# Fun with water bells

Explaining how surface tension makes water form surprising shapes

Water bell banner

Catch a spoon under the tap when washing up and everyone knows you’ll end up with water everywhere. What you might not know is what shape the water is trying to form. And what you probably won’t know is that if you let the water form this shape, it might end up connecting back into a closed 3D shape. Welcome to the exciting world of water bells.

## DIY water bell

Take a 10p coin and blu-tack it to the top of a pen. Take your contraption to the kitchen sink and turn it on a medium speed. Now hold the pen quite far down and put the pen under the stream, so that the water hits the coin flat. With very little adjusting of the position of the pen and speed of the water, you should be able to get the water to not just spread out, but to come back to form a water bell (see picture at the top).

Some things you might like to try:

• How big/small you can make the water bell?
• What if you use a non-circular coin at the top?
• What if you tilt the coin?

## Why do water bells come back on themselves?

The top half of the water bell is quite predictable, but it’s not at all obvious that the water should turn round and come back on itself. The problem was first attacked by leading physicist and mathematician GI Taylor in 1959 [Source]. Clearly gravity is pulling the water downwards, but it’s not clear that anything should be pulling the water bell back inwards. There are, however, a couple of candidates:

• Pressure difference: if the bell closes in on itself, does the bell suck itself in somehow?
• Surface tension: this force likes to make fluid shapes take up their smallest surface area

The water bell is being pulled in and out by the balance of these forces

By taking a vertical slice of the water bell, we can take each part of the slice and balance gravity, pressure difference and surface tension on it. By doing this we can predict the shape of the bell rather well, which ends up depending almost entirely on the speed of the water from the tap.

Since the water bell doesn’t form a tight seal at the bottom, we can ignore pressure differences. This leaves us with just balancing gravity and surface tension, but what exactly is the latter?

## Understanding surface tension

Surface tension allows insects, like this water strider, to walk on water [Source]

Surface tension is the force that causes fluid surfaces to take up their smallest possible area. It’s the reason bubbles are round. This is because the molecules of fluids – in this case water – attract each other. If they didn’t, they would be gases! The effect of this is that fluid surfaces act as slightly elastic surfaces, like the surface of a balloon. This means that some insects, such as the water strider on the right, can walk on water, despite being denser than it. Try dropping a paperclip flat in a glass of water!

In this case, the fact that the water bell is curved is very important, since surface tension is a force which acts along the surface of the water. Now consider a segment from a vertical slice of the water bell, looking top down.

The blue segment is a ‘fluid element’: a piece of a slice of the water bell. The upper surface is the outer wall of the bell, and the lower surface is the inner wall. Since surface tension (red) acts along the fluid surfaces, which are curved inwards, it produces an overall inwards force on the fluid element.

We know that surface tension (red arrows in the diagram) acts along fluid surfaces, here both inside and out. Since the fluid surface is curved inwards, the surface tension force also acts slightly inwards. This results in a (purple) resultant inwards force, which causes the water bell to pull inwards.

The water bell not only curves when you look top-down on it, but also when you look side-on at it, so we find that surface tension also pulls the bell in from this curvature as well.

This resultant inwards force is quantified in the Young–Laplace equation, developed in the early 1800s, which combines the surface tension strength with the curvature of the bell.

## Balance in the force

Having found the surface tension force, we then add it to the downwards gravity force. When we put it Newton’s second law,

$\boldsymbol{F} = m\boldsymbol{a},$

we end up a second order differential equation to solve for the shape of the water bell. It’s a bit ugly (equation 5.1 here if you’re interested) and as it’s second order, we need two boundary conditions. The radius of the coin and the angle that the water leaves the coin at are two measurable things which do the job.

The complication of the equation means we have to solve it numerically. If we enter in the parameters in the equation that the shape depends on (gravity, density of water, flow speed of the tap etc.), we get this wonderful prediction in blue. Is it any good?

Our prediction for the water bell fits our observations perfectly

Isn’t it amazing! There’s no trickery here – we considered the important forces, put them together, and then solved the resultant equations to give us such a spot-on prediction.

(In case you want to try, Wolfram Alpha gives you the surface tension and density of water, a 10p piece is 24.5mm wide, the flow of water from a tap is on average 10l/min, and the ejecting speed is on average 40cm/s.)

### What if you change surface tension or gravity?

Different fluids have different surface tensions. For fun, see what happens to the shape of the water bell if you increase surface tension or gravity:

Pause the video when the gravity is set to 0. Taylor found an explicit expression for the shape of the water bell in this case. In space, the water bell forms a catenary! More evidence, if any were ever necessary, that $y = \cosh(x)$ is the greatest function ever.

## Chocolate bells?

You can find the water bell shape in fluids other than water. Anywhere you have cylindrical falling liquids, you are likely to come across this phenomenon. Ever wondered why chocolate in a chocolate fountain falls inwards when it falls?

You can read through the equations of the water bell in chapter 5 of this excellent study of water bells.

Make your own water bell at home! Can you get the water bell to do more exotic shapes? (Try slowing the water down slowly…)

Adam is a postdoctoral researcher at Imperial College London, where he investigates weird, non-Newtonian fluids. If he’s not talking about the maths of chocolate fountains he is probably thinking about fonts, helping Professor Dirichlet answer your personal problems, and/or listening to BBC Radio 2.

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