# Oscillating path between self-similarities in liquid pinch-off

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Edited by Michael Marder, University of Texas, Austin, TX, and accepted by Editorial Board Member John D. Weeks October 24, 2018 (received for review August 17, 2018)

## Significance

Droplet pinch-off is one of the most commonly examined free surface flows displaying a finite-time singularity and can serve as the basis upon which to better understand similar singular behavior. A liquid filament connecting the droplet to the rest of the fluid thins before it breaks and passes through several self-similar regimes that have been thought independent of external conditions. These regimes are essentially asymptotic, and it is an open question how exactly the system passes through them. We find to our surprise that external conditions strongly affect the transient path connecting these regimes and even can temporarily prevent the fluid thread from evolving through self-similar profiles. Our results raise many questions about the influence of boundary conditions on such self-similar dynamics.

## Abstract

Many differential equations involved in natural sciences show singular behaviors; i.e., quantities in the model diverge as the solution goes to zero. Nonetheless, the evolution of the singularity can be captured with self-similar solutions, several of which may exist for a given system. How to characterize the transition from one self-similar regime to another remains an open question. By studying the classic example of the pinch-off of a viscous liquid thread, we show experimentally that the geometry of the system and external perturbations play an essential role in the transition from a symmetric to an asymmetric solution. Moreover, this transient regime undergoes unexpected log-scale oscillations that delay dramatically the onset of the final self-similar solution. This result sheds light on the strong impact external constraints can have on predictions established to explain the formation of satellite droplets or on the rheological tests applied on a fluid, for example.

From the greedy child who wants to detach the last drop of tomato sauce from its container to his loving parents who desperately try to fix the dripping faucet (1), drop formation surrounds us in our daily life. It also concerns various applications, from the usual inkjet printing to transistor circuits (2) or even bioprinting of mammalian cells (3), for instance. Drop detachment corresponds to a liquid breakup that is obtained by the pinch-off of the final thin liquid thread. This pinch-off is mathematically described by a finite-time singularity, the minimal radius of the liquid thread

We first place a circular aluminum cylinder at an oil–water interface. The cylinder is pinned at this interface and pulled downward with a stepper motor. We can thus control both the radius of the cylinder *B*). To move the cylinder quasi-statically (*Materials and Methods*). A stretched meniscus is formed between this moving cylinder and the layer of oil. The oil thread thins until it finally breaks up (Fig. 1*D* and Movie S2). We measure the evolution of the neck during this process. This experiment enables us to investigate the impact of a vertical velocity on the evolution of the detachment of an oil drop in a liquid bath. We then compare this situation to a rising oil droplet detaching from a circular nozzle immersed in water, for different needle radii *A*).

The vertical velocity of the cylinder in Fig. 1*D* imposes boundary conditions dramatically different from those of the rising droplet in Fig. 1*C* (Movies S1 and S2). Qualitatively, it is already clear that the shapes of the liquid threads are different between the two situations: In particular, the symmetry of the viscous filament at the detachment of the droplet is not recovered when the interface is pulled downward.

The characteristic lengthscale

We measure the time evolution of *C*). The thinning dynamics begin by the destabilization of a fluid cylinder due to the Plateau–Rayleigh instability (15, 16). Once the perturbation of the ideal liquid cylinder is large enough, the linear stability approach can no longer hold. Thinning is then governed by a nonlinear competition between surface tension and viscosity (17), where the minimum radius **3** (viscous regime),**4** (inertial–viscous regime):

In Fig. 2, we compare the evolution of the minimum radius **3** (slope **4** (slope *Inset*). But interestingly, for

We do recover the theoretical slopes predicted by Eqs. **3** and **4** (Fig. 2), but in the configuration of a stretched meniscus, we moreover observe oscillations from one slope to another. Several differences between the two experiments described here can be highlighted to account for these distinct behaviors: the larger diameter of the cylinder compared with the needle (approximately 10 times larger), its vertical velocity since only buoyancy helps the oil drop to rise whereas the cylinder pulls the interface downward at velocity ^{−1}, and finally the direction of motion of the oil droplet (downward for the cylinder and upward for the needle).

