How do Bubbles Freeze?

Freezing bubbles are not only beautiful, but also demonstrate incredibly complex physics. Here, Professor Jonathan Boreyko explains how bubbles freeze with examples of slow motion videos filmed in his laboratory at Virginia Tech.

This video is part of a collaboration between FYFD and the Journal of Fluid Mechanics featuring a series of interviews with researchers from the APS DFD 2017 conference.

Sponsored by FYFD, the Journal of Fluid Mechanics, and the UK Fluids Network. Produced by Tom Crawford and Nicole Sharp with assistance from A.J. Fillo.

Using maths to clean-up our oceans

Video of my ‘Teddy Talk’ at the 2019 St Edmund Hall open day.

Rivers are the major source of pollution in the oceans and if we are to clean them up, we first need to know where the majority of the pollution is concentrated. By creating a mathematical model for river outflows – verified by laboratory experiments and fieldwork – the goal is to be able to predict which areas are most susceptible to pollution from rivers and thus coordinate clean-up operations as effectively as possible.

What is the blast radius of an atomic bomb?

Picture the scene: you’re a scientist working for the US military in the early 1940’s and you’ve just been tasked with calculating the blast radius of this incredibly powerful new weapon called an ‘atomic bomb’. Apparently, the plan is to use it to attack the enemies of the United States, but you want to make sure that when it goes off any friendly soldiers are a safe distance away. How do you work out the size of the fireball?

One solution might be to do a series of experiments. Set off several bombs of different sizes, weights, strengths and measure the size of the blast to see how each property affects the distance the fireball travels. This is exactly what the US military did (see images below for examples of the data collected).

Picture1 Picture2

These experiments led the scientists to conclude that were three major variables that have an effect on the radius of the explosion. Number 1 – time. The longer the time after the explosion, the further the fireball will have travelled. Number 2 – energy. Perhaps as expected, increasing the energy of the explosion leads to an increased fireball radius. The third and final variable was a little less obvious – air density. For a higher air density the resultant fireball is smaller. If you think of density as how ‘thick’ the air feels, then a higher air density will slow down the fireball faster and therefore cause it to stop at a shorter distance.

Now, the exact relationship between these three variables, time t, energy E, density p, and the radius r of the fireball, was a closely guarded military secret. To be able to accurately predict how a 5% increase in the energy of a bomb will affect the radius of the explosion you need a lot of data. Which ultimately means carrying out a lot of experiments. That is, unless you happen to be a British mathematician named G. I. Taylor…

Taylor worked in the field of fluid mechanics – the study of the motion of liquids, gases and some solids such as ice, which behave like a fluid. On hearing of the destructive and dangerous experiments being conducted in the US, Taylor set out to solve the problem instead using maths. His ingenious approach was to use the method of scaling analysis. For the three variables identified as having an important effect on the blast radius, we have the following units:

Time = [T],       Energy = [M L2 T-2],      Density = [M L-3],

where T represents time in seconds, M represents mass in kilograms and L represents distance in metres. The quantity that we want to work out – the radius of the explosion – also has units of length L in metres. Taylor’s idea was to simply multiply the units of the three variables together in such a way that he obtained an answer with units of length L. Since there is only one way to do this using the three given variables, the answer must tell you exactly how the fireball radius depends on these parameters! It may sound like magic, but let’s give it a go and see how we get on.

To eliminate M, we must divide energy by density (this is the only way to do this):

eqn1.png

Now to eliminate T we must multiply by time squared (again this is the only option without changing the two variables we have already used):

eqn2.png

And finally, taking the whole equation to the power of 1/5 we get an answer with units equal to length L:

eqn3

This gives the final result that can be used to calculate the radius r of the fireball created by an exploding atomic bomb:

eqn4

And that’s it! At the time this equation was deemed top secret by the US military and the fact that Taylor was able to work it out by simply considering the units caused great embarrassment for our friends across the pond.

