View source for A Cascade of Structure in a Drop Falling from a Faucet

By Sung Hoon Kang

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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== 

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[[Image:Fig 1 (Shi et al).jpg|thumb|right|600px| '''Fig. 1'''. The shape close to the time of breakup of five drops with different viscosities. The liquids, with η=10<sup>-2</sup> P (A), 10<sup>-1</sup> 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 discontinous 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 L<sub>γ</sub> = (γ/ρg)<sup>1/2</sup>, 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 L<sub>η</sub> = η<sup>2</sup>/ργ (η: viscosity). For water, L<sub>η</sub> ~ 10 nm [3]. However, it is not possible to take the image of the shape of the interface whose thickness is smaller than L<sub>η</sub> 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 10<sup>3</sup> 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 L<sub>η</sub> by 10<sup>6</sup>. 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.

[[Image:Fig 2 (Shi et al).jpg|thumb|left|400px| '''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<sup>-3</sup> and 2x10<sup>-4</sup>, 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 L<sub>η</sub> = 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 their numerical analysis which showed profiles similar to experimental results as Fig. 4. THe sources of noise can be 

[[Image:Fig 3 and 4 (Shi et al).jpg|thumb|left|400px| '''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).]]

==References== 

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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).