2.1. Experimental set-up

The experimental apparatus is shown in figure 1. A cylindrical Plexiglas tank of diameter 11 cm and length 10 cm is rotated around its horizontal axis $z$. The tank is fixed in a cylindrical counter bore and the contact is made using ball bearings. The tank is entrained by a motor, via a tooth belt. For this experiment, the range of rotational velocities $\omega$ investigated is [600–900] r.p.m., i.e. from 63 to 94 rad s$^{-1}$. Three holes on the side of the tank are used to fill the tank with water, or to inject an air bubble. The water used here is demineralized water similar to the one previously used in Rastello et al. (Reference Rastello, Marié, Grosjean and Lance2009). It is characterized by a resistivity of 0.3 M$\Omega$ cm. This resistivity is in between that of the ultra-purified water of Duineveld (Reference Duineveld1995) (18 M$\Omega$ cm) and that of tap water (3 k$\Omega$ cm).

Figure 1.

(a) Sketch of the experimental set-up showing the positioning of the cameras relative to the rotating tank. (b) Configuration of the present problem, showing an approximately axisymmetric bubble lying close to the axis of rotation $z$.

Because of operating constraints (bubble injection, temperature measurements), it was extremely difficult to keep this water clean, which means it a priori contains contaminants. These contaminants are mainly solid impurities and/or traces of tensio-actives. Surface tension was measured with a pendant drop tensiometer (Attension Theta Flex, Biolin Scientic AB), and was close to $71.8 \pm 1.0$ mN m$^{-1}$ for all experiments. The liquid temperature was measured before each series of experiments (i.e. each bubble injection), with a digital Testo 106 thermometer. This temperature was comprised between 20 $^{\circ }$C and 21 $^{\circ }$C for each experiment. A small short-term increase of temperature, of at most one degree, was observed in the course of measurements. The corresponding uncertainty on viscosity is expected to be below 5 %. The flow without bubbles was characterized by particle image velocimetry (PIV) measurements in Rastello et al. (Reference Rastello, Marié, Grosjean and Lance2009) on the present experimental set-up. Averages of 100 flow fields showed that the mean flow profiles were linear over the whole section of the tank for every rotation rate and well matched with the velocity of the tank at the wall. This was illustrated by the profile at 400 r.p.m. in figure 3 of this reference. Given that the rotation speeds are much higher in the present study, it appeared useful to provide in figure 2 the profiles up to 900 r.p.m. to attest the quality of the flow at these high rotation speeds.

Figure 2.

Mean tangential velocity profiles scaled by the rotation speed of the tank.

Two cameras are used to record the bubble shape and position: a Phantom 4.3 V360 is used to record images normal to the axis of rotation (side view in figure 1), and in particular the stretching of the bubble along this axis of rotation. A second camera, Basler acA800 is positioned perpendicular to the first, along the axis of rotation (front view in figure 1). Lightning is achieved with two LED panels, one for each camera. The cameras are synchronized to record simultaneously the bubble at a frame rate of $F = 200$ Hz. The resolution is fixed at 600 $\times$ 800 pixels for both cameras. For each injected bubble volume and given $\omega$, a set of 255 synchronized images is recorded. The recorded images are then processed to extract the contour of the bubble on both side/front views, after appropriate calibration accounting for bubble position and refraction due to the cylindrical walls of the tank.

Bubbles can be injected when the cell is at rest, with three different fixed needle Hamilton syringes. In addition, the volume $V$ at a given $\omega$ can be measured via image processing, by assuming that the bubble is an ellipsoid and by measuring its axes on the front and side view projections. Measurements of volume at low $\omega$ are typically within 5 % of the injected volume. For the larger $\omega$ investigated, a small increase in the volume of the bubble with $\omega$ is typically observed, up to 15 % for most series. In the following, each series of data points, corresponding to a same injected bubble, is labelled by the mean volume $V$ measured with this method over the range of $\omega$. Note that, for all the series, the volume of the bubble is very small compared with the volume of the cell (the ratio of volumes goes from $10^{-7}$ to $3\times 10^{-4}$ for the biggest bubble). In order to be more explicit on the relevant bubble length scale for each series, we provide in table 1 a correspondence between $V$ and the mean radius $R_{eq}$ of an equivalent bubble with a spherical shape.