I love this story because it demonstrates the immense power of the technique of scaling analysis in mathematical modelling and in science in general. Units can often be seen as an afterthought or as a secondary part of a problem but as we’ve seen here they actually contain a lot of very important information that can be used to deduce the solution to an equation without the need to conduct any experiments or perform any in-depth calculations. This is a particularly important skill in higher level study of maths and science at university, as for many problems the equations will be too difficult for you to solve explicitly and you have to rely on techniques such as this to be able to gain some insight into the solution.

If you’re yet to be convinced just how amazing scaling analysis is, check out an article here explaining the use of scaling analysis in my PhD thesis on river outflows into the ocean.

And if that doesn’t do it, then I wish you the best of luck with those atomic bomb experiments…

Climate Change will increase Turbulence on Flights

We’ve seen many recent extreme weather events – from mudslides in Columbia to flooding in Australia – which scientists say are a consequence of climate change; but it’s not just the weather that is affected. The Earth’s atmosphere is made up of several layers of air which all flow around each other in patterns known as jet streams and an increase in temperature will cause these to speed up. This is bad news for air passengers, including the 1 million people currently airborne at this very instant, because an increase in the speed of the jet streams will cause more turbulence making flying less comfortable and potentially more dangerous. I spoke to atmospheric scientist Paul Williams…

  • Climate change will cause a 59% increase in light turbulence, 94% increase in moderate turbulence, and 140%  increase in severe turbulence.
  • Turbulence is measured on a scale from 1 to 7 where 1 means light turbulence, 3 means moderate, 5 means severe, and 7 means extreme.
  • Light turbulence is a slight strain against the seat belt, moderate turbulence causes unsecured objects to become dislodged and makes walking around difficult, and severe turbulence results in anything that isn’t strapped down being catapulted around the cabin.
  • Turbulence is caused by wind shear – the higher you go up into the atmosphere the windier it gets – and instabilities within these layers of shear generate turbulence.
  • As the atmosphere is heated, the temperature increase causes the jet streams to move faster, creating more wind shear and thus more turbulence.
  • The researchers hope that results such as this will encourage us to think more carefully about our carbon footprint as there are likely many effects of Climate Change that we do not know about.

You can listen to the full interview for the Naked Scientists here.

Do roast potatoes give you cancer?

The UK Foods Standard Agency issued a health warning in 2017 about the chemical acrylamide – found in starchy foods such as bread and potatoes – saying that it may cause cancer. The warning coincides with the launch of a new health initiative called ‘go for gold’ which encourages us to only cook foods to a golden yellow, rather than brown or black, to help to reduce the amount of acrylamide. I spoke to Jasmine Just at Cancer Research UK…

  • Acrylamide is produced naturally by starchy foods when they are cooked at high temperatures for a long period of time, such as when baked, fried, roasted or toasted.
  • It is created by the Maillard reaction that occurs between sugars and amino acids in the presence of water, which is also responsible for the brown colour and roasted taste.
  •  A number of animal studies have found that acrylamide has the potential to damage our DNA which can lead to cancer, but the same process has yet to be established in humans.
  • The risk is described as ‘probable’ but is certainly much less than that from smoking, obesity and alcohol.
  • The advice from Cancer Research UK is to maintain a healthy balanced diet, follow the cooking recommendations for baked or roasted goods, and to not store potatoes in the fridge as this increases the potential for acrylamide to develop when they are cooked.

You can listen to the full interview for the Naked Scientists here.

Alien maths – we’re counting on it

Are we alone in the universe? The possibility that we aren’t has preoccupied us as a species for much of recent history, and one way or another we need to know. The problem is, there is a lot of space, and only so fast you can move around in it, so popping over to our nearest neighbouring star for a quick look around is off the table. We simply don’t know how to communicate or travel faster than light. Nor have we picked up any signals which are identifiable as any sort of message from little green men.

Therefore, perhaps our best chance of making contact with an alien species is to announce ourselves to the universe. If we send out messages to promising-seeming parts of space in the hope that someone will be there to receive them, we might just get a response.

But supposing our signals reach alien ears (or freaky antenna things or whatever), what hope do we have of them being understood? Sure, we might make signals which are recognised as deliberate (and not mistaken for more literal ‘messages from the stars’), but how will they get anything across to aliens whose language is entirely unknown to us?

