{{\rm eq}}=12$ mm and the interfacial phenomena were similar when ${d}_{{\rm eq}}\ge 8.0$ mm; the bubble rises in an almost rectilinear path, but the significant wobbling induces the lateral deviations, called rocking motion. To analyze the bubble dynamics consistently, we defined the origin of the vertical (z) direction as being located on the undisturbed liquid–liquid interface and that of horizontal (x) direction agreed with the bubble centroid when it was first fully captured in the field of view (Fig. 1a). Depending on the combination of lower and upper liquids, the important non-dimensional parameters of the bubble (based on each liquid) are summarized in Table 2.

Table 2 Statistics and non-dimensional parameters of the rising bubble: $d_{{\rm eq}}$, equivalent bubble diameter; $U_{b}$, bubble rise velocity; $Bo ( = \rho gd_{{\rm eq}}^{2} /\sigma$), Bond number; $Re \left( { = \rho U_{b} d_{{\rm eq}} /\mu } \right)$, Reynolds number; $We ( = \rho U_{b}^{2} d_{{\rm eq}} /\sigma$), Weber number; ${\rm Oh} \left( { = {\rm We}^{0.5} /{\rm Re}} \right)$, Ohnesorge number

Since our main focus is to examine the effect of the bubble path, it is important to control the position of the interface relative to the oscillating rise path of the bubble. To achieve this, we changed the height of the lower liquid in the range of 150–170 mm. Here, we chose the minimum height of lower liquid to guarantee that the bubble reaches the terminal velocity. When the bubble centroid reaches the x-axis, we define the colliding angle ($\theta$) as the angle between the bubble major axis and the x-axis (see Fig. 1a for the illustration). For the linear (including rocking motion) path, we fixed the height of the lower liquid as 160 mm and change ${d}_{{\rm eq}}$ and $\Lambda$.

2.2 Simultaneous measurement of two-phase flow fields and interface

In the present study, we need to measure the two-phase flow fields and liquid interface simultaneously, and we used the combined two-phase particle image velocimetry and laser-induced fluorescence. As shown in Fig. 1a, we used a high-speed camera (Speedsense M310, Dantec Dynamics), which was slightly tilted upward (approximately $5^\circ$) to minimize the meniscus effect at the sidewall of the tank, to acquire the images at the speed of 1000 frames per second. As a light source, plane LED (wavelength of 675 nm) and continuous wave (CW) laser (RayPower 5000, Dantec Dynamics) with 532 nm wavelength were used to generate the bubble shadowgraph image and to illuminate seeding particles (PMMA Rhodamine B, 1–20 μm) and fluorescent dye (Rhodamine B), respectively. Here, we used the seeding particles to evaluate the liquid-phase velocity vectors, and the fluorescent dye was used to highlight the interface and lower liquid dragged by the rising bubble. The camera was equipped with an orange-colored optical filter (bp590, MIDOPT) to reduce the laser reflected by bubble surface and a 100-mm macro lens (Samyang). Depending on the tested conditions, the field of view was varied to assure the capture of detailed interfacial phenomena, and the resulting spatial resolution is 39–68 μm/pixel, i.e., ($8.63-9.07\times {10}^{-3}$)${d}_{{\rm eq}}$.

Once we obtained the raw images (Fig. 2a) from the setup explained above, we conducted a series of digital image processing to extract and quantify the liquid–liquid interface, bubble statistics, and liquid-phase velocity field. To highlight the liquid–liquid interface, assisted by the fluorescent dye (Rhodamine B) added into the lower liquid, we applied the contrast and median image filters (Lim 1990), which increase the contrast and blur the seeding particles, respectively, and calculate the intensity gradient and Laplacian value of the image (Tropea et al. 2007). In each step, we changed the gradient threshold value to optimize the threshold and the result is shown in Fig. 2b. Next, every pixel below the liquid–liquid interface was filled (Fig. 2c), by which we could obtain the shape of the interface, as shown in Fig. 2d. As shown, the selection of threshold would affect the binarization and the related quantification, e.g., interface height (${h}_{{\rm max}}$), as shown in Fig. 2c. By changing the threshold value within ± 25% of the optimal value (Piao and Park 2019), the uncertainty of evaluating ${h}_{{\rm max}}$ was calculated to be within 0.5% (about 3% in maximum).

Fig. 2

Image processing to extract the liquid–liquid interface a–d and bubble shape e–h. In h, red-colored * denote the dominant points of the fitted ellipse bubble

To extract the bubble shape, on the other hand, we applied the gray-level threshold binarization (Otsu 1979) to raw image; examples of raw image and result are shown in Figs. 2e and f, respectively. Additionally, we used the built-in MATLAB drawing function for making up the blurred region when the bubble locates on the interface. Finally, we used the least-squared ellipse fitting to find the dominant points (Fitzgibbon et al. 1999; Zhang et al. 2012), and determined how much its major axis is tilted against the horizontal axis (Fig. 2g). The result of extracted bubble shape is shown in Fig. 2h. The uncertainty of evaluating the bubble size, by changing the threshold value, is estimated to be 1% (about 4% in maximum). After obtaining the bubble shape, we calculated its trajectory by tracking the bubble centroid, which is further interpolated using the quintic spline [with the tolerance of ${\varvec{O}}({10}^{-6}-{10}^{-7})]$. By differentiating the interpolated trajectory (Choi et al. 2019; Kim and Park 2019a; Oh et al. 2020), we measured the bubble velocity. In all the cases tested, the maximum spline error is less than 5% of bubble equivalent diameter. Finally, to measure the liquid-phase velocity field, we applied the Laplacian of Gaussian (LoG) filter to the raw image to find the correlation peak more clearly (Lee and Park 2020; Maeng and Park 2021). For the velocity vector evaluation, we used the conventional cross-correlation algorithm with fast Fourier transformation (interrogation window of $32\times 32$ pixels with 50% overlap). During the evaluation, we masked the area occupied by the bubble.

