Equations Stripped: Navier-Stokes

A new series where I strip-back the most important equations in maths layer by layer so that everyone can understand them… First up is the Navier-Stokes equation.

Featured post

Naked Maths Trailer

Naked Maths is finally here!

Here’s the trailer for the new video series I’m making with the Naked Scientists taking a look at the maths that’s all around us.

 

Featured post

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.

 

You can read the first article about my thesis here.

Pokémon: the numbers

I love maths (as most of you will know by now) and I also love Pokémon (as some of you will also know), so I’ve decided to combine the two into a new project called Pokémaths. I’ll be doing the maths to answer such questions as: how many Pikachus does it take to power a lightbulb? How many calories would a Charizard need to eat every day to survive? And would Squirtle actually float?

But before I get to these important questions, let’s first talk numbers, specifically numbers of Pokémon. There are a LOT of Pokémon these days – 802 to be exact – and where better than to start with the original and best: Red and Blue. I’m a 90’s kid (though technically I was born in the 80’s which does mean Calvin Harris has love for me) and so I spent far too much of my childhood playing the original Pokémon – I was Red in case you asked. This first generation had 150 Pokémon for you to catch (or was it 151… did anyone ever find Mew?) and they ranged from everyone’s favourite little yellow mouse Pikachu to a big pink blob called Ditto.

The second instalment came with Pokémon Silver and Gold which brought with them exactly 100 new Pokémon. Most of them were still animal based, but there are a few more original ones… including some numbers! The Pokémon Unown (below) could take the form of any of the 26 letters of the alphabet, a question mark or an exclamation mark. So whilst not quite being a number itself, you can certainly collect a few of them together in your team to spell out O-N-E or T-W-O or T-H-R-E-E or F-O-U-R or…. I think you get the picture.

IMG_5194

Generation three meant another 134 new Pokémon, this time as part of the games Ruby and Sapphire, taking the total number to 386. Judging by the naming convention the Pokémon Company clearly put all of their creative effort into creating new Pokémon rather than naming the games… Some of my favourite Pokémon appeared in this generation – take Ludicolo for example, he’s basically a frog wearing a poncho and a sombrero. And don’t forget about Porygon-Z, the fully-evolved form of Porygon, who was one of the original 150. Porygon is great because it’s basically a 3D maths shape and also a play-on-words of polygon, which means any flat shape with at least three straight sides and angles.

Next up were Pokémon Diamond and Pearl – the first step away from colours – hooray! Though they will be back shortly. This generation introduced another 107 new Pokémon and they start to get really freaky… Take Drifloon for example. It’s a balloon. With a face. And a cloud for hair, obviously.

IMG_5195

For generation five we go back to colours, naturally, with Pokémon Black and White. These games were the first since the original ones to introduce a full 150 brand new Pokémon, plus 6 bonus special edition ones. With such pressure comes inspiration – or at least I assume that was the plan. A Pokémon that’s a pile of garbage called Trubbish? Or a candle called Litwick? Really? Let’s just make all inanimate objects into Pokémon… oh, wait they did that. Generation six brought us Pokémon X and Y (another step away from colours, well done Nintendo) and with them came the Pokémon Klefki. This is a set of keys with a face. Its either genius or madness, you decide…

IMG_5196

Generation six gave us 72 new Pokémon, taking the total up to a whacking great 721. Creating that many original characters is quite the feat, even if we have ice cream cones with faces… Vanillite I’m looking at you. The final installation came late last year with another non-colour-named set: Pokémon Sun and Moon. They were released to mark 20 years since the original games – wow, I’m old. The latest instalment brings us 81 new Pokémon giving a final total of 802. That’s 802 unique, original, entirely fictional characters. It’s seriously impressive (despite my complaints).

My plan with Pokémaths is to pick the most interesting and exciting Pokémon (obviously there a lot) and do some maths with them. It will combine my two favourite things and hopefully show you that maths can be used for absolutely ANYTHING. It’ll be fun too – after all who doesn’t want to know how much weight a Machamp can bench press?

 

For my interview with Biologist Matan Shelomi – the creator of the Pokémon evolution tree – click here. To check out the tree he created with real science click here.

