Fluid mechanics

Custard, paddling pools, and plasterers’ tools: all part of applied mathematics!

I don’t think I can list walking on custard as one of my hobbies – it’s not exactly a weekly activity – but I have found it a fun way to demonstrate the curious and counter-intuitive behaviour of some fluids. Yes, you can walk on custard. There’s no trick. You just need water and custard powder…and a lot of it!

The last time I did this was at the IMA Festival of Mathematics, a two day event organised by the Department of Mathematical Sciences at the University of Greenwich and the Institute of Mathematics and its Applications that featured many excellent talks and activities on the importance, usefulness, and pleasure of mathematics. Reviews of the festival can be found in PrimeTimes (a magazine written and edited by the University of Greenwich Mathematics Society) and in the latest edition of Mathematics Today. The festival also featured a family-size paddling pool, a plasterer’s bath, and about 200kg of cornflour. You can press “play” on the video below to see this in action.

This video (uploaded to the Twitter account of the Department of Mathematical Sciences, University of Greenwich, @Maths_GRE) shows Greenwich Maths students and staff jumping, running, mixing, and punching a mix of cornflour (the main ingredient of custard powder) and water. Nothing else, but a lot of mess.

Custard, or to be more precise cornflour and water mixed in the right proportions , is an example of a non-Newtonian shear-thickening fluid.  The “non-Newtonian” means the fluid does not follow the law postulated by Sir Issac Newton, which says that the “thickness” or “lack of slipperiness” of fluids remains the same no matter what you do to them (your natural intuition of fluids is probably in-line with  Newton’s law – you don’t feel the sea getting more or less slippery as you swim). The “shear-thickening” means the fluid does something really weird – it gets harder when you apply a force it.

If you slowly dip your hands into a shear-thickening fluid like the custard mix you see in the video then it will behave more-or-less like water. This is because the force you are applying (dipping you hands) is quite gentle. Likewise, if you slowly step into such a fluid you will sink – it is a fluid, after all! The interesting things happen when you hit or punch a shear-thickening fluid. Hitting and punching exerts a much stronger force and this force can actually cause these sorts of fluids to become momentarily thicker and harder. In some fluids this thickening can be so extreme that you will not be able to punch through their surface; you won’t even make a splash. Shear thickening is not a property of most fluids but it is a property of cornflour and water (i.e. custard) when mixed in the right ratio.

Another way to apply a large and (almost) continuous force to a shear-thickening fluid is to run or jump on it. As long as you keep running or keep jumping then you will not sink. If you stop running then stop applying a force, and if you stop applying a force then the fluid will go back to being runny, and if the fluids is back to being running when you are in it then….well, you are going to get wet and messy!

Sound impossible? It’s not – have a look at the video.

You can do this experiment with regular custard powder or cornflour. You must, however, use a high enough concentration of powder. About 2 parts custard powder (or cornflour) to 1 part water by volume will do it (that’s two cups of powder for every cup of water – closer to 1.5 parts powder should be sufficient but 2:1 puts you on the safe side). Remembering that 1 litre of water weighs 1kg, this means a lot of custard powder to fill the plasterers’ bath you see in the video.

The giant paddling pool was for protection only. It would have been great to fill a 2300 litre pool of custard but I’m afraid our budget wouldn’t stretch that far! And can you imagine disposing of it? (You can’t just tip a highly shear thickening fluid like this down a drain – the consequences are very problematic!). No, the pool was to protect the beautiful Old Royal Naval College, the home of the University of Greenwich. You can probably see from the video that custard walking is fun but messy – the powder gets everywhere. The pool simply contained the mess and protected the World Heritage site. If you want to try this at home then please remember to consider disposal and mess – you’ll regret it if you don’t.