Following previous studies that measured the velocity fields using the PIV (Lawson et al. 1999; Choi and Park 2018; Lee and Park 2020), the uncertainty of the measured liquid-phase velocity, ${u}_{p}\simeq\Delta s/(M\Delta {t}_{sep})$, is analyzed. Here, $M$ is magnification factor obtained by calibration, $\Delta s$ is the particle displacement and $\Delta {t}_{sep}$ is the time separation between two consecutive images. Then, the percentage error $\delta ({u}_{p})$ can be expressed as $\delta ({u}_{p})={({\delta (M)}^{2}+{\delta (\Delta {t}_{sep})}^{2}+{\delta (\Delta s)}^{2})}^{0.5}$, based on the error propagation analysis (Maeng and Park 2021). In the present setup, the percentage error of the magnification factor is calculated up to 0.5%. The percentage error of time separation, where shooting time interval error of high-speed camera is 500 nsec, is 0.05% in 1 ms actual time interval. Lastly, the percentage error of particle displacement, which is calculated pixel resolution divided by the mean particle movement, is less than 2% as the pixel resolution is 0.1 pixel. Collecting all these contributions, the percentage error of liquid-phase velocity measurement is estimated to be less than 2.1%. Since the maximum percentage error due to the deviation in the refractive index is assumed to be approximately 2%, the total uncertainty of the measured liquid-phase is estimated as 3%. To ensure the repeatability of measured variables, on the other hand, all the experiments were conducted at least 9 times for each case.

3 Result and discussion

3.1 General description of the bubble–interface interaction

Before we discuss the details of bubble–interface interaction, in Fig. 3, we have shown the representative interfacial phenomena in time (the dimensionless time is defined as ${t}^{*}=t{U}_{b}/{d}_{{\rm eq}}$) before and after the bubble–interface contact (${t}^{*}=0)$. Shown in the figure is the case (i) of 50.7 wt% water–glycerin mixture and 10 cp silicone oil ($\Lambda$ = 1.38). Since the working fluids in the present study are much more viscous (about 6–7 time) than water, the range of equivalent diameter and Reynolds number, at which the path instability occurs, is different from the typical ranges reported for water (Aybers and Tapucu 1969). However, we also confirmed the similar transition of rising trajectory from the straight, zigzagging to rectilinear rocking motion, while increasing Reynolds number (i.e., bubble size). In general, the bubble path instability is attributed to the wake instability, i.e., transition of standing wake vortex (for linear path) to periodic shedding of wake vortices (for zigzagging or helical path), which occurs as the bubble size (i.e., bubble Reynolds number) increases (Mougin and Magnaudet 2002). On the other hand, the rocking motion occurs when the bubble size becomes quite large such that the bubble deformation is involved and the wake vortices are irregular (Brücker 1999).

Fig. 3

Representative sequential ($t^{*} = tU_{b} /d_{{\rm eq}}$) bubble–interface interaction (left, raw image; right, two-phase flow field with (dimensionless velocities and vorticity ($\omega^{*} = \omega d_{{\rm eq}} /U_{b}$) contour) with a, b zigzagging path at $\theta = - 6.8^\circ$ a and $28.8^\circ$ b; c straight path; d rocking motion. Here, case (i) (50.7 wt% water–glycerin mixture and 10 cp silicon oil) is shown. The scale bar in each figure denotes 5 mm, if not specified otherwise

For the case of the zigzagging path, it is found that the deformation of interface and associated bubble dynamics change significantly depending on the colliding angle ($\theta$), as shown in Fig. 3a and b, which will be our main focus. When $\theta$ is small (e.g., $\theta =-6.8^\circ$ in Fig. 3a), the rates at which the wake vortices, which is roughly symmetric, around the bubble dissipate and the drift liquid flow induced in front of the bubble is decelerated are relatively slow. Associated with these flow fields, the deformed liquid interface rises higher, which forces the bubble to pass through the liquid interface instantly. Even following the same rise path, the bubble approaching the interface with a larger $\theta$ (e.g., $\theta =28.8^\circ$ in Fig. 3b), the bubble tends to be trapped under the liquid interface for a certain time duration and escape in a delayed manner after the liquid film rupture. As shown in Fig. 3b, this seems to be related to the direction of wake vortices behind the bubble and faster liquid momentum dissipation in front of the bubble. As time goes by, the bubble proceeds toward the direction set by its path instability but its wake is convected to the opposite direction, delaying the film drainage. When the bubble rises in a linear path (Fig. 3c), it cannot pass the liquid interface instantly because of its small size, i.e., small momentum of bubble and induced liquid flow. Being associated with this, the height of the deformed interface is the smallest and the liquid flow around the bubble dissipates fastest. Like the case of the zigzagging path with $\theta =28.8^\circ$, the bubble is trapped under the interface for a while and escapes after the film rupture. Finally, the bubble rising in a rocking motion passes through the liquid interface instantly and the interface height becomes the highest, powered by the largest induced momentum (Fig. 3d).

