Continuous generation of confined bubbles: viscous effect on the gravito-capillary pinch off

Drop and bubble formation at the end of a tube has been the subject of active investigations for a long time not only from fundamental but also from applicational interests in drop and bubbles DynamicsDroplets , which includes oil recovery HeleShawPetroleum2010 , soft electronics babatain2024droplets and energy harvesting xu2020droplet . As early as in 1864, Tate discussed a pendant drop at the tip of a tube starts falling when its weight surpasses the capillary force supporting the weight Tate , which remains a fundamental knowledge to measure surface tension vinet1993surface ; berry2015measurement . A technically more sophisticated modern version of Tate, the continuous generation of droplets becomes increasingly important in microfluidics umbanhowar2000monodisperse ; christopher2007microfluidic ; zhu2017passive due to recent demand for the manipulation of small amounts of liquids in various fields such as medicine, biochemistry, and pharmaceutical industries. However, basic physical principles governing the dynamics of the droplets formation at small scales and/or at low Reynolds numbers have yet to be elucidated. One of the difficulties in tacking with this problem in microfluidics is the smallness of the system. One possible strategy to cope with this difficulty might be to use highly viscous liquid on centimeter scales in confined space. By virtue of this strategy, we have unveiled a number of governing principles regarding drop and bubble dynamics in the form of scaling laws, focusing on viscous friction EriSoftMat2011 ; yahashi2016 ; murano2020rising ; tanaka2023viscous , coalescence EriOkumura2010 ; YokotaPNAS2011 ; koga2022inertial , breakup nakazato2018self ; nakazato2022air , and bursting murano2018bursting . In this study, we focus on the continuous formation of bubbles on centimeter scales in a confined geometry, which is much more directly relevant for microfluidics, to reveal physical principles governing the dynamics in the form of scaling laws. Using an original setup, we successfully obtained scaling laws through a clear data collapse with elucidating physical pictures behind the simple laws.

Salient features of the present study is as follows. (1) To provide an example of emergence of scaling laws from numerous possibility of viscous dissipations; in other words, we have obtained a remarkably simple physics from a seemingly very complex problem, demonstrating the power of scaling analysis at a high level not achieved previously. (2) To provide a condition of breakup in which viscous effect is crucial in addition to the conventional Tate’s mechanism of the balance of gravity and capillarity, revealing a novel and general mechanism of pinching-off. (3) To provide an example, in which physical principles relevant for microfluidics can be effectively elucidated by using a system on centimeter scales without losing the feature of the flow at low Reynolds numbers. The present results not only advance fundamental knowledge on drops and bubbles but also provide guiding principles relevant for numerous applications at low Reynolds numbers in various fields such as microfluidics and oil industry.

In this experiment, we fabricate a thin cell sometimes called a Hele-Shaw cell (of thickness $D=1.0$ to 2.0 mm, width 15 cm and height 20 cm) equipped with a gate (of width $D$ and angle $\theta=45$ to 60 deg.) and fill the cell with a viscous liquid (of kinematic viscosity $\nu=30$ to 50 cS), as in Fig. 1 (a). We inject air with a syringe through a brass tube (of inner radius 3.8 mm) at the bottom of the cell to observe a squashed chunk of air rising in the viscous liquid, which is trapped for a while under the gate with reducing its mass as a result of continuously generating bubbles, as in Fig. 1 (b). To characterize the dynamics, we measure the period of generation $T$ and the characteristic size $R$ ($\gg D$) of the bubble seen from front as a function of the height $H$ of the air trapped under the gate [see the rightmost photo of Fig. 1 (b)].

The density $\rho$ and surface tension $\gamma$ of the viscous liquid [polydimethylsiloxane (PDMS)] are $970$ to $980$ kg/m³ and $\gamma=20$ mN/m, respectively. To prevent cell deformation due to capillary adhesion, we use 5mm-thick acrylic plates for $D=1.0$, 1.5 mm and 3 mm-thick acrylic plates for $D=2.0$ mm. In fact, the cell thickness $D$ is precisely determined using the laser distance sensor (ZS-HLD2, Omron) and the precise value is used in the analysis, although, for simplicity, $D$ is represented by approximate values (1.0, 1.5, or 3 mm) as above (differences are within 3%).