To identify the different regimes correctly, we plot

The results for the configurations where a drop detaches from a needle (green circles) and an interface is pulled down quasi-statically with a cylinder of the same diameter (red triangles) are identical: an exponential increase of

Unlike the experiment where a drop detaches from a nozzle, when pulling the oil–water interface downward with a cylinder, we can strongly vary independently the parameters at play,

For every experiment, we can also measure the frequency of oscillation in log-scale *SI Appendix*, Fig. S1). This is consistent with the natural time dependence arising when a perturbation analysis is performed around the self-similar dynamics (*Materials and Methods*). The log-scale oscillation frequency depends only on the dynamical system and not on the external conditions.

Finally, we check whether the profiles are self-similar in the linear transient regimes observed during these oscillations. To that end, we report the profiles of the liquid thread and define rescaled variables according to Eqs. **3** and **4**,**7** (resp. Eq. **6**).

Fig. 5 shows the rescaled profiles for two cylinder diameters pulling down the interface, during each of the linear regions (*A* and *B*, *B*). Moreover, the collapse of the self-similar curves is valid only around the position of the minimum radius.

Fig. 5 *C*–*F* shows the rescalings for *C*), and the three profiles are well superimposed. We thus recover the final inertial–viscous regime, just like in the classic case of an oil droplet detaching from a needle. Similarly, after the exponential regime, the system is well described by the viscous regime (Fig. 5*F*), with the expected value of

Between these two regimes, *E*, the profiles do look like the self-similar profiles, they do not collapse satisfactorily (especially for positive values of *E* is comparable to the size of the data points and thus they cannot explain the differences between the three curves. In Fig. 5*D*, the situation is more obvious: The profiles are clearly not superimposed, and we even lose the symmetry of the profiles, even though this condition is essential for the calculation leading to Eq. **3**. It is clear that self-similarity is at least lost during the evolution of the neck close to breakup in Fig. 5*D*, whereas we can only speculate that it is also the case in Fig. 5*E*. Overall, these results suggest that self-similarity is lost during the oscillation of the neck radial velocity. After having identified the two self-similar regimes, we measure the duration of the transition between them. This duration increases dramatically both with the cylinder radii *SI Appendix*, Fig. S2), hiding somehow the self-similar features of the pinch-off for most of the process.

In conclusion, our results provide experimental proof of a log-oscillation of the slope of

## Materials and Methods

### Oil Droplets.

To perform the reference experiment of an oil droplet detaching from a needle of radius ^{−1}, density ρ = 966 kg⋅m^{−3}, oil–water interfacial tension γ = 42 mN⋅m^{−1}). The needles used have a circular orifice of known inner and outer diameters. Because the oil wets the surface of the needle, the important dimension is only the outer diameter.

### Stretched Meniscus Formed by a Moving Cylinder Pinned to the Oil–Water Interface.

For the cylinder experiment, we fill a tank of dimensions 0.2 × 0.2 × 0.25 m with deionized water and add on top of it a 2-mm–thick layer of the same silicone oil as for the needle experiment. We pull on the interface with aluminum cylinders within a range of diameters from 2

To ensure the reproducibility of our experiments, we check that the initial vertical position of the cylinder, when the motion is started, has no effect on the thinning dynamics. To that end, we reproduce several times the same experiment (with a given cylinder radius

### Acquisition.

A high-speed camera (Phantom v2511) records the dynamics with a recording speed up to 25,000 frames per second. To resolve perfectly the contour of the drop, we place between the backlight and the sample a mask, to increase the contrast with the ambient fluid.

### Log-Oscillation.

Without conducting the full calculation, we remind the reader how **9**,**9** into Eq. **8** leads to a system of equations almost similar to the one obtained by Eggers, with only an additional term:

## Acknowledgments

We gratefully acknowledge Matthieu Roché for his useful comments at various stages of this work.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: protiere{at}ida.upmc.fr.

Author contributions: A.L. and S.P. designed research; A.L. performed research; A.L., C.J., and S.P. analyzed data; and A.L., C.J., and S.P. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. M.M. 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.1814242115/-/DCSupplemental.

Published under the PNAS license.

## References

- ↵
- ↵
- Sirringhaus H, et al.

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- Sierou A,
- Lister JR

- ↵
- Shi X,
- Brenner MP,
- Nagel SR

- ↵
- Doshi P, et al.

- ↵
- ↵
- Castrejón-Pita JR, et al.

- ↵
- Li Y,
- Sprittles JE

- ↵
- ↵
- Rayleigh L

- ↵
- ↵
- ↵
- Rothert A,
- Richter R,
- Rehberg I

- ↵
- ↵

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