From the above observation, it is clear that the interfacial deformation and bubble dynamics are strongly related to the path instability of the rising bubble. In Fig. 4, we have plotted the variation of passage time, ${t}_{p}$, taken for the rupture on the interface to appear from $t=0$ and the maximum height, ${h}_{{{\rm max}}}$, with $\theta$. It is noted that ${h}_{\mathrm{max}}$ corresponds to the height of liquid interface at $t={t}_{p}$ (at the instant of the interface of rupture), when the bubble escapes the interface instantly, while it is defined as the maximum rise distance of deformed interface for the case of delayed escape. For the zigzagging path, the bubble instantly escapes the interface when it collides the interface at $-25^\circ \lesssim \theta \lesssim 18^\circ$. Here, the higher maximum interface height is attained, i.e., the lower liquid is dragged and penetrates the upper liquid more. With increasing $\theta$ ($\theta \gtrsim 18^\circ$ or $\theta \lesssim -25^\circ$), the passage time increases sharply and ${h}_{{\rm max}}$ becomes lower. Interestingly, it is measured that the bubble escape parameters show a slight asymmetry against $\theta$, not following our intuition. We will discuss the reason for this in detail in Sect. 3.3.

Fig. 4

Variation of a bubble passage time ($t_{p}$) and b maximum height ($h_{{\rm max}}$) of the liquid interface according to colliding angle ($\theta$). Here, the viscosity ratio of two liquids (50.7 wt% water–glycerin mixture and 10 cp silicone oil) is 1.38. ▲, zigzagging path (instant escape); ○, zigzagging path (delayed escape); ♦, straight path; ▼, rocking motion

When the bubble rises in a straight or rocking motion, on the other hand, the trend of bubble shape variation and interface disruption is not dependent on the colliding angle. For the case of a straight path, the colliding angle is scattered near $\theta =0^\circ$, while it covers a wider range for the bubbles in a rocking motion, owing to its substantial wobbling motion at the instant of bubble–interface contact. As we will discuss below, however, the effect of $\theta$ on the bubble–interface interaction is negligible for the bubbles in a rocking motion.

In Fig. 4a, the passage time of bubble in a straight path is distributed broadly in 0.1–0.4 s (see Sect. 3.2 for further discussion); however, it takes almost constant ${t}_{p}$ of 0.05 s for the bubble in a rocking motion to escape the interface. Comparing two cases of instant escape, interestingly, the bubble passage time (interface height) is longer (higher) for the bubble rising in a rocking motion, which is attributed to the strong liquid flow induced in front of the bubble. As the interface rises much faster by the bubble in a rocking motion, its height becomes much higher, but the liquid layer between the interface and bubble is retained to be thicker, delaying the drainage process.

3.2 Bubble dynamics and interface deformation

In this section, based on the visualization of bubble–interface interaction, we discuss the bubble dynamics and interfacial deformation, shown in the previous section, in detail. If not specified otherwise, the results are shown for the case (i) of 50.7 wt% water–glycerin mixture and 10 cp silicone oil (viscosity ratio, $\Lambda = 1.38$), and the case (ii) with larger $\Lambda (= 12.8)$ will be compared in Sect. 3.4.

Figure 5 shows the sequential evolution of bubble–interface interaction, for the case of a bubble rising in a straight path (${d}_{{\rm eq}}= 3.06$ mm), far below liquid interface (Fig. 5a). As the bubble approaches the liquid interface, it starts to deform when the bubble centroid is approximately $1{d}_{{\rm eq}}$ away from it (Fig. 5b), at which the liquid layer between the bubble and the interface starts to be drained sideways owing to the high pressure there. However, the bubble deformation is not discernible yet. At ${t}^{*}=0$, the liquid film becomes much thinner and the bubble starts to deform, i.e., its aspect ratio decreases (Fig. 5c), and reaches the most deformed state (minimum aspect ratio) at ${t}^{*}=0.76$, from its undisturbed shape (Fig. 5a), while it is trapped under the interface (Fig. 5d). As time elapses further, the bubble shape is flattened (aspect ratio increases) and moves downward with the interface (Fig. 5e). When the rising bubble does not escape the liquid interface instantly, Lhuissier and Villermaux (2012) claimed that the Poiseuille flow thinning mechanism, i.e., viscous flow within the liquid film is driven by the gravity only and the drained flow is assumed to be Poiseuille flow, is no longer applicable, and instead, the viscous drainage, occurring near the meniscus of bubble, becomes dominant. For the present bubble rising in a straight path, it takes quite longer ($t_{p}$ is scattered in a wide range of $O\left( {10^{ - 1} - 10^{0} } \right)$ second) for the bubble to pass through the interface (Fig. 4a), and at $t^{*} \simeq 20$, the rupture of the interface occurs near the meniscus of the bubble (Fig. 5f), which agrees well with the previous studies. After the liquid interface ruptures, the liquid interface over the bubble retracts and the bubble rises above the interface dragging a portion of the lower liquid (Fig. 5g). As the bubble rises in the silicone oil (upper liquid), the bubble shape becomes more oblate than in the water–glycerin mixture (Fig. 5h) and rises in a straight path without experiencing the discernible deformation, owing to the higher viscosity of the upper liquid and reduced inertia spent on escaping the interface.

