# Restoring universality to the pinch-off of a bubble

^{a}Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139;^{b}Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08540;^{c}Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139;^{d}Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139

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Edited by Osman A. Basaran, Purdue University, West Lafayette, IN, and accepted by Editorial Board Member John D. Weeks May 13, 2019 (received for review November 19, 2018)

## Significance

We observe the formation of bubbles and drops on a daily basis, from dripping faucets to raindrops entraining bubbles on the surface of a lake. The ubiquity of the phenomenon masks the fascinating underlying nonlinear dynamics that is such an important aspect of modern physics. Here, we report on the surprising observation that confinement makes the pinch-off of a bubble a universal process, as opposed to the unconfined case, where pinch-off is sensitive to the details of the experimental setting. We explain how the motion of the contact line, where the liquid, gas, and solid phases meet, leads to self-similar dynamics that effectively erase the memory of the system. Our observations have implications for immiscible flow phenomena from microfluidics to geophysical flows, where confinement, together with fluid–solid physicochemical interactions, play a key role.

## Abstract

The pinch-off of a bubble is an example of the formation of a singularity, exhibiting a characteristic separation of length and time scales. Because of this scale separation, one expects universal dynamics that collapse into self-similar behavior determined by the relative importance of viscous, inertial, and capillary forces. Surprisingly, however, the pinch-off of a bubble in a large tank of viscous liquid is known to be nonuniversal. Here, we show that the pinch-off dynamics of a bubble confined in a capillary tube undergo a sequence of two distinct self-similar regimes, even though the entire evolution is controlled by a balance between viscous and capillary forces. We demonstrate that the early-time self-similar regime restores universality to bubble pinch-off by erasing the system’s memory of the initial conditions. Our findings have important implications for bubble/drop generation in microfluidic devices, with applications in inkjet printing, medical imaging, and synthesis of particulate materials.

From dripping faucets to children blowing soap bubbles, we observe the formation of drops and bubbles on a daily basis. This seemingly simple phenomenon, however, has long puzzled and attracted scientists, from the early descriptions of da Vinci, Savart, Plateau, and Rayleigh (1⇓–3) to advanced experimental techniques that yield precise observations of the interface evolution leading to pinch-off (4⇓⇓⇓⇓–9). Most previous studies of singularities during bubble or drop formation have focused on unbounded fluid domains (10⇓⇓⇓⇓⇓⇓–17). Many natural phenomena and industrial processes, however, often involve flows under confinement (18⇓⇓⇓⇓–23), where the dimensionality of the confined geometry is known to strongly influence the pinch-off (24⇓⇓⇓–28). These studies have assumed that a continuous liquid phase coats all of the bounding surfaces. In many situations, however, one encounters partially wetting liquids, which naturally lead to the presence of contact lines, where a fluid–fluid interface meets the solid surface (29).

Here, we study the pinch-off of a bubble in confinement in the partial wetting regime. We show that the moving contact line singularity (30, 31) dominates the viscous dissipation at early times, leading to an axially dominated flow and the emergence of an early-time self-similar regime, which then crosses over to a late-time regime, where the flow is mainly radial and the viscous dissipation is dominated by the pinch-off singularity. While the observation of different self-similar regimes is expected when the balance of forces between inertia, viscosity, and capillarity changes (11, 32, 33), here we show that in our system, the cross-over between self-similar regimes occurs even though the entire evolution is controlled by a balance between viscous and capillary forces.

The separation of length and time scales in the vicinity of a singularity suggests that the local balance of forces should become independent of the details of the initial or boundary conditions, making the dynamics of the pinch-off universal (10). Surprisingly, in the case of the pinch-off of an inviscid bubble in an unbounded ambient viscous liquid, the local structure of the singularity is sensitive to the details of the experimental conditions, rendering the pinch-off nonuniversal (13, 34⇓–36). Here, we show that the presence of the early-time regime in a confined geometry establishes the tube diameter as the only length scale in the problem and erases the system’s memory of the experimental details and initial conditions, leading to the universality of the bubble pinch-off.

We study the bubble generation process in a microcapillary tube (diameter

In the case of bubble formation in a large quiescent tank, the balance of radial viscous flow and surface tension causes the bubble neck diameter to shrink linearly in time (35, 38). In contrast, here, during the process of bubble pinch-off in a microcapillary tube, the evolution of the diameter of the bubble neck indicates the presence of two distinct self-similar regimes (Fig. 2*A*), as illustrated with the results from 12 different experiments: the bubble neck diameter initially follows a *B*). The Reynolds number is defined as

To gain an understanding of this early-time self-similar regime for the time evolution of the bubble radius, we consider the dynamics of the growing dewetting rim (Fig. 1). This can be analyzed using a long-wave approximation (37), which assumes that the flow is mainly parallel to the tube axis. Near the point of pinch-off, we postulate that the shape of the profile becomes self-similar: *A*, *Inset*); here, all of the length scales are nondimensionalized by the tube diameter d, and the dimensionless time to the singularity is defined as *SI Appendix*, section 2):**1** represents the viscous forces, and the right-hand side represents the capillary forces: the first two terms represent the out-of-plane curvature and the last term represents the in-plane curvature. The only way for all of the terms to balance in time is to have *B*. The self-similar ordinary differential equation governing the neck profile in the early-time regime therefore has the following form:

