One important use of mathematics is to describe things that happen in the natural world around us. For hundreds of years now, mathematicians have worked together with physicists to explain, predict and quantify all manner of natural phenomena; from the solar eclipse and the movement of the planets, to the spread of diseases and infections.
But what happens when mathematics predicts something that seems too strange to be true? Is it a bug in the system, or is the world sometimes stranger than we might expect? This week, we take a look at some weird results from the field of fluid mechanics.
The hydraulic jump
A common example of a hydraulic jump. Image: Wikipedia, CC BY-SA 3.0.
When you switch on a tap, you might notice a circle directly underneath the tap where there seems to be no water. If so, congratulations — you’ve spotted a hydraulic jump! A hydraulic jump is when a moving fluid suddenly become a lot deeper, and this often happens when fluid has to slow down for some reason. When you turn on the tap, free-falling water suddenly collides with the sink, which slows is down and creates the jump. There is actually a very thin layer of water inside the circle, and a much deeper layer outside it. To understand how the sudden change in speed can lead to a hydraulic jump, it’s best to think about another place where they’re very common: rivers.
Imagine that you’re standing idly by a river, watching the water flow. If you study fluid mechanics, you might do this quite often, and pretend you’re doing science. How much water is flowing past you? If the depth of the water is $h$, the speed is $v$ and the width of the river is $w$, then the volume of water flowing past you is
$$ Q = h v w.$$
Now imagine that a bit further along the river (downstream, in “technical” terms) there is a big rock. As the water hits the rock it has to slow down dramatically, just like it did in the sink. That is, $v$ will get smaller. But the volume of water flowing in the river can’t change — everything that went past you hits the rock, and everything must carry on going after that. So $Q$ must stay the same.
If the width of the river doesn’t change between you and the rock, then there is only one way for both of these statements to be true. If $v$ gets smaller and $Q$ and $w$ stay the same, then $h$ must increase and the height of the water must rise just as suddenly and dramatically as the speed goes down. This is a hydraulic jump. If the thing that caused the fluid to slow down (the sink or the rock) isn’t moving, then the hydraulic jump will stay in the same place and there will be a part of the river that suddenly changes depth.
Why is this so weird? Normally, we think of liquid in one of two ways. A lot of the time, water is still and flat, like a glass of water. You can stir it, shake it or drop something into it, but after a while it will go back to being a flat, still surface. On the other hand, water is sometimes bumpy and moving, like the sea. The sea is full of waves, but we can see these moving around and going up an down. What’s interesting about a hydraulic jump is that it is both bumpy and still. The surface of the water is not flat, but the bump stays just where it is, which is different from what we normally expect it to do.
I was sat by a river in a park recently (told you people who work in fluids do it a lot…!) and noticed that the water was full of these hydraulic jumps. The bit of the river where I was sitting was full of rocks, and it had created loads of mini-jumps, one over each rock. Just five metres further on, the bottom of the river was flat and the surface was smooth. You can see the river in this video below:
OK, that’s a bit cool. But hydraulic jumps can get much bigger. Some of the most impressive jumps can be found on the river Eisbach in Germany, which is now the most popular place in the world for the sport of river surfing, something that seems to totally defy the laws of physics…
The Kelvin-Helmholtz instability
In fluid dynamics, instability is what happens when small perturbations (waves) get bigger and bigger. They’re an important area of study because they can drastically change the type of equations that you need to study a fluid, and so understanding when instabilities occur helps us understand which equations to use in which situations. One of the most famous, and most beautiful, types of instability is called the Kelvin-Helmholtz instability. This can happen when two different fluids are flowing next to each other at different speeds, and results in a train of evenly spaced waves with curly tops — just like the clouds in the banner image for this article!This video from Sixty Symbols does a great job of explaining how the difference in speed can create the waves. They also show footage of creating the instability in a laboratory. The slow-motion clip at 1:10 is definitely worth checking out:
Given that the conditions for the instability are quite simple, and should happen quite often, you might expect these waves to happen all the time. And you’d be right! As well as clouds, the Kelvin-Helmholtz instability occurs in oceans, both when wind blows across the surface of the sea and deep in the ocean, where different layers of seawater flow past each other. But, for me, there is one example of Kelvin-Helmholtz that is superior to all others.