Fig. 5

Temporal evolution of interface ($\Lambda = 1.38$) and bubble ($d_{{\rm eq}} = 3.06$ mm) rising in a straight path. The scale bar in each figure denotes 5 mm, if not specified otherwise

For the zigzagging bubble, the interfacial phenomena change depending on $\theta$, and thus, the representative cases of delayed ($\theta = 24.4^\circ$) and instant ($\theta = - 4.5^\circ$) escapes are shown in Figs. 6 and 7, respectively. Similar to the case of the straight rise trajectory, the liquid interface starts to deform as the bubble in a zigzagging path approaches it within the distance of $\sim1d_{{\rm eq}}$ (Figs. 6b, 7a). When the delayed escape occurs, the liquid interface is substantially deformed along the horizontal direction as well as along the vertical direction as the bubble first interacts with the interface (Fig. 6c, d). Due to the increased surface area of the deformed interface, thus, the restoring force owing to the interfacial tension would strongly act on the bubble to stay under the interface for a while. This effect is more dominant as the liquid interface surrounding the bubble becomes symmetric (Fig. 6e). However, the liquid layer between the bubble and interface becomes thinner slowly by the viscous drainage, and the film rupture occurs near the meniscus at $t^{*} \simeq 13$ (Fig. 6f). Then, the escape of the bubble is accelerated by dragging the lower liquid slightly (Fig. 6g) and its shape is transitioned to a very oblate (aspect ratio of 3.0) (Fig. 6h). In the upper liquid, the bubble rises in a straight path, despite its oblate shape. This is again owing to the fast viscous dissipation of the surface vorticity around the bubble, which will eventually be shed into the wake and causes the path instability (Mougin and Magnaudet 2002; Lee and Park 2017). As can be expected from the trend of bubble escape, the process of interface deformation, film rupture, and bubble shape transition are quite similar to those of the cases with a straight path.

Fig. 6

Temporal evolution of interface ($\Lambda = 1.28$) and bubble ($d_{{\rm eq}} = 4.32$ mm) rising in a zigzagging path ($\theta = 24.4^\circ$). The scale bar in each figure denotes 5 mm, if not specified otherwise

Fig. 7

Temporal evolution of interface ($\Lambda = 1.38$) and bubble ($d_{{\rm eq}} = 4.56$ mm) rising in a zigzagging path ($\theta = - 4.5^\circ$). The scale bar in each figure denotes 5 mm, if not specified otherwise

Even with the bubble in the same rise path, the resulting phenomena are quite different if the bubble approaches the interface with a smaller $\theta$. When the bubble first interacts with the interface, the bubble drags the interface (and lower liquid) almost along the vertical direction (Fig. 7c), which is different from the case of delayed escape (Fig. 6d). As the bubble inertia (buoyancy) prevails, the film drainage is accelerated much and the bubble starts to escape the liquid as fast as $t^{*} \simeq 1.1$ (Fig. 7d). It is noted that the film rupture occurs closer to the apex of the bubble, rather than near the meniscus. As the bubble rises in the upper liquid, the penetration height of the lower liquid, dragged by the bubble, is higher than the delayed escape (Fig. 7e). Similar to previous cases, the bubble in the upper liquid maintains a highly oblate shape and rises in a straight path (Fig. 7f).

In the case of the bubble in a rocking motion, substantial wobbling of the bubble surface is accompanied (Fig. 8). Similar to previous cases, the effective bubble–interface distance at which the interface starts to be affected is $\sim1d_{{\rm eq}}$ (Fig. 8a), agreeing with the previous report such that the liquid flow displaced away from the rising bubble due to the potential flow effect propagates upstream approximately $1d_{{\rm eq}}$ (Ellingsen and Risso 2001). The vigorous shape instability is still active as the bubble interacts with the interface (Fig. 8b, c), and the film rupture occurs near the apex of the bubble (Fig. 8d), similar to the case of instant escape of zigzagging path. Powered by the strong inertia, after the film rupture, a thick liquid jet penetrates the bubble (Fig. 8e), as the lower liquid continues to rise upward. After the dragged lower liquid is separated (pinch-off) from the bubble owing to the hydrostatic pressure, the bubble rising in the upper liquid still wobbles but its shape becomes flat as much as the bubble rising in a zigzagging path (Fig. 8f). After the pinch-off, a lump of lower liquid sinks downward and stays above the interface quite long without coalescing with the bulk lower liquid (Fig. 8g). It was observed that the liquid lump merges to the bulk, which recovers the initial undisturbed state at $t^{*} \sim{\varvec{O}}\left( {10^{3} } \right)$. On the other hand, the jet penetration of lower liquid through the bubble has been observed in previous studies, as well (Uemura et al. 2010; Mao et al. 2020). They showed that the liquid interface deforms downward owing to the interfacial tension and drives lower liquid upward after the film rupture occurs. The velocity of this jet is much faster than that of the bubble so that the jet penetrates the bubble. After some time, the jet becomes thinner and pinch-off occurs inside of the bubble, generating a droplet. When the bubble cannot pass through the liquid interface instantly, in the present experiments, the jet penetration does not occur; however, it was occasionally observed in the case of zigzagging bubble, despite its smaller bubble size, escaping the interface fast (for the details, see Fig. S1 in the supplementary material). For the case of instant escape in the oscillating path, the ruptured liquid film shrinks fast toward the bubble base and induces a vertical jet of lower liquid through the bubble. Since the velocity of the jet is as fast as ${\varvec{O}}\left( {10^{0} } \right)$ m/s, much faster than that of the bubble (~ ${\varvec{O}}\left( {10^{ - 1} } \right)$ m/s), it is possible to penetrate the bubble (Uemura et al. 2010). Since the jet width is much thinner in this case, mostly, the jet pinch-off tends to occur inside the bubble and a droplet of lower liquid is generated in front of the bubble, which moves upward together. Due to the induced bubble wake and droplet inertia, the liquid droplet may continue to rise for some time but eventually sinks slowly due to its higher density. Similar to the case shown in Fig. 8g, it takes quite a long for the droplet to merge with the bulk liquid and recover the initial state.