The technical word for the “thickness” or “lack of slipperiness” of a fluid is viscosity. All fluids have viscosity. Some fluids have a constant viscosity  (water), others have a viscosity that decreases under a shear force (shear thinning fluids, e.g.  ketchup), and others such as cornflour-based custard have a viscosity that increases under a force. Scientists who study such things are called rheologists (rheology is the science of deformation and flow). Rheology brings together people from many disciplines – mathematics, physics, chemistry, engineering, computer science…These people bring their expertise to the table and collaborate to better understand fluids and advance the technology around us. If you don’t think rheology is that important or common then have a look around. How many examples of fluid flow have you seen today? Maybe you are having a cup of coffee or a beer. You probably brushed your teeth today (I hope so!) Maybe you washed and conditioned your hair, or had a shave. Think about all the blood circulating your body. The oil that we use everyday, for better or worse, and it’s extraction process. Mud, cement, and lava. How many things made of plastic do you see around you? They were probably in a rheological state before you used them. Then there’s the more classical Newtonian fluid mechanics that’s relevant to aerodynamics (yes, air is a fluid), the design of sports equipment, weather, oceans, and climate….

And, of course, there’s custard.

Perhaps you are wondering why or how some fluids shear thickening. This has been a a source of confusion and interest for a long time. The answer is not a simple one but I’ll try to address it in a future post. In the meantime, try not to put a dense mix of custard into your washing machine. I did this once. It was a very costly experiment.

This activity would not have possible without the hard work and dedication of Tony Mann and Noel-Ann Bradshaw from the Department of Mathematical Sciences at the University of Greenwich. We also had great help and support from our students. Jonathan Histed deserves an special “thank you” for giving up his time (and tools!) to help make this happen.

Fluid mechanics

Waves and ripples: maths applied on holiday

Half way through a hike (well, more of a long walk, really) in the mountains of the beautiful La Cumbrecita, Córdoba, we spotted a little cove that looked the ideal place for a well-deserved rest (it may be winter here, but it was 30 degrees with relentless sun). After creeping past the cows and sliding down the stones we found ourselves at a mountain beach with clear, almost perfectly still, waters. The urge to throw pebbles into the stream was instinctive. I no longer needed to nap; I only wanted to make a splash! I believe Billy Connolly once said “Never trust a man who, when left alone with a tea cosy, doesn’t try it on”. I’m sure something similar (albeit less witty and revealing) can be said about people with small stones and still ponds.

I filmed some ripples and waves on my phone and I was reminded of the beautiful underlying mathematics…

The water is initially calm with some depth – this is its equilibrium state – and the introduction of the pebble (which plops into the pond) causes disturbances (waves and ripples) in the water.  The force of gravity tries to restore the equilibrium state  (the density of the water is greater that the density of the air above) and the disturbances propagate in all directions.  I am assuming that gravity is the only significant restoring force on the larger, prominent, waves (those that are clearly visible in my poor filming), so the ripples that form and spread are examples of surface gravity waves (I am ignoring surface tension here, but smaller capillary waves can be seen at the edge of the outer ring, if you look close enough – this is justified below).

The  linear theory of deep and shallow water waves can be found in most good books on fluid mechanics. My personal favourites are by Patterson and (of course) Batchelor, but there are many others. Without going through the details, the end result of some fun modelling and manipulation with PDEs and potential flow theory is the dispersion relation for linearised gravity waves

$\omega^2=gk\tanh{(kh_0)}$

where $\omega$ is the angular frequency of the wave, $k$ is the wavenumber (which is $2\pi/\lambda$, where $\lambda$ is the wavelength), $g$ is gravity, and $h_0$ is the depth of the undisturbed water.

The water in the video may not look very deep (and it isn’t – I got in and it only came up to my knees), but its not its absolute value that’s important but its size in comparison with other important measurements. Here the depth (a length) is a lot bigger that the wavelength (the distance between successive peaks). Since $h_0\gg\lambda$, we consider these waves to be deep water waves. Now, the wavenumber $k=2\pi/\lambda$, so we can say that $kh_0\gg 1$, and hence $\tanh{(kh_0)}\approx 1$. Therefore, the dispersion relation for the prominent waves in the film reduces to

$\omega^2=gk$.