Since the bubble has a tear-drop shape (of area $A$) as seen in Fig. 1 (b), the characteristic size $R$ is defined through the relation $A=\pi R^{2}$. The period of generation $T$ for a bubble is defined as the time difference between the moment of pinch-off the bubble of our focus and that of the previous bubble. Similarly, $H$ for a bubble of our focus is defined as the height at the moment of pinch-off of the previous bubble.

In Fig. 2 (a) and (b), we respectively show $T$ and $R$ as a function of $H$. The data corresponding to the label with the * mark contain data obtained on different days with refabricating cells on each day (those without * are obtained within two hours using the same cell). We see the data sets with * for a parameter set are well on a master curve, which demonstrates a reasonably good reproducibility of the experiment.

The insets shows that $T$ and $R$ as a function of $H$ start bifurcating (or oscillating with $H$) as $H$ decreases, where we analyze only the data on the upper branch (we do not use the data represented by the cross mark in the following). The reason of the bifurcation or oscillation is as follows. Due to the continuous bubble generation from air trapped under the gate, the volume of air under the gate (and thus $H$) keep decreasing, and the period of generation of bubbles $T$ decreases with time (and thus with decrease in $H$). This implies that the distance between created bubbles becomes short with time. As a result, at certain point, the upwards flow caused by a bubble just created could start to affect the creation of the next bubble. If the $n$th bubble drags the $(n+1)$th bubble, which results in $T$ and $R$ in the lower branch (represented by the cross mark), then the $(n+2)$th bubble is no longer affected by the $(n+1)$th bubble. However, the $(n+2)$th bubble does drag the $(n+3)$th bubble. In this way, we observe the alternate bifurcation (or oscillation with $H$), where the $(n+2m)$th bubbles (with $m=0,1,2,\ldots$) are of our focus because the creation of them is completed without the drag effect of the pervious bubble.

Figure 3 explains the relation between $T$ as a function of $H$ with the illustration in (a) summarizing the physical picture: the dynamics is determined by the balance between the gravitational energy change due to buoyancy and viscous dissipation in thin films whose thickness $h$ is determined by the theory not of Landau, Levich, and Derjaguin (LLD) LandauLevich ; Derjaguin1943 but of Derjaguin Derjaguin1943 ; derjaguin1993thickness . To justify this, we first confirm in (b) the relation

$$T/(\eta H)=k(D,\theta)(R/H)^{\alpha/2}$$

(1)

with $\alpha\simeq 1.2$, where the coefficient $k$ is dependent on $D$ and $\theta$ (Note that the former dependence is visible when the data of different colors with the same symbol are compared while the latter when those of different marks with the same colors). Then, we consider numerous possibilities of dissipation balanced with the energy change due to buoyancy, as explained in detail in Fig. 5 below. As a result, we remarkably find out that this form of scaling is possible only in the case of the dissipation in the thin film of thickness $h$, which scales as $\eta(V^{\prime}/h)^{2}$ multiplied per a volume $hH^{2}/\tan\theta$ per time. Here, $V^{\prime}$ characterizes the velocity inside the thin film, which should be smaller than $V$ and could be estimated by a volume conservation: $V^{\prime}TH/\tan\theta\simeq R^{2}$. We balance this dissipation energy with the change in gravitational energy per time: $\rho gR^{2}DH\simeq\eta(V^{\prime}/h)^{2}hH^{2}/\tan\theta$ with the assumption $D\gg h$ for the estimation of the volume of the bubble. If we further use the Derjaguin’s expression, $h\simeq\kappa^{-1}Ca^{1/2}$ with the capillary length $\kappa^{-1}=\sqrt{\gamma/(\rho g)}$ and the capillary number $Ca=\eta V^{\prime}/\gamma$, we obtain

$$\rho gTD^{2}/(\eta H)\simeq(R/H)^{2}\tan\theta.$$

(2)

This relation is convincing confirmed by a clear collapse of data shown in Fig. 3 (c) without any fitting parameters, although the agreement is not perfect: The slope of the straight line in (c) obtained by numerical fitting is $1.21\pm 0.04$, which is slightly larger than the expected value, unity. As discussed in Fig. 5 below, the theory of LLD fails to show a collapse of the data, which supports our present argument based on Derjaguin’s law.