Table 1.

Characteristics of the bubbles: mean volume $V$ and radius $R_{eq}$ of a spherical bubble of equivalent volume.

2.2. Bubble equilibrium – measurements

When the cell is rotated, the bubble migrates towards the axis of rotation. For moderate rotation frequencies, we observe that bubbles oscillate around their mean position with an amplitude that is large compared with the bubble size. This is illustrated in figure 3, with the red curve showing the vertical $y$ position of the centre of a bubble of volume $V=0.14$ cm$^{3}$ along the axis of the cell as a function of time, for $\omega =31$ s$^{-1}$. The value of $y$ is made dimensionless with $R_{eq}=3.2$ mm, the radius of a spherical bubble of equivalent volume $V=0.14$ cm$^{3}$. The bubble exhibits strong oscillations around its mean position, of an amplitude comparable to the bubble size. The frequency of the vertical oscillations corresponds to the frequency of the rotating cell. When $\omega$ is increased up to $\omega =89$ s$^{-1}$, the bubble moves closer to the axis of the cell: for this larger frequency, the amplitude of the bubble oscillations is strongly reduced, and becomes small compared with $R_{eq}$ (blue curve). The oscillations are also noticeably more regular.

Figure 3.

Variation of dimensionless vertical position of a bubble of volume $V=0.14$ cm$^{3}$ ($R_{eq}=3.2$ mm) as a function of time, for two rotation rates. The origin of $y$ is taken on the axis of the rotating cell, and the vertical position is made dimensionless with the equivalent spherical bubble size $R_{eq}$. The fluctuations are much smaller, and more regular, for the larger rotation rate.

A closer look at the motion of the bubble shows that it follows a limit cycle around its mean position (figure 4). This limit cycle is reminiscent of the behaviour observed for rigid spheres in the experiments of Bluemink et al. (Reference Bluemink, Lohse, Prosperetti and Van Wijngaarden2010) and Sauma-Pérez et al. (Reference Sauma-Pérez, Johnson, Yang and Mullin2018), even though the latter experiments have been carried out at much lower Reynolds number $Re$. Figure 4 shows the position of the centre of the bubble for the conditions of figure 3, namely $V=0.14$ cm$^{3}$ and $\omega =31$ and 89 rad s$^{-1}$, illustrating the associated limit cycles. The path of the bubble centre for the larger rotating rate spans a much smaller area than for the smaller rotating rate and exhibits a limit cycle with a regular pattern contrary to the more exploratory pattern for the case of the smaller rotating rate.

Figure 4.

(a) Path followed by the centre of the bubble with $V=0.14$ cm$^{3}$ and two experiments at $\omega =31$ and 89 rad s$^{-1}$. Time goes from dark to light colour in the gradient colour line. (b) Zoom on smaller cycle for the case $\omega =89$ rad s$^{-1}$.

The fact that larger and more chaotic fluctuations in position are observed for a reduced rotation frequency could be due to the different flow configuration in this case. In the $\omega = 31$ s$^{-1}$ case the bubble is much farther from the axis of rotation than in the $\omega = 89$ s$^{-1}$ case (mean distance from the axis of approximately $2R_{eq}$ in the first case, and of the order of $0.25R_{eq}$ in the second). This means that, at $\omega = 31$ s$^{-1}$, the bubble will be exposed to a different flow than in the vortex-like high frequency case, namely a configuration where a strong coupling is expected between the bubble and its own wake, via the rotating flow. The Reynolds number of the bubble of figures 3 and 4 is in this case of the order of 1000: we think the bubble, which is also not spherical and experiences moderate but significant shape fluctuations in this low frequency case, could be destabilized by its own wake.

A second possible reason for these fluctuations at low $\omega$ could reside in the mechanism evidenced by Phillips (Reference Phillips1960), which predicts that, if centrifugal forces are not strong enough to balance gravity, a cylindrical air cavity placed at the centre of a solid-body rotation flow will become unstable. The order of magnitude of the frequency $\omega _c$ beyond which a cavity of radius $R$ is stabilized is $\omega _c \sim \sqrt {g/R}$. For $R=3.2$ mm (the size of the bubble for which oscillations are observed in figure 3), this predicts $\omega _c \sim 55$ s$^{-1}$ which falls precisely between the stable and unstable cases.

