THINGS

Fall 2001
Volume 28, Number 1

My guess is that most physicists are Platonists at heart. We are taught from the very first that there are laws governing the way nature behaves. Although these laws are, we like to think, universal and immutable, they are not obvious. The laws of mechanics, of electricity and magnetism, of relativity and quantum mechanics are all remarkable, subtle, and of exceptional generality. Something that is not as often emphasized is that the idea of universality can extend to the objects themselves and is not merely relegated to the laws that govern their interactions.

[...]

All of this brings me back to the picture of the drop (figs. 1aÐb) that I mentioned at the outset. At its essence, this photograph displays another Platonic form--the drop at the moment of breakup. It shares with the example of the arc of the thrown ball that it is formed by the dynamics of the situation. However, it also shares with the example of the electron that it is a material object, however much the specificity of the object depends on the precise moment of its existence. What I want to argue is that this object is itself universal and that whenever we think of the thing a-drop-breaking-apart we necessarily allude to the scene depicted in the first photograph.

See Also Leonard B. Meyer: Concerning the Sciences, the Arts--AND the Humanities (September 1974) Joel Snyder and Neil Walsh Allen: Photography, Vision, and Representation (Autumn 1975) E.H. Gombrich: Standards of Truth: The Arrested Image and the Moving Eye (Winter 1980) Charles Spinosa and Hubert L. Dreyfus : Two Kinds of Antiessentialism and Their Consequences (Summer 1996)

The explanation for why this is true is so elementary (although it can look complicated when the mathematical formulae are included to make the argument rigorous) that I cannot resist mentioning it here.4 It depends on the realization that in the vicinity of the neck, where the drop eventually breaks apart, the thickness of the fluid must become arbitrarily small. After all, the neck decreases in diameter until its thickness goes to zero. Thus, at some point, it has become so much smaller than any other macroscopic length in the problem that those larger lengths can no longer matter for a description of the neck itself. It is smaller than the nozzle holding it. It is smaller than the size of the separating round drop at the bottom (whose size is determined by a competition of the downward pull of gravity against the force of surface tension holding the liquid together). If we ask what determines the dynamics in the vicinity of this thinnest point in the neck, we have to admit that it only makes sense that the dynamics should depend only on the thickness of the neck itself and cannot depend on the size of the nozzle or the size of the drop. So, the dynamics in the neck only depends on its thickness and we have arrived at a chicken-and-egg situation: The thickness of the neck depends only on the dynamics (since the dynamics is what pulls the neck into a thin thread in the first place), which in turn depends only on the thickness, and this repetition continues ad infinitum. As time goes on toward the point of pinch off, the neck conforms to a very special shape--one that must be similar to its own shape at both an earlier and a later time. The only thing that happens as the drop gets closer to the point of snapoff is that the neck gets thinner and more stretched out, but the overall shape must remain the same--only scaled by different amounts in the radial and axial directions at different times. It is thus a universal shape.5

The point I want to emphasize is that the object shown in the photograph of the drop breaking apart is, like the electron, a Platonic form. The photograph has fixed on paper a shadow that is precisely the shadow of the Platonic metaphor. Every drop of water falling will look exactly like this in the vicinity of snapoff. It does not matter that this drop was held by a nozzle and was pulled down by gravity. As we can see in figure 1b, the drop actually breaks up in at least two points, one near the bottom as shown in the first picture and again when the satellite drop breaks off near the nozzle. The second image shows that the cone-into-sphere shape of the snapoff is the same pointing upward near the nozzle as it was pointing downward near the initial drop. The point of breakup would have looked the same even if it had been formed from a wave tossed into the air at the shores of Lake Michigan.

4. The ideas of scale invariance in fluid dynamics go back a long way and appear in many manifestations. The work on drop breakup is only one example of where these ideas arise. The work described in this article is from Xiangdong Shi, Michael P. Brenner, and Sidney R. Nagel, "A Cascade of Structure in a Drop Falling from a Faucet," Science, 8 July 1994, pp. 219Ð22; Brenner et al., "Breakdown of Scaling in Droplet Fission at High Reynolds Number," Physics of Fluids 9 (June 1997): 1573Ð90; and Itai Cohen et al., "Two Fluid Drop Snap-Off Problem: Experiments and Theory," Physical Review Letters, 9 Aug. 1999, pp. 1147Ð50. Much of the work on the breakup of drops is reviewed in Jens Eggers, "Nonlinear Dynamics and Breakup of Free-Surface Flows," Reviews of Modern Physics 69 (July 1997): 865Ð929.

5. Making this argument rigorous requires finding a scaling description for the interfacial shape near the point of snapoff. If h(z,t) is the radius of the drop at a vertical position, z, and time, t (where z and t are each measured with respect to the position and time of the snapoff singularity), a scaling solution asserts that h(z,t) is related to H(_), a function of a single variable, _ = z t-_:

where _ and _ are exponents that need to be determined. The idea of universality classes is that over a wide range of physical parameters, these exponents _ and _ will not vary and that all systems in the same class will have the same shape at snapoff. Such a solution is verified by inserting it into the Navier-Stokes equations describing the motion of the fluid interface. The scaling equation for the shape near the singularity, using the single variable _, provides a much simplified description of the shape and the dynamics of the drop. The understanding of the process that I detailed in the text is obtained from an analysis of the implications of such scaling phenomena.

Sidney R. Nagel, the Stein-Freiler Distinguished Service Professor of Physics at the University of Chicago, has worked on problems dealing with nonlinear and disordered phenomena appearing in macroscopic systems far from equilibrium. He has recently coedited with Andrea J. Liu Jamming and Rheology: Constrained Dynamics on Microscopic and Macroscopic Scales (London, 2001).

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