Fig. 8

Temporal evolution of interface ($\Lambda = 1.38$) and bubble ($d_{{\rm eq}} \simeq 8.0$ mm) rising in a rocking motion. The scale bar in each figure denotes 5 mm, if not specified otherwise

In Fig. 9, we have quantified the dynamics of bubble and interface in time, comparing four cases that we have discussed above, which is non-dimensionalized using size ($d_{{\rm eq}}$) and rise velocity ($U_{b}$) of the bubble. In the figure, the uncertainty range of each symbol denotes the standard deviation obtained by collecting more than 9 times of independent measurements. From the time trace of vertical bubble position, it is clearly distinguished between the cases with an instant escape (bubble in a zigzagging path with a lower $\theta$ and rocking motion) and delayed escape (bubble in a zigzagging path with a higher $\theta$ and straight path) from the interface (Fig. 9a). Along the horizontal direction, the rising bubble in the upper liquid shows almost no movement, i.e., rises in a straight path without a path instability (Fig. 9b). It is noted that as shown in Fig. 8f, the bubble still wobbles after escaping the liquid interface and the horizontal bubble position oscillates slightly. Due to the increased viscosity of the upper liquid and the loss of kinetic energy during the interaction with the liquid interface, the bubble Reynolds number in the upper liquid decreases to O(10¹) from O(10²) in the lower liquid (Table 2). Therefore, the shear layer around the bubble becomes quite short (being attached to the bubble) and thick, which does not experience instability. At the flow around the sphere and circular cylinder at a low Reynolds number of O(10¹), it is well known that wake is steady and attached without an instability (shedding) (Zdravkovich and Bearman 1997), and we have also measured the standing wake structure behind a rising bubble in the upper liquid when viscosity ratio is ${\Lambda } = 1.38$ (Fig. 10). When the bubble (zigzagging path in the lower liquid) escapes the interface instantly, a pair of symmetric (standing) counter-rotating vortex is attached to the bubble surface (Fig. 10a). For the bubble rising in a rocking motion, owing to the larger momentum of the bubble and substantial wobbling, the wake vortices are slightly asymmetric but still attached to the bubble (Fig. 10b).

Fig. 9

Dynamics of rising bubble and interface ($\Lambda = 1.38$): a vertical bubble position; b horizontal bubble position; c velocity bubble velocity; d horizontal bubble velocity; e aspect ratio; f maximum interface height. black diamond, straight path ($d_{{\rm eq}} \simeq 3.14$ mm); circle, zigzagging path with an instant escape ($d_{{\rm eq}} \simeq 4.37$ mm); black triangle, zigzagging path with a delayed escape ($d_{{\rm eq}} \simeq 4.27$ mm); black inverse triangle, rocking motion ($d_{{\rm eq}} \simeq 8.0$ mm)

Fig. 10

Wake structure behind a rising bubble in the upper liquid: a bubble rising in a zigzagging path ($\theta = - 4.5^\circ$); b bubble rising in a rocking motion. Here, the viscosity ratio is $\Lambda = 1.38$

By differentiating the bubble trajectory, we obtain the vertical and horizontal components of the bubble velocity in Fig. 9c and d, respectively. Following the bubble position, the trend of rising velocity variation is classified according to the passage time ($t_{p}$); it is decelerated substantially at $t^{*} \simeq 0$, for the cases with a delayed escape. As shown in Fig. 11a, for the case of delayed escape ($\theta = 24.4^\circ$, for example), the region of higher vertical liquid velocity ($u_{l,z} )$, induced by the wake vortices, behind the bubble is formed away from the bubble base pushing the bubble along the horizontal direction more (Fig. 9d). On the other hand, with a smaller $\theta = - 4.5^\circ$, the higher-$u_{l,z}$ the region is maintained right behind and in front of the bubble (Fig. 11b), emphasizing that the orientation of the wake vortices, i.e., bubble, is an important key. Interestingly, when the bubble rises in the upper liquid, whether from the delayed or instant escape, the terminal velocity is almost identical (Fig. 9c). Previously, Raymond and Rosant (2000) showed that the terminal velocity of a bubble, rising in a liquid of which the viscosity is about 10–40 cp (= mPa·s), is almost identical when the bubble size is over 3.0 mm.

Fig. 11

Liquid-phase flow field (velocity vectors and vertical velocity contour) evolution (left, $t^{*} = 0$; right, $t^{*} = 0.47)$ during the interaction between the interface ($\Lambda = 1.38$) and bubble rising in a zigzagging path: a delayed escape ($\theta = 24.4^\circ$); b instant escape ($\theta = - 4.5^\circ$)

In Fig. 9e, we plotted the temporal variation of the bubble aspect ratio (AR), defined as the ratio of maximum spanwise length to the maximum streamwise length, as an indirect measure of bubble deformation. Here, the streamwise and spanwise direction is parallel and perpendicular to the direction of bubble movement, respectively. For the ellipsoidal bubble, rising in a straight and zigzagging path, thus, AR is identical to the ratio of major to minor axis length. As we have shown, the bubble deformation is closely related to the bubble–interface interaction and bubble escape dynamics. All the cases show that AR starts to decrease at $t^{*} \simeq 0$, as it is shown in Figs. 5, 6, 7 and 8. As the bubble escapes the liquid interface, AR increases sharply (its timing is different depending on the passage time) and saturates to a certain value (except the case of rocking motion), which is larger than the initial value in the lower liquid. It is further found that the terminal aspect ratio is smallest for the case of a straight path and largest for the case of a zigzagging path with an instant escape. According to Liu et al. (2015), the bubble aspect ratio depends on Weber and Bond numbers, when the viscous force is negligible compared with the surface tension and inertial forces (i.e., $Oh \ll 1.0$). According to their model, $AR = { }1/\left( {0.9948 + 0.1535We^{1.7047} Bo^{ - 0.4052} } \right)$, it is estimated that the aspect ratio of the bubble in a straight path is approximately 1.86 and 2.14 in lower and upper liquids, respectively. In the case of the oscillating path, the aspect ratio is estimated as 2.17 and 2.6 in the lower and upper liquids, respectively, which reasonably agrees with our measurement. Finally, the variation of dimensionless maximum height of lower liquid ($h_{\max } /d_{{\rm eq}}$) in time is plotted in Fig. 9f. In the cases of a delayed escape, it is found that the maximum liquid height is approximately the bubble diameter; maximum value occurs before the film rupture since the bubble loses the vertical momentum significantly after the film rupture. However, for the case of an instant escape, $h_{\max } /d_{{\rm eq}}$ reaches up to 2.0–5.0, after the film rupture, because the bubble still has the vertical momentum large enough to draw the lower liquid.

