Plateau-Rayleigh instability

Black holes in your kitchen

LIGO photo
Evidence for the collision of two black holes was found a few years ago at the LIGO detectors. But where can you find a link between your coffee and a black hole? An aerial photo of the LIGO detector at Hanford. Image courtesy of Caltech/MIT/LIGO Laboratory

Where, in your kitchen, would you find a link between the physics of the everyday and the physics of black holes? It’s a question with many answers, and maybe you could think of a few. But one involves a process you may see while brewing your coffee, though you may have to slow down to see it.

The connection is in the way a gentle stream of water breaks up into droplets as it falls. Brewing coffee using a swan necked kettle in a V60, it is something that I see as I slow the rate of pour. Is this a good way of preparing a coffee? Possibly not, but it does allow me to experiment with the physics. You could also see the effect from a slowly dripping tap or in a few other places around the home. It occurs when the cylinder of flow is much longer than the radius of the flowing water.

The question is really, why would a cylinder of flowing water seemingly spontaneously break up into a broken stream of raining droplets? The answer is in a phenomenon now known as the Plateau-Rayleigh instability.

To see why it may occur, we can think about how water flows out of a kettle or a tap. In any cylinder of fluid there will be regions of the flow that are a bit fatter and regions that are a bit thinner. These can be imagined as a series of waves on the surface of the cylinder (you can see a schematic of this effect here). At small wavelengths, the water cylinder remains stable, so for very rapid (but small) fluctuations in the diameter of the flow, you will not notice any difference to the way you pour. But as the wavelengths become larger, and beyond a critical wavelength, the amplitude of these oscillations increase rapidly with time (the maths describing the ‘why’ is here).

kettle, V60, spout, pourover, v60 preparation
Pouring water from a swan necked kettle offers a perfect opportunity for observing Plateau-Rayleigh instabilities.

As the amplitude of the oscillations grows, there will come a point at which the bulges are so large and the necks of the stream so thin (relative to the stream’s diameter) that surface tension effects will cause the necks of the cylinder to break resulting in the stream of droplets that you see. When Plateau first observed this in 1873, he thought that the continuous stream became a flow of droplets when the length of flow was just over 3 (around π) x the radius of the flow. In fact, the break up seems a little more complex, and from my V60 kettle I’d estimate that the length at which it occurs is greater than 3x the radius of the pour, but the experiments of Plateau and the theory of Rayleigh did rather explain what was going on with the stream.

How is this related to black holes? Black holes are massive objects that exist within a very small region of space. Many black holes are thought to be the result of the collapse of a massive star at the end of its life, although there are examples of smaller and more massive black holes. The sort that result from a collapsed star can have a mass around 20x that of the Sun but fit into a space with a diameter of just 10 miles, which is about the distance from Heathrow to Hammersmith (still not central London!). Every planet, moon, star or black hole has an “escape velocity” associated with it that is a function of the object’s mass. The escape velocity is the speed at which you would need to move away from the object in order to avoid being pulled back to the object’s surface. For the earth you need to travel at more than about 11 km per second in order to escape the earth and enter into orbit around it (or move beyond that). For the moon, because it has much less mass, the escape velocity is far lower. For a black hole, the escape velocity is much higher and actually exceeds the speed of light.

The “event horizon” of a black hole is the point at which the escape velocity from the black hole is so high that it exceeds the speed of light. We cannot see into the black hole, because the light cannot escape from within the event horizon.

a heat sensitive coffee mug
What other astronomical connections can you find in your coffee cup? Do let me know what you think.

It turns out that for certain mathematical reasons, it can be useful to consider the event horizon as a stretched fluid membrane with elastic like properties much the same as the surface tension causes to water. At this point it gets a little complicated because not all black holes are spherical*, some indeed can be cylindrical. So we have a cylindrical object with an event horizon with properties that cause it to behave in a manner similar to a fluid with surface tension.

You may well have seen where this is going already. Because yes, it turns out that such cylindrical “black branes” are susceptible to breaking up into many smaller objects exactly analogously to the Plateau-Rayleigh instability in a stream of water. Exactly how they broke up (eg. did they break into spherical objects) was left to further investigation, but the maths was developed in a 2006 study to explore this phenomenon further, you can read more about it here.

It is a bit of a bizarre connection to realise in your kitchen. But the world is often weirder, more beautiful, and more connected than we are sometimes tempted to think. Do let me know of other astronomical connections to your kitchen that you can see. I can think about one or two more related to black holes, but I’m sure you can think of many more. Please just leave a comment below, on Twitter or on Facebook.

*This is certainly true in the maths of black holes, it’s too far outside my subject field to know if such objects have been observed or thought to have been observed in reality.