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

Hanging out at J+A Cafe, Clerkenwell

Exterior of J and A cafe (the bar is on the other side of the passageway)

Exterior of J and A cafe (the bar is on the other side of the passageway)

Tucked down a little alley, in the back streets of Clerkenwell is the J+A Cafe. Not just a cafe, but also a bar, J+A is a satisfying place to find, particularly if you happen to find it serendipitously. As you head down the alley, the café is on your right whereas the bar opens up on your left. The café is simply furnished, with bare brick walls adorned with a few impressionist paintings. There are plenty of seats at which to enjoy good coffee and home-made cake. Their website suggests that J+A specialise in Irish baking and so we dutifully had a slice of Guinness and chocolate cake with our coffees. Importantly, the dreaded “does it contain nuts?” question was met with a knowledgable answer and without the ‘frightened bunny face’ that I often encounter when I ask this question. J+A definitely gets a tick in the ‘cafe’s with good nut knowledge’ box on my website.

Lights were suspended from the ceiling, connected by wiring that was allowed to hang down, a section of electrical wire held at both ends and freely hanging. While I’m sure that this was done for aesthetic reasons (and certainly it works on that level), such hanging wires are in fact far more than merely pleasing to the eye. Such hanging wires were a mathematical puzzle just four centuries ago. Indeed, these simple hanging wires form curves that are so important they get their own name; they are catenary curves, from catena, the Latin for chain.

lights at J and A coffee Clerkenwell

Between each lamp, the electrical cord formed a catenary curve.

Galileo had thought that a wire hanging under its own weight and suspended at its two end points formed a parabola. A fairly simple curve that is easy to describe mathematically. It was natural for Galileo to assume that these catenary curves were really parabolic. He had earlier shown that objects that fell with gravity followed parabolic paths, and after all, the hanging wires did look almost parabolic. It fell to Joachim Jungius to show that the curve was not parabolic and then to Huygens, Bernoulli and Leibniz to derive the equations determining the form of the curve. Although the differences between the parabola and the catenary curves are subtle, they have profound consequences.

When a chain, or a wire, is suspended and allowed to hang under its own weight, it forms a catenary. Flipping this around, quite literally, a catenary arch will be self-supporting. This means that a vault made of a series of catenaries or a dome that is made into the shape of a catenary will be self-supporting with no need for buttresses. This property of the catenary curve was used by Antonio Gaudi in his designs of the Casa Mila in Barcelona and also by Christopher Wren. The famous dome of St Pauls is not a catenary, but it is not one dome either. It is in fact 3 domes stacked together. The outer dome is spherical (which is weak from a structural point of view) while the inner dome is a catenary. The dome between these two was designed, using the mathematics of the day, to support the impressive outer dome (more info here and here). Wren, was not just an architect, he was also a keen mathematician, there is maths, physics and beauty throughout many architectural designs.

Mathematics in the city reflected in the lights of J+A.

J+A is at 1+4 Sutton Lane, London EC1M 5PU