Scientists in the ‘70s were asking themselves these very questions, and the most promising approach they came up with to get around this problem was one which used maths. In fact, it used an ingenious trick dating back all the way to the Ancient Greeks. The fruit of their labour, broadcast in 1974, was called the Arecibo message.

So, what is it? First off, the Arecibo designers gave up on the hope of sending a written message the aliens could read. Better to stick with pictures – you have to assume aliens will be pretty low down on the reading tree. But this still leaves a conundrum.

When you’re sending a message to space, you have to send a binary signal – a series of ‘1’s and ‘0’s (aka bits) which you hope will start to mean something when it’s processed on the other end. This is precisely how sending pictures over the internet or between computers works too – your message is turned into bits, beamed to the other computer, and then turned back.

And herein lies the problem; the aliens receiving the binary signal won’t have any idea what they’re supposed to do with the bits or how to piece the message back together to make a picture again. You’ve posted them a Lego set but no instructions, and even though they’ve got the bricks there’s no way they’ll figure out whether it was supposed to be built into a race car or a yellow castle. After all, they might not even know what those are!

The way around this is to make the process for turning the message into a picture as simple as possible, so the aliens will be able to guess it. And the way you turn the bits into a picture really is very simple – just write them out in a 23×73 grid, and colour in any square with a ‘1’ in it. Below is what you get (with added colour-coding – see below for what the different parts mean).

aricebo

White, top: The numbers 1 to 10, written in binary

Purple, top: The atomic numbers for the elements in DNA

Green: The nucleotides of our DNA

Blue/white, mid: A representation of the double helix of DNA. The middle column also says how may nucleotides are in it.

Red: A representation of a human with the world’s pointiest head, with the average height of a man to the left, and the population to the right.

Yellow: A representation of the solar system and the sizes of the planets, with Earth highlighted

Purple, bottom: A curved parabolic mirror like the one used to send the message, with two purple beams of light being reflected onto the mirror’s focus, and the telescope’s diameter shown in blue at the bottom.

Image credit: Arne Nordmann 

But how, you might ask, are the aliens supposed to figure out the 23×73 dimensions of the grid? Here is where Ancient Greek maths comes to save us.

The Arecibo message is 1679 bits long. That sounds random, but it is anything but – 1679 is actually the product of two numbers, 23 and 73. Sound familiar? That’s the dimensions of the picture! It’s precisely the fact that 1679 equals 23 times 73 that lets you write out the 1679 bits in a 23×73 grid.

You might be wondering why we used such weird numbers for the sizing. Couldn’t we have chosen nicer, rounder numbers for the picture, like 50×100 say? No. If we did that, the aliens might make a mistake like writing out the bits in a 5×1000 grid or a 500×10 grid, and this would still work numbers-wise because 50×100 = 5×1000 = 500×10.

The key here is that unlike 50 and 100, 23 and 73 are prime numbers. Primes are numbers which can only be divided by one and themselves, like 3 and 5. And most importantly, any number can be split up into primes in a unique way – for instance, 15 is 3×5, and there is no other way to get 15 by multiplying together prime numbers. Likewise, there is no other way to get 1679 than as 23 times 73. So, it is impossible for the aliens to make a mistake when they have to draw out the grid. The Lego set you posted may have no instructions, but you were careful to include parts which can only go together the right way.

An Ancient Greek called Euclid knew this key fact, that numbers split uniquely into primes, over two thousand years ago. The Arecibo designers are banking on the aliens being at least as good with numbers as he was, to be able to decipher the message. Given these are aliens who are capable of picking up a radio signal from space, it seems like a pretty safe bet that they can manage better than an ancient society which believed women have fewer teeth than men because a . It’s a gamble, and it relies on assumptions that the maths we’re interested in is what all species will be interested in – but then what part of blindly shooting intergalactic friend requests into space in the hope someone we’d want to know finds them wasn’t going to be a gamble?

Joe Double

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