3.3 Further discussion about the delayed and instant escape

To relate the bubble escape (interfacial deformation) and its path (including the wake-induced flow) in detail, first, we calculate the momentum transferred from the rising bubble. Previously, the ratio of inertia force to the surface tension was used to study the criteria for the rising bubble or sinking particle to pass through the liquid interface (Greene et al. 1988; Magnaudet and Mercier 2020). In the present study, based on the two-dimensional velocity field, we compare the momentum flux of the liquid flow. To consider the effect of bubble orientation (i.e., path instability) at the instant of bubble–interface interaction, we define the reference area, as shown in Fig. 12. Here, $x^{\prime}$- and $z^{\prime}$-axes correspond to the direction of major and minor axes of the bubble, respectively, and the rectangular area of interest is bounded by $- 2 \le x^{\prime}/d_{{\rm eq}} \le 2$ and $- 2 \le z^{\prime}/d_{{\rm eq}} \le 1$, which was determined by the typical size of the bubble-induced flow (Ellingsen and Risso 2001). Along $z^{\prime}$-direction, then we calculate the liquid-phase momentum flux (per unit length) as $\dot{M}_{z^{\prime}} = \rho_{l} \smallint u_{z^{\prime}}^{2} dx^{\prime}$ (Heathcote et al. 2004; Park et al. 2016), which is the main source to drive the thinning of liquid film between the bubble and the liquid interface.

Fig. 12

Definition of the reference area to calculate the momentum flux of the liquid flow and bubble momentum

The profiles of liquid momentum flux along $z^{\prime}$-direction, normalized as $\dot{M}_{z^{\prime}}^{*} = \dot{M}_{z^{\prime}} /\dot{M}_{b}$, where $\dot{M}_{b} = \rho_{g} gA_{b}$ denotes the bulk bubble momentum, are shown as a function of time ($t^{*}$), in Fig. 13. For the case of a straight path (Fig. 13a), $\dot{M}_{z^{\prime}}^{*}$ decreases sharply in all positions as time goes by, which indicates that the dissipation of liquid momentum occurs fast both at the front and rear of the bubble. In Fig. 13b, it is clear that the liquid momentum for the delayed escape (in a zigzagging path) decreases significantly after $t^{*} = 0$, which is attributed to the misalignment between the moving direction of bubble and wake behind bubble such that the liquid momentum acts perpendicular to the bubble advancement. This makes the right-hand side of the liquid interface rises upward (see Fig. 15b) and disturbs the film thinning at the apex of the bubble. On the contrary, for the instant escape (Fig. 13c), the larger liquid momentum is retained at $z^{\prime}/d_{{\rm eq}} = 0.5$ further up to $t^{*} = 0.5$, which assists the bubble to rise upward without a significant deceleration (Fig. 9c); it is almost zero at $t^{*} > 0$ for the delayed escape (Fig. 13b). When the bubble rises in a rocking motion (Fig. 13d), $\dot{M}_{z^{\prime}}^{*}$ in front of the bubble is quite small; however, the momentum of the bubble itself is quite larger than in other cases so that the bubble can rise without a significant deceleration (Fig. 9c). Also, a non-negligible $\dot{M}_{z^{\prime}}^{*}$ is retained over $t^{*} = 0.5$, by which the liquid interface keeps rising upward even after the film rupture.

Fig. 13

Profiles of dimensionless liquid momentum flux in time: a straight path b zigzagging path with a delayed escape; c zigzagging path with an instant escape; d rocking motion. black circle, $t^{*} = - 0.5$; black square, $t^{*} = 0$; black triangle, $t^{*} = 0.5$, diamond, $t^{*} = 1.0$; black inverse triangle, $t^{*} = 1.5$. For each case, the bubble momentum varies as 0.074–0.087, 0.120–0.160, 0.137–0.182, and 0.378–0.468, respectively

Based on the above results, we estimate the total momentum ($\dot{M}_{T}$) that the liquid interface receives from the bubble and flow induced by it, as a function of time. Based on the same reference area shown in Fig. 12, it is calculated as $\dot{M}_{T} \left( {t^{*} } \right) = \dot{M}_{b} \left( {t^{*} } \right)\cos \theta + \mathop \sum \limits_{{z^{\prime} = - 2d_{{\rm eq}} }}^{{z^{\prime} = d_{{\rm eq}} }} \dot{M}_{z^{\prime}} \left( {t^{*} } \right)$. As shown in Fig. 14, it is clear that the magnitude of the momentum added to the liquid interface increases with increasing the bubble size before colliding the interface. For the bubble in an oscillating path, on the other hand, total momentum varies even with the similar bubble size, which is attributed to the deviation between the direction of bubble advancement and wake-induced flow (Fig. 11). For the bubble with an instant escape (in a zigzagging path), $\dot{M}_{T}$ remains almost 1.0 before the film rupture occurs ($t_{p}^{*} \simeq 1.1$). For the bubble with a delayed escape, $\dot{M}_{T}$ is similar to the instant escape case before the bubble–interface collision, but it decreases sharply in time at $t^{*} > 0$; in particular, it is smaller than that of straight path case after $t^{*} = 0.5$. In the case of a rocking motion, the total momentum is quite large compared to other cases owing to its large size but the amount of momentum reduction is also the largest. It is because the larger amount of energy is lost to the substantial deformation of the liquid interface.