Figure 4 explains the relation between $R$ as a function of $H$ with the illustration in (a) summarizing a surprising physical picture: buoyancy opposed not only capillarity but also viscosity determines the condition of pinch-off, different from Tate’s law. If we consider a natural form of viscous stress $\eta V/D$ acting on the circumference of the disk-shaped bubble whose area scales as $RD$ (with the assumption $D\gg h$), we obtain a force balance

$$\rho gR^{2}D=\alpha_{0}\gamma D+\beta_{0}\eta VR$$

(3)

with dimensionless coefficients $\alpha_{0}$ and $\beta_{0}$. To remove $V$ from this equation, we use an equation for the bubble velocity $R\simeq VT$ and Eq. (2) for $T$, we arrive at the following relation based on the unexpected pinch-off condition:

$$R/\kappa^{-1}=\sqrt{\alpha+\beta HD/(\kappa^{-2}\tan\theta)}$$

(4)

This equation reveals that the dimensionless quantity $R/\kappa^{-1}$ should be a function of a dimensionless quantity $HD/(\kappa^{-2}\tan\theta)$, which is convincingly confirmed in Fig. 4 (b) without any fitting parameters. We further use Eq. (4) to fit the data to obtain $\alpha=0.665\pm 0.04$ and $\beta=0.411\pm 0.02$ by taking averages of numerical fitting for each parameter. An excellent agreement between this result of fitting and the data is shown in Fig. 4 (c).

Throughout the present study, we ignored the effect of inertia, which is justified as follows. We can estimate the upper bound for Reynolds number by $Re=$ $\eta VL/\eta$ once a relevant characteristic length scale $L$ is identified. Judging from Eqs. (2) and (4), it is natural to consider that $L$ would scale as $\kappa^{-1}$, which is comparable with $D$. Then, considering the range of parameters in the present study, we confirmed $Re$ is less than 0.0005 (for $L=1.8$ mm), which means $Re\ll 1$, as we assumed.

We remark that the use of Derjaguin’s law for the thickness $h$ of the thin film is supported not only by Fig. 3 but also by Fig. 4, since Eq. (4) is based on Eq. (2). In addition, as previously mentioned, we see in Fig. 5 (a) that the theory of LLD, which predicts $h\simeq\kappa^{-1}Ca^{2/3}$ and replaces Eq. (2) with the expression $\rho gTD^{2}/(\rho H)\simeq$ $(R/H)^{2}\tan\theta/D$, fails to explain our results: we could not see a collapse of the data when $\rho gTD^{2}/(\rho H)$ is plotted as a function $(R/H)^{2}\tan\theta/D$. In general, LLD is valid only for $Ca^{3}<1$, while Derjaguin law is valid for larger values of $Ca$ CapilaryText ; maleki2011landau . In the present study, $Ca$ is in the range from 0.00661 to 0.146, with the average 0.045 and the standard deviation 0.03, to confirm $Ca^{3}>1$, which supports the appropriateness of the use of Derjaguin’s law, beyond the agreement shown in Figs. 4 and 5.

We examine the validity of the assumption $D\gg h$. Considering the range of parameters in the present study, we can confirm $h/D$ to be in the range from 0.12 to 0.19 with the average 0.16 and the standard deviation 0.01, to reasonably well confirm $h/D\ll 1$.

As announced previously, we considered various possibilities for dissipation as specified in Fig. 5 (b1) to (b7), where the dissipation in thin films suggested in red was considered except for (b3), by which the dissipation developed around the bubble characterized by $\eta V/D$ is represented. For the film thickness we considered both cases of LLD and Derjaguin. In total, we considered 13 possibilities, as suggested in the illustration. As a result, we found that Case 3 to 7 fail to show the dependence on $\theta$ observed in experiment, while Case 8 to 13 could not be put into the form in Eq. (1). The remaining possibilities were then Case 1 and 2, which were the cases already examined in the text. See further details for Appendix.

As far as we know, there are no previous studies in which a governing dissipation is singled out from numerous possibilities to results in simple and clear scaling laws as in the present study. In this sense, our case is a remarkable example, in which a simple physics emerges from complexity.

In addition, the effect of viscosity on the Tate’s condition of pinch-off uncovered in the present study should be a general mechanism to be considered in many other cases in microfluidics or/and at low Reynolds numbers. Together with this, the present study provides a clear and fundamental physical understanding for the dynamics of continuous bubble formation relevant for numerous applications, advancing and impacting on the field, demonstrating a system on centimeter scales could be useful.