The planet Jupiter is the largest in the solar system and, thanks to modern high-resolution satellite photography, we know more about it than ever before. It turns out that Jupiter is a windy place: the whole surface is covered in violent storms that rush round the planet at great speeds. Without any land to disrupt their flow these storms keep going round and round, creating the band patterns that we see in photographs. Because not all the storms go at the same speed, there are lots of opportunities for the Kelvin-Helmholtz instability, and sometimes the waves get so big that they can be seen from space, as in this image taken by NASA’s Cassini probe in 2000 (you can best see the Kelvin-Helmholtz waves just to the right of the great red spot, which is another amazing fluids feature that you can find on Jupiter)
The surface of Jupiter, filmed over 24 Jupiter days in 2000. Image: Wikimedia commons.
To me, this is one of the best things about fluid dynamics. You study some (fairly dry) mathematics, use it to prove a slightly strange result, and then learn something about beautiful natural things happening everywhere in the galaxy. Amazing.
Reversibility of Stokes’ flow
So far we have discussed fluids which a fluid dynamicist would usually consider inviscid. This means that the friction and stress forces between the fluid particles is very small, and doesn’t need to be considered in huge technical detail when we come to study the mathematics of it all. Typically a fluid dynamicist who wants to study these fluids will ignore the term in the governing equations which accounts for viscosity (the name for friction in fluids) — a very common solution in applied mathematics! However, everyone knows that there are much more complicated fluids to be found everywhere. Examples where we can’t ignore viscosity include oil, tar, honey… and I could go on!
Mathematicians of course sometimes want to study these more complex fluids, and this gives rise to a new approximation of the equations governing them. This approximation is called Stokes’ flow, and in contrast to the earlier mentioned approximation, Stokes’ flow involves the assumption that the term governing viscosity is much more important than the other terms involving the fluid velocity, and so the remaining terms can be ignored.
One particularly interesting feature of Stokes’ flow is that it is reversible. This means that by reversing the forces applied to the fluid, the velocity of the fluid (and hence the trajectory for each particle, or the way that the fluid moves) is also reversed. Mathematically, this is because the Stokes’ flow equations have a kind of symmetry. The equations are
$$ \nabla \cdot \mathbf{u}=0, $$
$$ 0=-\nabla p +\mu \nabla^2 \mathbf{u}+\mathbf{F}. $$
It’s not important to understand what these equations mean, all we need to notice is the following: if we reverse the force ($\mathbf{F} \rightarrow -\mathbf{F}$), then to keep the equations the same requires that we compensate by reversing the velocity and pressure fields ($\mathbf{u} \rightarrow -\mathbf{u}$ and $p \rightarrow -p$). This means that the fluid will have to travel in the opposite direction, and retrace its path. Put simply: if you apply a force to a fluid that can be described with Stokes’ flow, the fluid will move. If you then apply the equal and opposite force, it will exactly reverse the motion that it just did. If this sounds boring, then I highly suggest watching the video below… You’ll change your mind, I know it!
Whew! That’s about enough weirdness for now. Let us know if we’ve missed anything; send your favourite freaky results (or just say hi!) to contact@chalkdustmagazine.com.
This post is part of a series for the UCL year 12 maths research summer programme, where year 12 students investigate an area of maths under the guidance of a Chalkdust member and PhD student at UCL. The closing event will celebrate the work of the year 12 students, and will feature a talk from Prof. Lucie Green, a space scientist and TV & radio presenter. The closing event is on Thursday 12th July and is open to the public. If you would like more information about the programme or to attend the closing event please contact Dr Luciano Rila l.rila@ucl.ac.uk.
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