shape of a stirred coffee surface

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