Fig. 14

Temporal change of the total momentum transferred to the liquid interface ($\Lambda = 1.38$): black diamond, straight path ($d_{{\rm eq}} \simeq 3.14$ mm); black triangle, zigzagging path with an instant escape ($d_{{\rm eq}} \simeq 4.37$ mm); circle, zigzagging path with a delayed escape ($d_{{\rm eq}} \simeq 4.27$ mm); black inverse triangle, rocking motion ($d_{{\rm eq}} \simeq 8.0$ mm)

In Fig. 4, it was interesting to see that the variation of passage time ($t_{p}$) and the maximum height of liquid interface ($h_{\max }$) are not symmetric about the colliding angle, $\theta$, and we are to discuss this further. We chose two cases of $\theta = - 18.5^\circ$ and $21.5^\circ$, of which the absolute value is similar but belongs to the instant and delayed escape, respectively (Fig. 4). In Fig. 15, we show the temporal variation of the bubble advancement and interface shape with the wake structure behind the zigzagging bubble, according to $\theta$. For both cases of $\theta$, the initial ($t^{*} \simeq 0$) shape of interface deformed by the rising bubble looks to be similar, i.e., the left-hand and right-hand side of the bubble becomes convex downward for $\theta = - 18.5^\circ$ (Fig. 15a) and $\theta = 21.5^\circ$ (Fig. 15b), respectively. As the time goes on, however, the curvature of the interface changes to concave one for $\theta = 21.5^\circ$, while the convex shape is retained further for $\theta = - 18.5^\circ$. The maximum height of the elongated liquid interface is slightly higher for the case of $\theta = - 18.5^\circ$. As shown in Fig. 15a, for the negative $\theta \left( { = - 18.5^\circ } \right)$, the direction of the wake-induced flow tends to be aligned parallel to the bubble advancement direction, which adds the momentum along that direction and does not push the liquid interface upward. With a positive $\theta \left( { = 21.5^\circ } \right)$, on the other hand, the pair of wake vortex becomes highly asymmetric and the wake-induced flow is directed upward at the right-hand side of the bubble, forcing the interface upward instead of the bubble (Fig. 15b). Based on this comparison, it is understood that the evolution of bubble orientation, i.e., the direction of the wake-induced flow, is the key of making different escape dynamics of the bubble rising along the same zigzagging path. As shown in Fig. 15c, in the same zigzagging path, the bubble with a negative $\theta$ at the instant of bubble–interface interaction proceeds toward decreasing $\left| \theta \right|$, i.e., the wake-induced flow is more aligned to the vertical direction. However, with a positive $\theta$, the bubble migrates with increasing $\left| \theta \right|$ so that the bubble-induced momentum from the highly asymmetric wake structure acts more toward the horizontal direction.

Fig. 15

Temporal variation of two-phase flow (vorticity contour) and interface ($\Lambda = 1.38$) shape for the bubble in a zigzagging path with a an instant ($\theta = - 18.5^\circ$) and b delayed ($\theta = 21.5^\circ$) escape. c Schematic for the bubble orientation, rising along the zigzagging path

To sum up, when the bubble reaches the liquid interface with a positive colliding angle, due to the misalignment between the advancing direction bubble and the wake-induced flow, the delayed escape occurs at the relatively smaller $\theta \simeq 18^\circ$ (Fig. 4). With a negative colliding angle, on the contrary, the liquid-phase flow induced by the par if wake vortex is well aligned with the direction of bubble movement, so that the instant escape can be retained further up to $\theta \simeq - 25^\circ$.

3.4 Effect of viscosity ratio

So far, we have considered 50.7 wt% water–glycerin mixture and 10 cp silicone oil (viscosity ratio, ${\Lambda } = 1.38$) as the lower and upper liquids, respectively, and we change the upper liquid to 100 cp silicone oil. To keep the condition of refractive index matching, the composition of lower liquid was modified from 50.7% water–glycerin mixture to 53.3% water–glycerin mixture (Tables 1 and 2), which results in the viscosity ratio of $\Lambda = 12.8$. Since the change of the amount of glycerin added to the water did not change the mixture property significantly, we confirmed that the bubble statistics including the rise velocity, equivalent diameter, and non-dimensional parameters are almost identical for both cases (Table 2). Thus, it is understood that the initial conditions (e.g., the path instability) of the bubble interacting with the liquid interface of ${\Lambda } = 1.38$ and $12.8$ are the same.

As the viscosity ratio increases, the major difference in the bubble–interface interaction is that the bubble rising in a zigzagging path does not escape the interface instantly, even with a smaller colliding angle. Only the bubble in a rocking motion escaped the interface instantly. In Fig. 16a, we find that the passage time of the bubble rising in a zigzagging or straight path is quite widely scattered at ${\varvec{O}}\left( {10^{ - 1} - 10^{0} } \right)$ second with $\theta$. Comparing with the case of instant escape (Fig. 4), it is judged that all the bubbles belong to the case of delayed escape. However, in a rocking motion, bubble passage time is below 0.1 s, indicating the instant escape. We can also find that the maximum height of liquid interface is reduced compared with the case of lower viscosity ratio, and almost constant across the range of measured $\theta$ in zigzagging and straight paths (Fig. 16b). The enhanced viscous dissipation becomes more dominant than the effect of wake-induced flow behind the bubble, and thus, the dependency on the bubble orientation is mitigated. Similar enhancement of viscous dissipation owing to the increase of upper liquid viscosity has been reported by Feng et al. (2016). However, in the case of a rocking motion, the enhanced viscous dissipation reduces the total momentum that the liquid interface receives so that the passage time increases, which causes the maximum height of the liquid interface to increase compared to the case of ${\Lambda } = 1.38$.