Our objective in the following is to focus on the mean position of the bubble as a function of rotation frequency and bubble size: we therefore chose to focus on $\omega$ in the range [63–94] s$^{-1}$, for which the amplitude of the bubble oscillations remains moderate compared with the bubble dimensions. The upper value of 94 s$^{-1}$ corresponds to the maximum $\omega$ that can be reached with the motor entraining the cell. All the quantities introduced in the following sections related to the bubble size and position are measured for each image, and then averaged over the total number of images recorded by each camera. The standard deviation around these averaged values will be indicated by the error bars.

We will assume in the modelling that the bubble is axisymmetric, of characteristic lengths $L$ and $D$ (figure 1b). As illustrated in figures 5 and 6, for the larger bubbles investigated this assumption is not strictly valid, but the aspect ratio measured on front view projections remains smaller than 1.4 even for the larger bubbles and for all $\omega$ investigated. For each image the equivalent bubble diameter $D$ for a given injected volume and $\omega$ is then defined from the front view projection as the mean value between the minor axis and major axis dimensions (as measured with the Matlab regionprops function). The length $L$ is directly measured as the major axis from the side view projection, with the same Matlab function.

Figure 5.

(a) Front view for a bubble of volume $V=0.25$ cm$^{3}$ ($R_{eq}=3.9$ mm), at $\omega \approx 63$ s$^{-1}$. (b) Same bubble and same conditions simultaneously recorded from side view.

Figure 6.

(a) Front view for a bubble of volume $V=0.25$ cm$^{3}$ ($R_{eq}=3.9$ mm), at $\omega =89$ s$^{-1}$. (b) Same bubble and same conditions simultaneously recorded from side view.

2.3. Dimensionless parameters

We now wish to identify the parameters, and corresponding dimensionless numbers, needed to describe the equilibrium position and the shape of a bubble of volume $V$ placed in a solid-body cylindrical rotational flow. As mentioned before, we assume the bubble is axisymmetric, and characterize its shape with two length scales: a length scale $L$ corresponding to the (supposedly larger) dimension of the bubble along the axis of rotation $z$, and the smaller length scale $D$, corresponding to the diameter of the bubble projection in a plane normal to the axis of rotation (see figure 1).

We introduce two dimensionless numbers to describe the shape of the bubble: the aspect ratio $X=L/D$, which measures the stretching of the bubble, and the ratio $\alpha = V/(LD^{2})$. The latter characterizes the form of the bubble in a section containing the rotation axis $z$: it is for example expected to be equal to ${\rm \pi} /6$ if the bubble is an ellipsoid, or to ${\rm \pi} /4$ if the bubble is a cylinder. Note that the cylindrical shape is the limit shape predicted for very large $\omega$ in the model of Rosenthal (Reference Rosenthal1962).

The physical control parameters characterizing this problem are the liquid density $\rho$, the acceleration due to gravity $g$, the rotational velocity of the tank $\omega$, the gas–liquid surface tension $\sigma$ and $\mu$, the dynamic viscosity of the liquid. In addition, we must consider the bubble position in the cross-section, which determines the flow around the bubble and hence the force exerted by the liquid upon on the bubble: we characterize this position with the coordinates of the bubble centre in polar coordinates, namely the distance $r_e$ to the axis of the cell, and the angle $\theta$ with the vertical direction (figure 7).

Figure 7.

Bubble position in the cell cross-section: the centre of the bubble is characterized by $r_e$ and $\theta$. The action of the liquid on the bubble is modelled as the sum of drag $\boldsymbol {F}_{\boldsymbol {D}}$, lift $\boldsymbol {F}_{\boldsymbol {L}}$, pressure and added mass $\boldsymbol {F}_{\boldsymbol {A}}$ contributions.

The above parameters can be grouped into five additional independent dimensionless numbers, which we choose to be the Rossby number $Ro=r_e/D$, the Reynolds number $Re=\rho \omega r_e D/\mu$, the Froude number $Fr=\omega ^{2} r_e/g$, the Weber number $We=\rho \omega ^{2} D^{3}/8\sigma$ and the angle $\theta$.