Gratifyingly elegant! The speed of a wave, or the phase velocity, is the velocity a wave crest is travelling at and is given by $c=\omega/k=\pm\sqrt{g\lambda/(2\pi)}$. This equation tells us that longer waves (those with larger $\lambda$) move faster that shorter ones. This is demonstrated in the video above (filmed in slow motion on my phone – nothing technical!), where we can see that the outer waves are larger (have longer wavelengths) than the smaller ones they encapsulate. This was quite a basic experiment (more of a game than a test!) but I find the link between pebble throwing and mathematical fluid mechanics quite pleasing – yes, I am a proud nerd!

(It’s worth noting in passing that if $\lambda\gg h_0$ the waves are said to be shallow water waves. In this case, $\tanh{(kh_0)}\approx kh_0$ and $c=\pm\sqrt{gh_0}$. From this we can see that the wavespeed in shallow water depends on the depth of the water, but is independent of the wavelength. Thus, waves in shallow water do not have frequency dispersion.)

I then threw two stones, one after the other, into the centre of the pond (see the video above). I think I can just about see the large wavelength, fast moving, outer waves caused by the second stone catch up and overtake the shorter wavelength inner waves of the first splash. Perhaps not the most detailed and controlled experiment, but I am on holiday, after all!

Are they waves or are they ripples?

Well, I wasn’t certain at first.  Ripples in water are often capillary waves, meaning their behaviour is dominated by surface tension.  Their wavelength is typically less than the order of a centimetre and their wavespeed more than 20 centimetres per second. From the videos, the prominent waves seem to fall outside of these ranges while the smaller disturbances outside the “rings” appear to be capillary waves, or ripples. A more convincing argument is based on looking at the group velocity of the waves.

The group velocity, $c_G$ is the velocity with which a group, or packet, of waves propagates. In the video, the packet of the prominent waves is the expanding ring. Within this group we can see several individual waves, each moving with their wavespeed (or phase speed). For deep water gravity waves, the group velocity is  $c_G=c/2$. So the group velocity is smaller than the phase velocity. This means that gravity waves at the back of a packet will catch up with the front (the outer ring) before vanishing. This seems to happen here. If we try to track an individual crest in the slow-motion video (at the top of the page) we find we lose track of it when it is at the front and notice other waves are behind it. So it seems reasonable to conclude that there are surface gravity waves. Yippee.

If you have a very keen eye, you should be able to see that the smaller ripples in front on the outer ring behave differently. These ripples move slower with increasing wavelength, meaning the ones at the front of a packet of ripples end up at the back.  Furthermore, individual ripples seem to pop-up at the font of the packet and vanish at the back. This suggests the group velocity is faster than the phase velocity – a characteristic of capillary waves and ripples.

Fluid mechanics

Toys and fluid mechanics

I visited my 10 year old nephew this week for the the first time in a year (I’m not avoiding him – we live on opposite sides of the the world).  His tia and I gave him several presents, most of them educational as well as fun (naturally!). Lego tends to be his toy of choice but this time we  included a rubber bands vehicles set. It’s great!! My nephew made the aeroplane (one of several options) and, rather embarrassingly,  was much batter at it than me (even though the instructions were written in a language he doesn’t understand). My practical engineering skills may not be as sharp as a 10 year old’s, but at least I could talk to him about the mathematics of flight and airfoil theory.

My nephew is very bright but, to be honest, I think I went too far with the Milne-Thomson Circle Theorem! But it’s been a while since I last studied and taught inviscid flow theory and viscous boundary layers – perhaps I got a little over excited!

This is such a wonderful example of applied maths: beautiful theory combined with powerful applications. ODEs and PDEs, Cauchy-Riemann and conformal mappings, asymptotics and separation…and then there’s CFD! I even think the common misconceptions/misinterpretations of Bernoulli’s principe, although sometimes frustrating, add a little extra spice to the matter.  What a great subject to start studying as an undergraduate mathematician…or a 10 year old Lego enthusiast.

Experiment is the acid test for theory and I’m pleased to say that this plane took off smoothly from the dinning room table, glided elegantly through the patio doors…and off the balcony of the first floor flat before an ungraceful fall to Earth. Fortunately, it lives to fly again.