 

The Penalty Kick Equation

A couple of weeks ago I talked about the science behind the perfect penalty kick on BBC Radio Cambridgeshire, and lo and behold a few days later the Championship Playoff final went to penalties. I may have jinxed it – sorry. The penalty shootout was actually of a pretty high standard with 7 out of 10 penalties being scored, two saves and only one missing the target. Clearly Reading’s Liam Moore did not listen to my interview…

For all of the footballers out there who have missed penalties recently, I thought I would explain the idea of the science behind the perfect penalty a little further, and in particular the maths equation that describes the movement of the ball. On the radio of course I couldn’t really describe the equation, so here it is:

Screen Shot 2017-06-05 at 10.09.22

If you’re not a mathematician it might look a little scary, but it’s really not too bad. The term on the left-hand side, D, gives the movement of the ball in the direction perpendicular to the direction in which the ball is kicked. In other words, how much the ball curves either left or right. This is what we want to know when a player is lining up to take a penalty, because knowing how much the ball will curl will tell us where it will end up. To work this out we need to input the variables of the system – basically use the information that we have about the kick and input it into the equation to get the result. It’s like one of those ‘function machines’ that teachers used to talk about at school: I input 4 into the ‘machine’ and it gives me 8, then I put in 5 and I get 10, what will happen if I input 6? The equation above works on the same idea, except we input a few different things and the result tells us how much the ball will curl.

So, what are the inputs on the right-hand side? The symbol p just represents the number 3.141… and it appears in the equation because footballs are round. Anytime we are using circles or spheres in maths, you can bet that p will pop up in the equations – it’s sort of its job. The ball itself is represented by R which gives the ball’s radius, i.e. how big it is, and the ball’s mass is given by m. We might expect that for a smaller ball or a lighter ball the amount it will curl will be different, so it is good to see these things are represented in the equation – sort of a sanity test if you will. The air that the ball is moving through is also important and this is represented by r, which is the density of the air. It will be pretty constant unless it’s a particularly humid or dry day.

Now, what else do you think might have an effect on how much the ball will curl? Well, surely it will depend on how hard the ball is kicked… correct. The velocity of the ball is given by v. The distance the ball has moved in the direction it is kicked is given by x, which is important as the ball will curl more over a long distance than it will if kicked only 1 metre from the goal. For a penalty this distance will be fixed at 12 yards or about 11m. The final variable is w – the angular velocity of the ball. This represents how fast the ball is spinning and you can think of it as how much ‘whip’ has been put on the ball by the player. Cristiano Ronaldo loves to hit them straight so w will be small, but for Beckham – aka the king of curl- w will be much larger. He did of course smash that one straight down the middle versus Argentina in 2002 though…

So there you have it. The maths equation that tells you how much a football will curl based on how hard you hit it and how much ‘whip’ you give it. Footballers often get a bad reputation for perhaps not being the brightest bunch, but every time they step up to take a free kick or a penalty they are pretty much doing this calculation in their head. Maybe they’re not quite so bad after all…

 

 

 

 

 

 

 

The Game of Life

The simple mathematical game that inspired the pattern on the facade at Cambridge North station – live interview with BBC Radio Cambridgeshire.

 

Examples (image credits: Wikipedia)

Stable ‘still-life’ patterns that remain fixed for every turn.

Block                 Beehive             Boat                Loaf                    Tub

Game_of_life_block_with_border       98px-Game_of_life_beehive   82px-Game_of_life_boat     98px-Game_of_life_loaf     Game_of_life_flower

‘Oscillators’ that cycle through a number of designs repeating every few turns.

Beacon (period 2)     Blinker (period 2)     Toad (period 2)

Game_of_life_beacon                 Game_of_life_blinker                Game_of_life_toad

Pulsar (period 3)          Pentadecathlon (period 15)

Game_of_life_pulsar                I-Column

Spaceships that travel across the board forever.

Glider             Lightweight Spaceship

Game_of_life_animated_glider        Game_of_life_animated_LWSS

You can play the game of life for yourself online here.

WordPress.com.

Up ↑