Hugh Simmons

Have you ever made a cup of tea or coffee and marvelled at the pretty patterns you get when you add the milk? Well, you’re not alone… Over the last few days I’ve spent what is, quite frankly, an unhealthy amount of time thinking about the maths in a cup of tea to try to pin down some of what’s going on. Here goes…

Chaos

As you’ve probably noticed, you get a different pattern every time you add the milk and we can put this down to the chaotic nature of the processes involved (which we’ll get into later). But what do we actually mean by the word ‘chaos’?

Well, mathematicians define something to be chaotic if a very small (infinitesimal) change in initial conditions leads to significantly different behaviour. So in our case that means a slightly different angle of the milk, or the temperature of the tea, or a slight change in any other variable, can have a big impact on what the patterns evolve into. And being human, we’re unfortunately never going to be able to replicate exactly the same pouring technique, which explains why it feels like we always get something a little different.

A typical example of chaotic behaviour is what is commonly referred to as the ‘butterfly effect’. The idea is that a butterfly beating its wings in South America could cause a hurricane in Texas three weeks later as a result of those initial small perturbations to the air pressure. Really, I suppose the atmosphere’s just a big cup of tea in a lot of ways, when you think about it…

Another common example of chaotic motion is the double pendulum:

A double pendulum is very similar to an ordinary one you can find in a grandfather clock, but instead of having one rod between the weight and the centre of rotation, it has two connected together at a hinge, which creates an angle between the two rods. This extra quantity or ‘degree of freedom’ we need to describe the system compounds the effect the initial conditions have on the motion, and we see completely different outcomes.

I’d hazard a guess that pouring milk into tea is chaotic for the same reason that physicists don’t try to describe gases in terms of the paths of individual molecules – a tiny change in the angle of a collision will have knock-on effect and there are a huge number of collisions happening per second.

Ok, so that’s all well and good, but it doesn’t really give any insight into the processes that form the types of patterns we see in the mug, or indeed why they happen at all, so let’s dig a little deeper.

In preparation for this article, I made quite a few cups of tea, which isn’t out of the ordinary, but what was out of the ordinary is that I decided to film them. I then slowed the video down and added a filter to make it easier to see what was going on. And surprisingly, I did actually notice some general patterns in the behaviour which were repeated again and again.

The general setup of the ‘experiments’ was to pour the milk whilst trying to keep it as central as possible. Unfortunately, this proved to be much more difficult than I initially suspected, and in most cases it was a little off-centre which I think was represented in the results. But, before we get to the results, I suppose the first question to ask is why do we see anything on the surface at all? If the milk has a higher density than the tea then why doesn’t it just sink to the bottom of the mug and stay there? Well, the answer is something called a convection current.

Convection Currents

As if the situation wasn’t complicated enough as it is, there are actually already flows within the tea before we add the milk! As the surface of the tea makes contact with the much colder air, there is a transfer of heat, and the tea on the surface cools down. Because it cools, the surface tea becomes more dense and so sinks to lower the gravitational potential energy. However, as it moves down through the warmer tea it heats up again and expands, causing it rise again. This process is constantly repeating so long as there’s a temperature difference between the tea and the air, and it leads to well-defined flows called ‘convection currents’ throughout the tea.

Therefore, when we add some cold milk into the mix, what we are seeing is the motion created by these convection currents. The milk is only brought back to the surface once its density has been lowered enough by the heat of the tea.

But what about the shapes we see at the surface? In my experiments I often saw two mushroom clouds appearing either side of the line defined by the direction of my pour (as shown in the image above).

Some simple analysis leads me to believe that this is likely caused by the shape of the mug. If I’d managed to pour exactly in the centre such that the milk descends vertically downwards it stands to reason that we’d see something with circular symmetry. You can think of the situation much like the following puzzle.

Imagine we are playing a game on a circular coffee table with the rule being we each take turns placing a coin onto the table. The last person able to fit a coin on the table keeps the lot. What would your strategy be? And how would you react if I told you there was a guaranteed way to win if you went second?

The answer, funnily enough, is to do with circular symmetry; if I put a coin down on any part of the table, you can always reflect that point about the centre of the table and have somewhere to put your next coin. This is because a circle is a shape that doesn’t change no matter what angle you rotate it through. So rotating through 180 ° is always guaranteed to still be on the circle and therefore you can always mirror your opponent’s move.

The two clouds in our cup of tea form because the milk isn’t poured exactly in the centre, and so we don’t have this nice property of circular symmetry. Instead, we only have symmetry about the line formed by the centre of the mug and where the milk is poured in, which is why there are two clouds – one on either side of the line.

So that explains the mushroom clouds, but what about the pretty swirls? We’ve seen above the effect that density can have on the fluid motion, and in fact the differing densities of the tea and the milk lead to a whole host of interesting and complex interactions. Here are two of my favourites.

Rayleigh-Taylor Instability

Rayleigh-Taylor is defined as the instability of a boundary between two fluids of different densities. The classical example is when two fluids of the same velocity are layered one above the other in an effective gravitational field.

The key insight from Taylor in helping to understand the process, was to observe the instability is physically equivalent to a lighter fluid being accelerated into a heavier one. This is in fact what we have at the base of our milk stream as it hits the surface of the tea, just not in quite such a nice sheet! As the milk descends into the tea, the heavier fluid is below the lighter fluid, and thus the total gravitational potential energy is lowered.

The simplified (linear) equations predict the roughly sinusoidal shape of the first two pictures in the experiment above. The rate at which they form is related to the density difference between the two fluids. On the edges of the stream we can see wave-like patterns that come from the difference in velocity between the yellow and blue fluids. This in in fact a second type of instability called the ‘Kelvin-Helmholtz instability’.

Kelvin-Helmholtz Instability

Kelvin-Helmholtz is the name given to the behaviour seen when there is a velocity difference at the interface of two fluids. It’s just like wind blowing over the sea but with gravity acting in a different direction.

In our cup, the sides of the milk stream exhibit this behaviour on their way down through the tea, but the nice wave-like pattern is quickly lost as small fluctuations are amplified into large eddies like in the video below, which are similar to what we observe on the surface.

So there you have it! More detail – and certainly more maths – than you ever thought possible in a cup of tea. Now, who’s putting the kettle on?