Nailing Science: The Maths of Rivers

Creating scientifically accurate nail art whilst discussing my research in fluid dynamics with Dr Becky Smethurst and Dr Michaela Livingston-Banks at the University of Oxford.

We recorded 1h30mins of footage, so this is the heavily edited version of our chat ranging from the fluid dynamics equations needed to describe the flow of water in a river, the Coriolis effect, the experimental set up replicating this, and how these experiments can help with the clean up of pollution.

How to build a river in the lab

My thesis is based on experiments. A weird thing for a mathematician to say you might think, but that’s the truth. It was always planned to be experimental in nature – it even says so in the title – and that’s because it isn’t practical to go to the nearest large scale river outflow (for my work that would be the Rhine in the Netherlands) and start trying to measure things. Fieldwork works well on a Geography trip: you put your wellies on and start splashing around in a stream, measuring the depth with a metre ruler and the speed of the stream by timing a little paper boat as it sails downstream… But things aren’t so easy when you’re talking about a river several kilometres wide and tens of metres deep. The bigger rivers are harder to measure, but the big rivers are precisely the ones that we need to look at, because they’re the only ones big enough to be affected by the earth’s rotation. But we’ll come to that. First up, how do we recreate a big river in the lab?

The trick is to scale things down, as you may have guessed, but there’s a little more to it than just building a scale model of a river. Those little wooden models of a city or building that architects use to help see how their plans will come to life are scale models of the real thing: they are built the same, just with all measurements at a ratio of 1:500 say of the real distances. For example, if a real-life football pitch is 100m in length and 75m wide, then for a 1:500 scale model it would be 20cm long and 15cm wide. A scale model is a good idea in principal, but when working with rivers that are 1km wide and 10m deep, once you scale it down to lab-size, your depth is about as thin as a piece of paper, which isn’t practical. We have to be a little cleverer as mathematicians and think about what properties of the river are the most important and then only include those in our lab model.

I’ll give you an example. Let’s suppose we are trying to work out how fast Usain Bolt travels when he runs the 100m. For ease of the numbers, we can say he runs 100m in 10.0 seconds (a slow day for Usain – he’d had a few too many chicken nuggets before the race). There are many other factors that will affect his speed:

  • The wind was blowing at a speed of 1m/s against him
  • It was raining
  • He was wearing a waterproof coat (he forgot to remove it)
  • His bodyweight was 2kg higher than normal (those damn chicken nuggets)
  • One of his shoes was missing a spike
  • The race was in Brazil

We know that all of these things will affect Usain’s speed, but which do we actually think are important enough to include them in our model? If we ignore them all then at a first guess, we can just use the speed = distance/time triangle from school which gives 100/10 = 10m/s. This would be a first order estimate using just two properties: time and distance. If we want to include more information and get a more accurate answer, then maybe we can include the wind speed: 1m/s against him means he must run at 11m/s to cover 100m in 10 seconds. This is an increase of 10% on our first estimate of his speed, and so probably quite important. Because of the direction of the wind the rain will act against him too, though probably by only a very small amount. The coat will increase the air resistance slowing him down, but again probably quite small.

The key point here is that there are many factors that will affect the speed of Usain Bolt as he’s running the 100m, but as mathematicians our job is to figure out which ones are the most important and to only consider those. If we tried to model every small effect things would get very complicated very quickly and we don’t want that (trust me I’ve tried). Simple is good – so long as you don’t ignore the important bits…

For Usain’s speed we can probably keep the distance, time and the wind speed and that’s about it. Even if we included everything else I doubt the value would change very much from 11m/s, certainly by less than 10% which is a nice acceptable error that we can live with. For scaling rivers down to work with them in the lab we have to do the same thing: pick the important parts of the problem and ignore the rest. The key thing is picking the right bits – which we’ll come onto next time.

 

All of the articles explaining my PhD thesis can be found here.

Where does river water go?

It might seem like a simple question, but just think about it for a second… Water falls from the sky as rain, it flows over and under the ground and enters into a river. The river flows downstream, maybe passing through a few lakes along the way, until it reaches the ocean. Now what happens? The water has to enter the sea and will eventually be evaporated by heating from the sun and end up back in the atmosphere to form rain again. A lovely full-circle route called the water cycle – you probably learnt about it in Geography class. But, what if we could track a raindrop from the sky, into a river and then into the ocean. What would happen? Does it just flow into the ocean and then get mixed up and thrown around in the wind and waves? Or do tides drag it out further to sea? And what about the fact that the Earth is rotating? Perhaps not so simple after all…

Water-Cycle-Art2A.png

Image credit: https://pmm.nasa.gov/education/water-cycle

This is essentially what my thesis is about. When water from a river enters the ocean, where does it go? It took me almost 4 years to figure it out – or more accurately to understand more about what’s actually happening… I certainly do not have all of the answers (as you’ll see). My plan is this series of articles is to try and explain 4 years’ worth of laboratory experiments, fieldwork, computer simulations and of course maths, so that anyone can understand what I’ve done and why it is important.

That’s probably as good a place to start as any: why is it important to understand more about where river water goes? Also known as the classic question faced by all researchers: why should we care? The grand big-picture answer is of course that by understanding more about the world around us, then and only then, can we begin to answer the fundamental questions about the meaning of life, the universe and our very being. That all sounds a little too Brain Cox to me so let’s try a simpler reason… pollution.

From waste outflows leaving factories to fertilisers used by farmers, it all flows into our rivers. And we want to know where this pollution will end up so that we can try to stop it causing too much damage. If we know where river water goes, then we know where pollution goes – easy (or so we hope). Pollution from fertilisers is a particular problem because it’s difficult to stop. If a factory is pumping out pollution into a river we can tell them to stop and to dispose of the waste properly by some other means. If we tell farmers to stop using fertilisers then they produce less crops, which means less food for us – hopefully you can see the issue.

The fertilisers used on crops seep into the soil and enter the underground water supply. This then flows into rivers, which flow into the sea. Fertilisers contain lots of nitrogen and this is great for growing plants – they love the stuff. The problem with having lots of nitrogen in the ocean is that it causes huge plankton blooms. Plankton are little plant-like things floating everywhere in the ocean, basically tiny sea plants. If you get lots of plankton blooming at the surface of the ocean, it blocks the sunlight from reaching the plankton beneath the surface and so they can’t photosynthesise to produce food, which means that they die. Lots of dead material means lots of bacteria. These little critters break down the dead plant material and when doing so use up oxygen in the water until eventually there’s none left. This is very bad news for fish – they need oxygen to breathe – and so they end up dying too. It’s basically a big circle of death which we scientists call eutrophication.

The good news is that if we know where the river water ends up and therefore where the fertiliser ends up, we can put measures in place to stop eutrophication from happening. So no more dead fish – hooray! At least not until they are caught in giant nets, but that’s a whole other kettle of fish… (pun very much intended).

So there you have it: knowing where river water goes means we can control pollution and stop fish from suffocating to death. And of course by understanding more about the world around us we can begin to answer the fundamental questions… nope I can’t do it. I’ll leave the star-gazing to Brain Cox.

 

All of the articles explaining my PhD thesis can be found here.

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