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

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.**265**(1994) 219-222, via jstor.)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.

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!

Tags: Physics, Singularity, Drop formation

## 11 comments:

In fluid dynamics there is onset of turbulence and, much later, shocks. Cavitation. Smooth goes rough.

Also true for the Casimir effect and contingent Scharnhorst effect at

verysmall separations. Graphite's planes are unremarkably held together by van der Walls forces not 1/r^4 Casimir. MgB_2 planes are not especially naughty even when they are being otherwise naughty as a high temp BCS superconductor.Aren't there ray optics singularities?

Wave catastrophes:

http://arxiv.org/abs/physics/0309036

Hi Stefan,

This is very interesting to me that you two would solidify the theoretics into a metaphor.

Over the years I tried to look for comparative examples as well. One of these was sonoluminescence, not that I would say free energy is ever produced( has been falsified), but the example of the bubble collapse itself. How, it is collapsed.

But your droplet as well too. I tried to capture it in abstraction here in raindrops

But the point you are making here,

Stefan: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.Is exactly the point I have been trying to stress to Moshe, Clifford and others for a while now. My pet theory perhaps?:)

Moshe writes,"

Nature is relativistic, and this fact is crucially important!"It's located at the cross over point and and I have taken Moshe to it in a most general way. The QGP is a fluid and relativistic in interpretation by Navier Stokes. That's my point about how to avoid the singularity. No one wants to listen from a such a layman.

And then of of course such an example is revealed in the cosmos...Yada yada yada...:)

Thanks for continuing to let me comment here unconstrained, of course aware of your policy.

Best,

Hi Stefan,

An interesting post that draws analogy between singularities that we can deal with, to those that we currently can’t. For me this has always related back to the central issue that you point to, which is to ask if a continuum can exist in physical reality and then to further ask if both matter/energy and space can be considered the same in this respect?

To me it has always appeared as possible for space to be imagined as a continuum, which also would concede to it being homogeneous. With matter/energy however this is more difficult, especially with quantum physics revealing its before hidden atomic nature. So at a black holes singularity it’s easy to picture space drawing down to a infinite point over infinite (uncountable distance or time perhaps), yet difficult to have matter/energy as doing the same.

So perhaps a singularity acts like a strainer of sorts, where the two are simply separated locationally (dimensionally) from one another as they have in your more worldly examples. From a GR perspective it has always appeared that energy/matter form to be some sort of an irritant for space, as if the two really didn’t belong together. From this standpoint the singularity represents being just another symptom of this incompatibility. I realize this is simply fanciful conjecture, yet perhaps not an unreasonable one.

Best,

Phil

really slow drop:

http://www.physics.uq.edu.au/pitchdrop/pitchdrop.shtml

Hi Arun,

thanks for the hints!

Aren't there ray optics singularities?Argh, don't try to google for "ray singularity" - it produces only Kurzweil hits ;-)

I am not so sure which equations dynamically produce finite-time singularites in ray optics? Or is this more of a geometrical effect? Ah, but Michael Berry comes to my mind in this respect - he quotes as research interests "Singularities of bright light (caustics)" and "Singularities of faint light ... phase singularities", and there are lots of papers by him online.

Hi Plato,

ah, hadn't seen Moshe's post and the following discussion, thanks... Well, the problem with singularities General Relativity is obviously much more difficult to resolve than in fluid dynamics ;-)

Cheers, Stefan

Hi,

I've just seen, Dmitry at NEQNET has a post on Physics of turbulence: four puzzles, which also touches on hydrodynamical singularities. Interestingly, he mentions that removing the viscosity term from the Navier-Stokes equations even worsens the situation about singularities:

A typical solution of the Euler equations corresponding to a flow with a large number of vortices wants to blow up in finite time.Best, Stefan

Stefan:Well, the problem with singularities General Relativity is obviously much more difficult to resolve than in fluid dynamics ;-)To be "lead by science" for sure.

IN the experimental process where is such a crossover point located? Firstly, to define the QGP "as a fluid" is current? How did "that information" get to the "other side?" Again, experimental processes are revealing that such collision processes are producing results, and we have the experimental backdrop that is measuring it.

The characteristic of the "fluid itself" are then in question. Where does this take you? I am not sure how ruling out the viscosity will help you there. The character and nature of the viscosity allows for relativistic interpretations in that microscopic process?

Even though the information is badly scrambled, it is evidenced that such a "cross over" takes place?

We are made of star stuff?:)

Best,

Yes interesting of course and thanks Stefan. Continuity and smoothness allow for "an object of cosmological appeal" and a "rejuvenation of the cosmos" to exemplify it's current state.

Increase the number of blackholes cosmologically and what happens?

Hmmmm.....

Best,

Hi Stefan,

I thought this post of yours should be linked in more ways, then just one.

Again I thought it would be good to pull this "theme in mind" about droplets form an earlier comment.

CliffordMarch 3, 2008 at 10:30 pm

Stefan… actually it was a coincidence, and from what you point out, a nice one. I don’t know anything about the mathematics of droplets, but it seems likely that there’d be a singularity, indeed. No, Dave did not talk about this, as far as I could tell.Best,

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