Fig. 16

Variation of a bubble passage time ($t_{p}$) and b maximum height ($h_{{\rm max}}$) of the liquid interface according to colliding angle ($\theta$), for the zigzagging path. Here, the viscosity ratio of two liquids (53.3 wt% water–glycerin mixture and 100 cp silicone oil) is 12.8. ○, zigzagging path (delayed escape); ♦, straight path; ▼, rocking motion

To investigate the effect of increased viscosity in detail, we quantified the dynamics of the bubble and liquid interface (Fig. 17). It is seen that only the bubble in a rocking motion can pass through the liquid interface instantly (Fig. 17a, b), but the overall trend of bubble trace in time, whether it is a delayed or instant escape, is quite similar to the case of lower viscosity ratio. In terms of the bubble velocity, only the case of rocking motion does not experience a sharp deceleration (Fig. 17c), but the oscillation of the bubble velocity, which was observed in the case of ${\Lambda } = 1.38$, is not observed here because of the enhanced viscous dissipation (Fig. 17c, d). While rising in the upper liquid, the terminal velocity of the bubble is faster with a larger bubble size, agreeing with the numerical result of Hua and Lou (2007). Contrary to the trace and velocity of the bubble, in Fig. 17e, the bubble aspect ratio shows quite a different trend compared with the case of lower viscosity ratio (Fig. 9e). As the viscous force becomes comparable to the surface tension and inertial forces (i.e., $Oh \sim {\varvec{O}}\left( {10^{ - 1} } \right)$), the bubble aspect ratio is almost the same in lower liquid; however, it becomes quite different in the upper liquid. For the case of ${\Lambda } = 1.38$, except the rocking motion, the shape of the bubble was oblate ellipsoidal and AR was different depending on the rising path, but with increasing ${\Lambda } = 12.8$, the shape of the bubble becomes closer to a circle with the same aspect ratio regardless of the rising path. For the rocking motion, when ${\Lambda }$ changes from 1.38 to 12.8, the shape of the bubble changes from a spherical cap with substantial wobbling motion to an oblate ellipsoidal cap bubble without a wobbling motion. As we have explained, the maximum interface height has decreased in all cases, especially in a rocking motion, the maximum height has decreased approximately 30% compared with the lower viscosity ratio case. Associated with this, the penetration of the lower liquid jet through the bubble, which has been observed for the cases with an instant escape, has not been observed for the higher viscosity ratio. Finally, we calculated the momentum transferred to the interface from the bubble (see Fig. S2 in the supplementary material). As expected, the liquid momentum flux dissipates quite faster than the case of ${\Lambda } = 1.38$. Although the bubble-induced momentum flux is reduced a lot for the case of rocking motion, the liquid interface still receives a momentum sufficiently large for the film rupture to occur instantly.

Fig. 17

Dynamics of rising bubble and interface ($\Lambda = 12.8$): a vertical bubble position; b horizontal bubble position; c velocity bubble velocity; d horizontal bubble velocity; e aspect ratio; f maximum interface height. ♦, straight path ($d_{{\rm eq}} \simeq 3.02$ mm); ○, zigzagging path ($d_{{\rm eq}} \simeq 4.45$ mm); ▼, rocking motion ($d_{{\rm eq}} \simeq 8.0$ mm)

4 Concluding remarks

In the present study, we experimentally investigated the conditions with which the rising bubble may escape the interface between two immiscible liquids instantly or delayed. By varying the bubble size (i.e., its rise path as straight, zigzagging, and rocking motion) systematically under the different viscosity ratio of two liquids, we simultaneously measured the spatiotemporal variation of bubble dynamics, interface morphology, and the liquid-phase flow, using the combined two-phase particle image velocimetry and laser-induced fluorescence. The key to this successful measurement was a rigorous process of refractive index matching between two liquids. Obviously, it is more likely that the larger bubble tends to escape the liquid interface instantly; however, the bubble rising in an oscillating path (path instability) may undergo the delayed escape as well as the instant departure, even with the same bubble size. Based on the detailed analysis of the bubble–interface interaction and estimation of the bubble-induced momentum flux transferred to the liquid interface, we found that the key to classifying the escape pattern is the alignment of the wake vortices behind the bubble. It is well understood that the wake instability, that is the asymmetric shedding of the wake vortices, is the main origin of the bubble path instability. Along the oscillating rise path, thus the direction of the liquid-phase flow induced by these wake vortices changes together. As a result, depending on the orientation of the bubble at the instant of bubble–interface collision, the wake-induced flow may push the bubble upward (instant escape) or only deform the interface (delayed escape). On the other hand, with a higher viscosity ratio, it is found that the stronger viscous dissipation makes the effect of the bubble path negligible; in this case, only a very large bubble (rocking motion) was able to escape the interface instantly.

While the disruption of the liquid–liquid interface by the rising (or falling) bubble (or droplet/particle) has been investigated for a while, the comprehensive analysis based on the bubble dynamics, interface morphology and liquid-phase flow structure has not been performed so far. We believe that the present approach would be very useful in studying the interfacial phenomena occurring in a complex-fluid system, and we are currently extending the present work toward the interaction between the rising bubble and miscible liquid interface.