Note that we have chosen to introduce a Reynolds number based on the mean velocity seen by the bubble. An alternative choice could be to introduce a Reynolds number based upon the shear seen by the particle

(2.1)\begin{equation} Re_{shear} = \dfrac{\rho \omega D^{2}}{\mu}=Re/Ro. \end{equation}

We will discuss in § 4 the relevance of this choice. The liquid–gas density and viscosity ratios can also be introduced. All experiments are here carried out with air and water, and since these two parameters are constant in the present study, we will not discuss them in the following.

The main control parameter driving the stretching of the bubble is expected to be $We$, and similarly the forces acting on the bubble are expected to be mostly controlled by $Re$. We will show in the following sections that corrections in $Ro$ have to be introduced when the bubble approaches the axis of rotation of the cell.

4.1. Drag coefficient

We plot in figure 14(a) the drag coefficient, deduced from (4.6b) via measurements of the mean values of $r_e$ and $\theta$, as a function of $Re$. The red solid line indicates the prediction of Schiller & Naumann (Reference Schiller and Naumann1933) for a solid sphere in a uniform flow. We have included in this graph the results of an additional measurement for a smaller bubble (grey asterisk, $V=0.50$ mm$^{3}$, $R_{eq}=0.5$ mm). For this particular bubble, $\omega$ is varied between 10 and 30 s$^{-1}$, and $Ro$ varies between respectively 10 and 3 in this interval, i.e. this bubble remains relatively far from the cell axis. We recover in this particular case previous results also obtained with demineralized water (Rastello et al. Reference Rastello, Marié, Grosjean and Lance2009, Reference Rastello, Marié and Lance2017); the points of this series fall close to the curve for solid spherical particles due to the inevitable presence of contaminants on the bubble interface. On the contrary, for larger bubbles closer to the cell centre, and which for most of them straddle the axis of rotation ($Ro<0.5$, see figure 13), we measure much larger drag coefficients, reaching values of up to 10. In addition, the scatter of the different series shows a strong influence of $Ro$ on $C_D$: for a given $Re$, $C_D$ is larger for a larger bubble, i.e. for smaller $Ro$.

Figure 14.

(a) Drag coefficient $C_{D}$ as a function of $Re$. (b) Drag coefficient $C_{D\varDelta }$ as a function of the shear Reynolds number $Re_{shear}$.

The large values of $C_D$ show that expression (4.5) does not capture correctly the order of magnitude of the drag force at low $Ro$. A spinning wake is expected to envelop the bubble at this low $Ro$, and therefore the configuration is different from that at large $Ro$ where different orders of magnitude of the fluid force are expected to coexist perpendicular or along the wake direction. It seems reasonable to expect that, in the present vortex-like low $Ro$ limit, both lift and drag will be of similar orders of magnitude.

Another way to put it is to consider that the drag caused by the spinning wake around the bubble will be dominated by the dynamic pressure difference around the bubble; the order of magnitude of this difference is the difference between the (maximum) dynamic pressure at the point farthest from the origin and the (minimum) dynamic pressure at the point closest to the origin (see $P^{+}$ and $P^{-}$ in figure 11), namely

(4.7)\begin{equation} \Delta P_d \approx P^{+}-P^{-}=\tfrac{1}{2}( \rho \omega^{2}(R+r_e)^{2}-\rho \omega^{2}(R-r_e)^{2})=2\rho \omega^{2} R r_e, \end{equation}

instead of $\rho U(r_e)^{2}/2$ for the uniform flow limit. Instead of the classical form of the drag force in uniform flow (4.5), the proposed expression for the drag force is then

(4.8)\begin{equation} F_D=C_{D \varDelta}A_b\rho \omega^{2}Dr_e \hat{\boldsymbol{\theta}}, \end{equation}

where we have introduced a new drag coefficient $C_{D \varDelta }=C_D Ro/2$. Note again that the order of magnitude introduced by (4.8) is larger than that of (4.5), since $r_e$ is smaller than $R$ for almost all our conditions (low $Ro$ limit). The scaling law for the drag force is then similar to that introduced for the other forces exerted by the fluid (added mass, pressure and lift force), in (4.3).