An important point here is that the long-wave approximation is developed for the dewetting rim, and not for the bubble neck. The relevant length scales are therefore the height (

To estimate the cross-over time between the two regimes, we compare their corresponding radial velocities. In the early-time regime, the growth rate of the dewetting rim is proportional to the velocity of the receding contact line, i.e., *SI Appendix*, section 3). Note that very close to the point of pinch-off (*B* shows that, indeed, using the visco-capillary time scale and the tube diameter as the characteristic time and length scales leads to the collapse of all data corresponding to 12 different experiments with different tube diameters (

To test the self-similarity of the bubble neck profile, we probe its evolution in time in Fig. 3*A* (for the experiment with *B*, we show that, indeed, scaling both the neck diameter and axial dimension with the the minimum neck diameter collapses the profiles in the early-time regime (blue symbols). The self-similar solution of the long-wave model Eq. **2** (black dashed line) fits the data in this regime. This observation further confirms the validity of the long-wave model in the early-time self-similar regime. The data in the late-time regime (green symbols), however, deviate from the predictions of the long-wave theory.

Very close to the point of pinch-off, in the late-time self-similar regime, we can simplify the balance of normal viscous stresses and the surface tension to obtain *SI Appendix*, section 4).

In the local neighborhood of the minimum neck radius, we expect the profile to be parabolic: **3** indicates the axial extent of the profile scales as *D*. Using ζ as the axial length scale, we can therefore collapse the neck profiles over the entire pinch-off process onto a single parabolic curve (Fig. 3*E*).

In the early-time self-similar regime, both the neck radius, *B*), and the axial scale of the profile scale as *C*), indicating that the axial radius of curvature of the neck profile becomes independent of time:

In our system, however, this late-time self-similar regime is preceded by an early-time self-similar regime of the first kind, which sets the axial length scale of the late-time regime at the cross-over between the two regimes. This characterization is captured in Fig. 3*D*, which shows a cross-over in the scaling of the axial length scale at *SI Appendix*, section 5). In the early-time regime, we have *A* shows a plot of the axial radius of curvature as a function of time to the pinch-off for each experiment. When nondimensionalized, we observe that, indeed, the data corresponding to all 12 experiments collapse onto a single curve (Fig. 4*B*), with an asymptotic universal radius of curvature of

We therefore conclude that the combined effect of geometric confinement and contact-line motion leads to the emergence of an early-time self-similar regime of the first kind, which at late times crosses over to a regime of self-similarity of the second kind. While the balance between viscous and capillary forces controls the dynamics of interface evolution in both regimes, the cross-over occurs due to a change in the dominant contribution to the viscous dissipation; from the spatially localized moving contact-line singularity at early times to the temporally localized bubble pinch-off singularity at late times. This change in the dominant contributor to the viscous effects is accompanied by a change in the direction of the flow, from axially dominated in the early-time regime to radially dominated in the late-time regime. The late-time regime is also observed in the pinch-off of bubbles in unbounded fluid domains, where the flow is mainly radial and the axial length scale characterizing the bubble neck is sensitive to the details of the experimental system, making the pinch-off nonuniversal (13, 34⇓–36). Here, in the case of bubble pinch-off in confined domains, however, we observe that the axial length scale of the neck region is set by the early-time self-similar regime, effectively erasing the system’s memory of the initial conditions and restoring universality to the pinch-off process.

While we have focused on the case of bubble pinch-off in a viscous liquid, we expect our observation of the universality of the pinch-off process to persist for any other (at least viscously dominated) fluid–fluid displacement process in a confined medium involving moving contact lines. The restoration of universality has important consequences for controlled generation of bubbles, drops, and emulsions in microfluidic devices with a myriad of applications in medicine (44, 45) and material science (46⇓⇓⇓–50), as well as for understanding of multiphase flows in geologic media (22, 51), where geometric confinement and liquid–solid physicochemical interactions play a key role.

## Materials and Methods

Materials and Methods are described in *SI Appendix*.

## Acknowledgments

We thank Denis Bartolo and Jens Eggers for insightful discussions, and Benzhong Zhao for advice on the experimental setup. This work was funded by the US Department of Energy (Grant no. DE-SC0018357).

## Footnotes

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^{1}To whom correspondence may be addressed. Email: pahlavan{at}princeton.edu or juanes{at}mit.edu.

Author contributions: A.A.P. and R.J. designed research; A.A.P. performed research; A.A.P., H.A.S., G.H.M., and R.J. analyzed data; and A.A.P., H.A.S., G.H.M., and R.J. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. O.A.B. is a Guest Editor invited by the Editorial Board.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1819744116/-/DCSupplemental.

Published under the PNAS license.

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