A Cascade of Structure in a Drop Falling from a Faucet
By Sung Hoon Kang
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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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 . 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 : 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 . 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% . 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.
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 their numerical analysis which showed profiles similar to experimental results as Fig. 4. THe sources of noise can be
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