stirring coffee

A reason to add milk to your coffee

stirred cup of coffee with streak lines
It is astonishingly difficult to photograph the swirls of a stirred black coffee, still harder to capture the shape of the surface. This was an attempt with a strong light reflected on the surface.

You sit down to savour a well rounded, freshly roasted and just brewed pour over. Is there a good reason to add milk to it? Well, besides anything else, it may be a good test of an idea suggested by coffee-cup physics.

It’s about what happens as we stir our coffee. Many of us have contemplated our drink as we have stirred it either to cool it down or indeed to add milk or sugar. The surface of the coffee forms a depression at the centre, while at the walls of the mug, the surface forms a fairly steep slope. What is causing this shape and could it have any influence on how we appreciate our brew?

The shape of the stirred coffee surface is a consequence of the balance of forces acting on the surface. In addition to the force of gravity, there is the centripetal force on each bit of coffee swirling around the centre of the drink. These two forces have to balance at the surface of the water (assuming constant air pressure above the surface). If you make the further assumption that the coffee liquid rotates as one mass, so that the coffee at the edge of the mug rotates at the same angular velocity as the coffee at the centre, the centripetal force increases with increasing distance from the middle. This means that gravity dominates in the middle of the coffee whereas, towards the edge, the larger centripetal force is having a far greater influence. It is this that leads to the depression at the centre of the coffee and indeed the parabolic shape of the surface (click here for a mathematical derivation). The parabolae formed by such rotating liquids can be so perfect that liquid mirror telescopes have been developed to closely scrutinise specific parts of the sky. One problem with these liquid mirror telescopes is that the rotation of the liquid (often mercury) has to be perpendicular to the force of gravity. Which means that the telescopes are not able to move to different regions of the sky but instead only look ‘up’. Nonetheless, this does mean that they scan the same region of sky each night and so can be used very effectively to compare changes in that region of sky.

vortices, turbulence, coffee cup physics, coffee cup science
When a cup of water is first put onto a rotating platform, the liquid at the centre does not rotate at the same speed as the walls of the mug (that comes later). During these times, turbulent boundary layers appear at the walls of the mug which can be visualised with ink as has been done here.

Stirred coffee in a mug though is not a rigidly rotating liquid. Instead, the friction at the walls of the mug means that the coffee at the outer edge is slowed down and so the rotation is faster at the middle of the coffee than the edge. To form a parabola on the surface of a mug of coffee, it would be better to put the whole mug onto a record player played slowly. How does the shape of a stirred coffee differ from the surface of a coffee placed on a record player?

Initially, as the spoon is forcing all the liquid around together, the curvature will be approximately paraboloid. The interest comes once the spoon is removed and the friction between the coffee liquid and the sides of the mug becomes important. Towards the walls of the mug, the rotation will be slowed down which means that the centripetal force will decrease. Gravity will then dominate the combination of forces and the coffee surface will become flatter. As more of the coffee slows down, progressing from the edge of the cup towards the centre, the coffee surface will further flatten until the central depression is all that is left. As the friction slows more of the liquid down, so the depression at the centre of the coffee will also eventually disappear.

This is where the milk comes in. Assuming that you add cold milk to the centre of the rotating (hot) coffee, what should happen is that the milk (which is denser than the coffee because it is cold) will sink down towards the bottom-middle of the cup. As it sinks, so it will drag some of the swirling coffee down with it causing the coffee at the centre to accelerate and rotate faster around the centre of the cup*. The faster rotation will increase the centripetal force and so the central depression will become a bit more obvious again. This is the prediction anyway. So far, using chilled water and food dye, I have not been able to convince myself of the effect. But perhaps you will have more luck. Do let me know in the comments or over on social media, what results you get with this.

vortices in coffee
Vortices behind a spoon dragged through coffee. Experimental physics is a great excuse for playing with coffee.

Returning to the just stirred coffee, there may be one more thing to notice. At the interface between two moving fluids, a turbulent layer can form. We can see this when we first put a coffee on a record player (link here), or with the appearance of certain clouds (link here). This leads to a suggestion. As the coffee will be rotating faster at the centre of the cup than at the edge (owing to the resistance of the mug walls), the turbulence in the air over the centre of the cup will be greater than that at the sides. Fast moving fluids flow at lower pressure than slow moving fluids (Bernoulli’s equation). And although strictly speaking this is only valid for non-turbulent air flow, the principle can explain how planes fly and it may also have a consequence for our coffee.

As the air above the coffee at the centre of the mug will be moving faster than the air outside the mug, the air above the centre should be at an ever so slightly lower air pressure than that outside the mug. We know that water evaporates more quickly at lower atmospheric pressure. Consequently, more coffee aromatics will be evaporating from the centre of a just-stirred cup of coffee than from one you have left to sit still for a similar amount to time. To phrase this in a slightly different way, stirring your coffee should make it more aromatic and fragrant.

