A Cascade of Structure in a Drop Falling from a Faucet

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Original entry: Sung Hoon Kang, APPHY 226, Spring 2009

Title: A Cascade of Structure in a Drop Falling from a Faucet

Reference: X. D. Shi, M. P. Brenner, S. R. Nagel, Science 265, 219-222 (1994).

Soft matter keywords

Viscosity, Capillary length, Drop cascade

Abstract from the original paper

A drop falling from a faucet is a common example of a mass fissioning into two or more pieces. The shape of the liquid in this situation has been investigated by both experiment and computer simulation. As the viscosity of the liquid is varied, the shape of the drop changes dramatically. Near the point of breakup, viscous drops develop long necks that then spawn a series of smaller necks with ever thinner diameters. Simulations indicate that this repeated formation of necks can proceed ad infinitum whenever a small but finite amount of noise is present in the experiment. In this situation, the dynamical singularity occurring when a drop fissions is characterized by a rough interface.

Soft matter example

Fig. 1. The shape close to the time of breakup of five drops with different viscosities. The liquids, with η=10-2 P (A), 10-1 P (B), 1 P (C), 2 P (D), and 12 P (E), were allowed to drain slowly through a glass tube with a nozzle diameter D of 1.5 mm. (A) and (E) show pure water and pure glycerol, respectively. They used an 80 nm Hasselblad lens attached to a bellows and a still camera; the drop was illuminated from behind by a fast (~ 5 us) flash from a strobe (EG&G model MVS 2601, Salem, MA) that was triggered with a variable delay from the time the drop intersected a laser beam incident on a photodiode.

Every morning, one goes to a restroom to wash his/her face by using water from a faucet. Some of you may have noticed that when water drips from a faucet, its shape changes from a single mass of fluid into two or more drops. This phenomenon is one of the examples of a singularity in which physical quantities become discontinuous in a finite time [1]. The authors of this paper studied the shape of the singularity for fluids of different viscosity dripping through air from a cylindrical nozzle. We can consider three independent length scales that characterizes the hydronamics of the dripping faucet [2]: i) the diameter of the nozzle (D); ii) the capillary length Lγ = (γ/ρg)1/2, which is obtained from the balance between the surface tension γ, and the gravitational force ρg (ρ: the fluid density, g: the acceleration of gravity); and iii) the viscous length scale Lη = η2/ργ (η: viscosity). For water, Lη ~ 10 nm [3]. However, it is not possible to take the image of the shape of the interface whose thickness is smaller than Lη due to the resolution limit of the current optical techniques.

In order to find the asymptotic shape of the singularity, the authors investigated the droplet snap-off as a function of viscosity. Instead of using water, they mixed glycerol with water and could increase the viscosity by a factor of 103 while the surface tension was not varied by more than 15% [4]. The increase of the viscosity by a factor of 1000 resulted in the increase of the Lη by 106. Then, they observed that the shapes close to the breakup point of liquid draining from a nozzle of diameter D = 1.5 mm were varied for five liquids with different viscosities as shown in Fig. 1. As the viscosity increases, the neck of the drop was elongated and structures not observed in dripping water were formed.

Fig. 2. Photograph of a drop of a glycerol in water mixture (85 wt%). (A) The initial stages of the breakup as the droplet separates from the nozzle by a long neck. A secondary neck is visible just above the main drop. (B) A magnified view of the region near the breakup point (obtained from a different photograph). The thickest region in this picture corresponds to the long neck of (A). Only one secondary neck is visible. (C) The same region as in (B) but at a slightly later time. The second neck and a well-developed third neck are clearly visible in this picture. Using the Eggers similarity solution, we estimate that (B) and (C) are 2x10-3 and 2x10-4, respectively, before the time of breakup. The photographs were taken as in Fig. 1, except with a 35 mm lens. Because the clearest demonstration of the structure of the singularity was obtained with the use of the largest drop, they found it advantageous to replace the glass nozzle by a brass plate riddled with numerous small holes on which the liquid would slowly collect before falling. The diameter of the drop at the plate was approximately 20 mm.

Then, they focused on the structure of a liquid drop with intermediate viscosity and did experimental and numerical studies of the singularity. When a drop of a 15% water and 85% glycerol mixture (η = 1 P and Lη = 0.13 mm) breaks, it forms a long thin neck of fluid connecting the nozzle exit plane to the main part of the drop as shown in Fig. 2A. Fig. 2B shows a closeup view of the region just above the main drop where a secondary neck forms. Shortly after, the secondary neck also has instability and forms a third neck. From high speed camera images, they observed that the second and the third necks were formed in a similar way: an initial thinning near the drop was followed by a rapid extension of the neck upward away from the drop.

From numerical analysis, they reported that such a phenomenon can happen ad infinitum (with as many as seven successive necks at the resolution limit) as long as a noise source is present. However, if there is no noise, their simulation results showed that near the singularity the interface is smooth as shown in Fig. 3. Then, they discussed numerical analysis of the profile for a drop in the presence of noise, which showed profiles (Fig. 4) similar to experimental results. They argued that there can be many sources of noise such as surfactants on the interface, pressure fluctuations in the nozzle, air currents in the room, shear heating in the liquid and thermal fluctuations.

Fig. 3. Simulation of the profile for a drop of a glycerol in water mixture (85 weight %; viscosity = 1 P) falling from a nozzle of diameter D = 4.5 mm. (A) A view of the entire drop, at approximately 10-8 s before snap-off, shows that near the bottom of the long neck there is a region where the thickness decreases and forms a secondary neck. (B) An enlargement of the region in (A) enclosed by brackets. Note that the scales are different along the two axes, so that the shape can be easily seen. The inset shows another, greater magnification.
Fig. 4. Simulation of the same drop as in Fig. 3 in the presence of noise. A view of the entire drop is indistinguishable from that in Fig. 3A. (A) An enlargement of the first neck (as in Fig. 3B) shows a third, fourth, and (very thin) fifth neck sprouting from the second neck. (B) An enlargement of the bracketed region of (A) shows the fourth and fifth necks of the previous figure, as well as the sixth neck. In the inset, a magnification of the sixth neck shows a seventh neck.

As a summary, they showed that the singularity found in the fission of a viscous drop of liquid is different from that of less viscous fluids. They observed a rough interface with necks growing other necks instead of a smooth interface that they expected from a numerical analysis result without noise. For possible reasons of the rough interface instead of a smooth one, they discussed it could be due to experimental noise and/or approximation of their simulations.

This paper was quite interesting to me because the authors did a nice analysis of a soft matter example which often happens in our life, but we miss details. Before reading this paper, I didn't know that a drop of liquid coming from a faucet can form a hierarchical structure due to instability. It helped me to learn how soft matter principles work in an interesting phenomenon observed in our daily lives.


1. A. Pumir and E. D. Siggia, Phys. Rev. Lett. 68, 1511-1514 (1992).

2. J. Eggers and T. F. Dupont, J. Fluid Mech. 262, 205-221 (1994).

3. D. H. Peregrine, G. Shoker, A. Symon, J. Fluid Mech. 212, 25-39 (1990).

4. G. W. C. Kaye and T. H. Laby, Tables of Physical and Chemical Constants (Longman, New York, 1986); J. B. Segur and H. E. Oberstar, Ind. Eng. Chem. 43, 2117-2120 (1951).