We plot in figure 14(b) the variations of $C_{D \varDelta }$ as a function of $Re_{shear}$. The values of $C_{D\varDelta }$ are mostly in the range [0.6–0.9], which shows that the chosen definition captures the correct order of magnitude of the drag force. In addition, the data of figure 14(a) appear much better collapsed when plotted as in figure 14(b): for $Re_{shear}$ varying in the range [500–4000] we find an approximately constant $C_{D\varDelta }$ with $C_{D\varDelta } \approx 0.75$. This means that the classical drag coefficient can be estimated as $C_D\approx 2C_{D\varDelta }/Ro \approx 1.5/Ro$ at low $Ro$. An even simpler estimate for $C_D$ can then be built from this expression, by further assuming that $r_e\approx g/\omega ^{2}$ (valid for all series except the smaller $V=0.69$ mm$^{3}$ bubble, see figure 10), which yields $C_D\approx 1.5 D\omega ^{2}/g$. We plot in figure 15 the measured $C_D$ as a function of this simple prediction: the proposed expression manages to provide a relatively good estimate of the drag coefficient for the large range of conditions we investigate here. We believe the larger dispersion observed for large bubbles and large rotation rates is caused by larger fluctuations for these conditions. These fluctuations will be discussed in the following § 4.2.

Figure 15.

Drag coefficient measured from bubble position, plotted as a function of proposed simplified model $1.5D\omega ^{2}/g$: this prediction provides a good estimate of $C_D$ for the large range of conditions investigated here.

The data for the smaller bubble (black cross points) show much larger values of the drag coefficient $C_{D\varDelta }$ in figure 14(b). This results from the fact that the measured $r_e\omega ^{2} /g$ for this series of points is smaller than for the other bubbles investigated, as mentioned above (figure 10). Indeed, for such a small bubble ($V=0.69$ mm$^{3}$, $R_{eq}=0.55$ mm), the values of $Re$ are significantly smaller: the reasoning behind the expression of (4.8) is not expected to be valid.

4.2. Lift

We show in figure 16 the variations of $2C_L-C_A$, deduced from (4.6a), as a function of $Re$. In order to isolate the lift coefficient, and compare it with the results of the literature, we need to estimate the added mass coefficient. This coefficient can be computed analytically based on the measured shape of the bubble, provided the bubbles are assumed to be ellipsoidal. As mentioned above, the model of Rosenthal (Reference Rosenthal1962) predicts a small departure from the ellipsoidal shape, but this is assumed to be negligible here given the moderate deformation of the bubbles (coefficient $\alpha$ expected to increase from 0.52 to at most 0.56). The stretching of aspect ratio $X$ along the axis of the cell, which has been discussed in § 3, tends to increase the value of the added mass coefficient along the $\hat {\boldsymbol {r}}$ direction. However, the bubbles also tend to flatten slightly along $\hat {\boldsymbol {\theta }}$ due to the rotational flow when $Ro$ is not too small (figure 5a), and are therefore not strictly axisymmetric. The aspect ratio in the ($\hat {\boldsymbol {r}},\hat {\boldsymbol {\theta }}$) plane can reach values up to 1.4 for the largest bubbles and largest values of $We_{re}$, the Weber number based on the mean velocity $r_e\omega$: this flattening, even at moderate aspect ratios, is expected to decrease the added mass coefficient along $\hat {\boldsymbol {r}}$ compared with the axisymmetric assumption. We chose here to compute $C_A$ numerically from the mean bubble dimensions deduced from the visualizations: we assume that the bubbles are ellipsoidal, and integrate the expressions proposed by Lamb (Reference Lamb1934) (see Appendix A). The balance between the stretching along the axis of rotation (which increases $C_A$) and the more moderate flattening along $\hat {\boldsymbol {\theta }}$ (which reduces it) results in added mass coefficients relatively close to 0.5, i.e. close to those of a sphere. The increase of $C_A$ observed in figure 23(a) when bubble size is increased corresponds to the increase in $X$ for larger bubbles.

Figure 16.

Variations of $2C_L-C_A$ as a function of $Re$, from (4.6a). Same legend as in figure 8.