There are of course questions. Would the air pressure really decrease so significantly to affect the evaporation rate? How do you account for the fact that stirring coffee cools it relative to a coffee that is left to sit and wait? (Though why stirring a coffee should cool it is a whole other conversation). Nonetheless, it would appear to be a perfect excuse to brew and enjoy more coffee. Inhale deeply, stir contemplatively and, perhaps, add a little milk.

*In “Vortex flow in nature and technology”, HJ Lugt, John Wiley and Sons, 1983

Causing a stir

coronal hole, Sun

Where it all begins. The dark object is a Coronal hole on the Sun. Image credit and copyright NASA/AIA

What’s the difference between your cup of coffee and the solar wind (the fast stream of charged particles emanating from the Sun)? Perhaps this seems a strange question, we ought first to ask what connects your coffee with the solar wind. But, when we look at what connects them, you may be surprised to find the reason that they are different.

The solar wind is a flow of charged particles that streams past the Earth at roughly 400 km/s. To put this figure into some perspective, 400 km/s is 24, 000 km/min which means that the wind travels from the Earth to the Moon in 16 minutes. In comparison it took  Apollo 11 over 3 days between leaving Earth’s orbit and entering the Moon’s (over 4 days between launch and landing). The particles in the solar wind originate in the Sun’s Corona where temperatures get so hot that the gases have enough energy to escape the gravitational pull of the Sun itself. As these particles reach the Earth, they encounter the Earth’s magnetic field and, being rapidly slowed down by the Earth being in the way, a shock wave forms which is known as the Earth’s Bow Shock.

We must all have dragged a spoon through coffee and watched as the vortices form behind the spoon. It is a low-speed example of turbulent behaviour in the coffee. So it is perhaps not surprising that when the very hot and very fast solar wind hits the magnetic field region of the Earth, we find turbulence there too.

vortices in coffee

Vortices behind a spoon being dragged through coffee are an example of turbulence.

Now when we stir our coffee, we will see that there is one big rotation of fluid in the direction of the spoon but we may also notice smaller eddies in the drink. Some of these form from the fact that the coffee is rotating but the mug’s walls are staying motionless, friction forces the fast moving coffee to slow down at the walls. You can actually see this effect if, rather than stirring your coffee, you put it on a record player (or other rotating platform) as has been featured on Bean Thinking previously. Similarly, when you have a large vortex in the form of a smoke ring, it can decay into many smaller vortex “smoke rings” in what is known as a vortex cascade. This too is an effect that you can see in coffee (but rather than smoke rings you can make milk rings with a straw). Very often these milk rings will decay into many smaller rings in the same sort of vortex cascade as you get with the smoke, you can see a video of the effect here or at the bottom of this post. Big vortices decay into smaller vortices until they (to our eyes) disappear entirely.

vortices, turbulence, coffee cup physics, coffee cup science

Vortices created at the walls of a mug when the whole cup of coffee is placed on a rotating object (such as a record player). This is an image of water in a rotating mug with a drop of ink placed next to the mug’s wall.

The important thing is that this type of vortex cascade has also been observed in the solar wind. Rather than a giant spoon though, the solar wind stirs itself as the fast wind encounters the (relatively) slow Earth. We are used to stirring our coffee as a way of cooling it down, perhaps we blow on it gently to speed up the cooling process. But this is the difference between your coffee and the solar wind. When the solar wind is stirred up, it gets hotter. To examine how this occurs, scientists have been examining data from the Cluster set of satellites. Launched by the European Space Agency to study the magnetosphere of the Earth, Cluster has provided clues as to how the solar wind differs from a cup of coffee. Back in 2009, scientists analysed the data from Cluster looking at precisely how the turbulence produced as the solar wind meets the magnetosphere cascades into different sorts of eddies, different levels of turbulence. Comparing the data to theoretical models, they showed how the turbulence started off on large length scales (of the order 100 000 km), and decayed into smaller and smaller length scales until it reached 3km. At this point, all that energy, all that motion was dissipated as heat. Stirring the solar wind heated it up.

Why does stirring the solar wind heat it up whereas stirring your coffee cool it down? It’s to do with the environment of the coffee and the wind. On the Earth, the coffee will be surrounded by a cooler atmosphere. Stirring the coffee brings the hot liquid into contact with the cooler air and so the heat from the coffee can escape more efficiently into the atmosphere. They say in space, no one can hear you scream, which is another way of saying that there is no atmosphere through which sound waves can travel¹. No atmosphere means that there is no way of the heat generated by all that turbulence getting dissipated into a cooler air around it. So, as heat is energy, all that energy involved in stirring up the solar wind gets dissipated as heat in the wind which then has a higher temperature to that which we would naively expect.

So, next time you are waiting for your coffee to cool and stir it to hasten the process, take a moment to think about what is happening approximately 90 000 km above your head where the solar wind is being effectively stirred, and heated, by our planet’s magnetic field.

Seeing a vortex cascade in coffee:

 

¹The origin of the phrase however suggests that this was not quite the meaning that was intended, it was a promotional phrase used for the film Alien.