Singularities in your Kitchen

When Sabine was preparing her talk about black holes and information loss, we thought about other examples of singularities in physical theories besides the centres of black holes in General Relativity. Somehow, the topic seems to pursue me since then - the current issue of the Scientific American welcomes me with Naked Singularities on the title page, and there is even a newly created Singularity University.

Photograph of a drop of a mixture of glycerol in water. The diameter of the drop is about 20 mm. The photo of the right shows the neck in detail. From "A Cascade of Structure in a Drop Falling from a Faucet" by X. D. Shi, Michael P. Brenner, and Sidney R. Nagel, Science 265 (1994) 219-222, via jstor.)

But I was fascinated most by what I've learned since then about singularities in fluid dynamics - singularities that actually occur in the kitchen, every time a drop of water falls off the tap.

A singularity in the mathematical formulation of a physical theory means that a variable which represents a physical quantity becomes infinite within a finite time. This is, actually, not that rare a phenomenon in non-linear theories. For example, in General Relativity, Einstein's field equations when applied to the gravitational collapse of a very massive star develop infinities in density and curvature at the centre of the system. Another famous example of a non-linear theory is fluid dynamics as described by the Navier-Stokes equations - and this is also a habitat of nice singularities.

For example, when a thin jet of water decays into drops, the breakup is driven by surface tension which tries to reduce the surface area. Such a reduction can be realised by diminishing the radius of the jet. Shrinking, triggered by tiny fluctuations of the surface, becomes more and more localised, end eventually, the jet breaks in finite time. The local radius goes to zero, local flow velocity and surface curvature diverge, and the surface is not smooth anymore. Something very similar happens when a drop forms and pinches off from a tap, as can be seen nicely in the photograph taken from the paper by Shi, Brenner, and Nagel. Breakup occurs just above the spherical droplet, where the radius of the thread of the fluid shrinks to zero and the surface becomes kinky.

Of course, a singularity in the Navier-Stokes equations at the pinch-off of a droplet doesn't mean anything mysterious. But it is a hint that in this situation and at small enough length scales, the equations do not make sense anymore, or at least disregard essential physics. In this case, we know of course that the molecular structure of matter becomes important, replacing the continuum description of matter implied by the Navier-Stokes equations. On the scale of molecules, the concept of a sharp and smooth surface is ambiguous, but already at length scales between 10 and 100 nanometer, van der Waals forces between molecules come into play which are not considered in the continuum formulation.

It's a bit of a stretch to say that some similar effect might remove the singularity at the centre of a black hole, but on a very general level a similar breakdown of the theory that predicts a singularity might occur. In this case it would be General Relativity to be replaced by a theory of quantum gravity that accurately describes the region of strong curvature and high density.

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Here are a few paper about singularities in fluid dynamics I found interesting:

• Hydrodynamic Singularities by Jens Eggers, arXiv:physics/0110087v1


• A Brief History of Drop Formation by Jens Eggers, arXiv:physics/0403056v1


• Theory of Drop Formation by Jens Eggers, Phys. Fluids 7 (1995) 941 and arXiv:physics/0111003v1


• Sink Flow Deforms the Interface Between a Viscous Liquid and Air into a Tip Singularity by S. Courrech du Pont and J. Eggers, Phys. Rev. Lett. 96 (2006) 034501 and arxiv:physics/0512095v1 - That's a different kind of "singularity" than in the drop breakup, even extending along a line, and with nice data showing the "approach" towards the singularity.

If you know of other examples of singularities in fluid dynamics, or in other physical systems, I'll be glad to collect them in the comments!

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Tags: Physics, Singularity, Drop formation