These computed added mass coefficients are then used to deduce $C_L$: figure 17 shows the variations of the lift coefficient $C_L$ as a function of $Re$, and the comparison with some of the correlations introduced in Rastello et al. (Reference Rastello, Marié and Lance2017) for much larger $Ro$. The lift coefficient appears smaller than that measured by Bluemink et al. (Reference Bluemink, Lohse, Prosperetti and Van Wijngaarden2010) or Rastello et al. (Reference Rastello, Marié and Lance2017). The fact that $C_L$ is overestimated by these correlations could be caused by the smaller $Ro$ in our experiments: a decrease of the lift coefficient when $Ro$ is reduced has been observed by Rastello et al. (Reference Rastello, Marié and Lance2017) (see figure 20 of their paper). Note, however, that this decrease has been observed for $Ro>6$ i.e. bubbles located much farther from the axis. Bluemink et al. (Reference Bluemink, Lohse, Prosperetti and Van Wijngaarden2010) observed similar trends for spheres and investigated a few locations closer to the axis. They noted that $C_L$ decreased with the shear rate $Sr_{\omega }=Ro^{-1}$ up to $Sr_{\omega }=0.4\ (Ro=2.5)$ and that, after, the decrease stopped and $C_L$ became constant. This suggests that the influence of $Ro$ could also be limited in our case. The question of the limit of $C_L$ when $Ro$ becomes small therefore remains an open question.

Figure 17.

Lift coefficient as a function of Reynolds number. The dashed and solid lines correspond to equations (B1)–(B2) and (B5) in Rastello et al. (Reference Rastello, Marié and Lance2017): black line, the bubble does not spin. The lift is only induced by the rotation flow, equations (B2)–(B5). Red line: the bubble is supposed to spin with the same rotation speed as the tank. This scenario is possible as the bubble is contaminated and located on the rotation axis. Rotation adds a contribution equal to 3/16, equation (B1). Equation (B2) is the correlation of the numerical results of Bluemink et al. (Reference Bluemink, Lohse, Prosperetti and Van Wijngaarden2010) for non-spinning sphere (code Physalis, $Re \leq 200, Sr_{\omega }=Ro^{-1} \leq 0.1$); equation (B5) is the correlation of the experimental data of these authors for a non-spinning sphere ($Re >274, Sr_{\omega }=Ro^{-1} \leq 0.4$), with a different additional constant: 1.82 against 1.99 in Bluemink et al. (Reference Bluemink, Lohse, Prosperetti and Van Wijngaarden2010).

Besides the effect of $Ro$, we know from the literature (Kariyasaki Reference Kariyasaki1987; Adoua, Legendre & Magnaudet Reference Adoua, Legendre and Magnaudet2009; Rastello et al. Reference Rastello, Marié and Lance2017; Hayashi et al. Reference Hayashi, Legendre and Tomiyama2020) that bubble deformation can yield a decrease of the lift force and can even make this force change sign from positive to negative. Even though the main bubble stretching is aligned with the rotation axis, larger bubbles are actually also deformed in the $(r,\theta )$ plane, see figure 5 for example: for these non-axisymmetric cases, bubble deformation certainly affects the value of the lift coefficient. Note finally that the motion of the bubble around its equilibrium position may increase the inertia of the surrounding fluid, and hence lead to an increase in the effective added mass: the data of figure 17, which do not take this effect into account, are therefore expected to represent an underestimation of $C_L$.

When experiments are performed in demineralized water, as used here, the surface of the bubble is contaminated and in this case it is expected to ‘spin’. This has been shown for small bubbles at equilibrium far from the axis of rotation ($Ro \geq 6$) by tagging the flow near their surface with small fluorescent particles, in Rastello et al. (Reference Rastello, Marié, Grosjean and Lance2009) and Rastello et al. (Reference Rastello, Marié and Lance2017). Visualizations show that fluorescent particles located very close to the bubble surface (the distance is smaller than 50 $\mathrm {\mu }$m in figure 14 of Rastello et al. Reference Rastello, Marié, Grosjean and Lance2009) rotate around the bubble. These authors assimilate the motion of the fluid close to the surface to the rotation of the surface itself, and speak of a ‘spinning’ surface. For bubbles with $Ro\geq 6$, the mean spinning rate is higher than the rotation rate: up to 1.6, depending on $Re$ (figure 10 in Rastello et al. Reference Rastello, Marié and Lance2017). This results in an extra Magnus-like lift force and a separated wake behind the bubble whose description can be found in Rastello & Marié (Reference Rastello and Marié2020).

In the present case, bubbles stabilize on average on, or very close to, the rotation axis ($Ro<1$). It is probable that, if spinning, they will spin at most with the rotation rate of the tank and not faster (see figure 3 in Bluemink et al. Reference Bluemink, Lohse, Prosperetti and Van Wijngaarden2008 for spheres), which will result in an extra Magnus-like lift coefficient of at most the order of 3/16. Figure 17 shows that the present lift coefficients mostly lie below the values expected for non-spinning bubbles (indicated by the black curve): we believe this discrepancy with previous models is due to the effect of $Ro$, which is much smaller in our case than in previous studies.

When the bubble moves closer to the axis of rotation at low $Ro$, and in particular when it straddles the axis of rotation of the cell, the flow configuration changes: the relative importance of the shear increases, and the relevant Reynolds number is expected to become at some point the shear Reynolds number $Re_{shear}=Re/Ro$ introduced in §§ 2.3 and 4.1. We plot in figure 18 the variations of the lift coefficient $C_L$ as a function of $Re_{shear}$: the data of figure 17 appear better collapsed. The decrease of $C_L$ observed in the range $Re_{shear}=500 - 2500$ is consistent with the decrease observed by Rastello et al. (Reference Rastello, Marié and Lance2017) when $Ro^{-1}$ is increased, and when the Reynolds dependence is accounted for.

Figure 18.

Lift coefficient as a function of shear Reynolds number, $Re_{shear}=Re/Ro$. The greyed region corresponds to conditions where the bubble experiences strong shape fluctuations.

A sharp transition seems to occur for large bubbles when $Re_{shear}$ becomes larger than 2800 (greyed region in figure 18). Imaging of the bubble motion just before and after this transition indicates that the apparent increase in $C_L$ is associated with a strong periodic stretching of the bubble, synchronized with the rotation of the tank. We show in figure 19 how the increase in the lift coefficient appears correlated with the strong increase of the variance of the longitudinal aspect ratio $X$. The strong increase of these fluctuations appears related to a forcing of the bubble stretching by the rotation of the tank (see supplementary movie 1 available at https://doi.org/10.1017/jfm.2021.683). We introduce $\omega _0=\sqrt {12\sigma /\rho L^{3}}$ as an estimate of the eigenfrequency of the stretched bubble (Risso Reference Risso2000): as $\omega$ is increased, $\omega _0$ decreases due to the stretching of the bubble and $\omega$ and $\omega _0$ eventually coincide. Figure 19(b) shows that the transition observed around $Re_{shear}=3000$ in figure 18 corresponds precisely to $\omega /\omega _0 \approx 1$. The resonance expected for these conditions may explain the strong increase observed in the fluctuations of the aspect ratio.

Figure 19.

(a) Lift coefficient as a function of the variance of the longitudinal aspect ratio $X$. (b) Lift coefficient as a function of the ratio between the tank rotation frequency and the eigenfrequency of the stretched bubble $\omega _0$.

The question is then why the resonance provokes this steep apparent variation in the lift coefficient. A possibility could be that this apparent increase is caused by the much larger fluctuations in the position for these larger bubbles. Relative variations of $r_e$ can reach up to 30 %–40 % when resonance occurs, compared with less than 10 % for all series of bubbles with $Re_{shear}<2500$. These fluctuations in position are probably themselves triggered by the strong fluctuations in the shape of the bubbles. At any rate, because of these large fluctuations in $r_e$, mean values of $C_L$ for $Re_{shear}>2500$ deduced from the mean position result in fact from the averaging of a strongly non-stationary dynamics, something which is not accounted for in the equations leading to system (4.6).

Imaging of the bubble motion shows that the strong oscillations caused by the resonance strongly distort the bubble, which may then exhibit non-axisymmetric deformations. As illustrated in figure 20, these oscillations can even lead to the rupture of the bubble into two fragments of similar size; see also supplemental movie 2. This is only observed for the largest bubbles ($V>0.25$ cm$^{3}$) and largest $\omega$ investigated. The study of the conditions for break-up, and more generally of the dynamics of the bubble when the rotation of the cell forces its eigenmodes, is left for future work.

Figure 20.

Series of images showing the break-up of a bubble of volume $V=0.27$ cm$^{3}$ ($R_{eq}=0.4$ cm), at $\omega =84$ s$^{-1}$.