91É«ÇéƬ

Have you ever found yourself dispensing wine from a box, or beer from a standing keg, and wondered how much liquid is left in the container? If you were thinking about it like a mathematician, you might have noticed that the fuller the keg is, the faster the beer comes out.

This might seem obvious, but this is a known mathematical relationship, and it uses a versatile piece of maths that crops up in many other real-world situations. Fluid dynamics is the study of flow: how liquids and gases behave under forces and interact with their environment, which we can model mathematically. Bernoulli’s principle tells us that for a given particle of fluid, the total sum of its energy and pressure remains constant as it moves through space.

This means the sum at one point must match the sum in another location along the same path. For a typical fluid like air or water, the quantities involved are pressure, and energy in two forms: potential energy (height) and kinetic energy (velocity). If we keep one of these values fixed, the other two will then have a proportional relationship, which we can take advantage of.

In the case of a wine box, there will be a path that the wine moves along that connects the top of the liquid’s surface inside the box, through to a point just past the spout where the liquid flows out. As both of these points are touching the surrounding air, we can say that their pressures are roughly equal. Bernoulli’s principle then gives a relationship between the height difference and the speeds at these two points – namely, the more wine that is left in the box, the greater the height difference between the two points and the faster the fluid will come out of the spout.

The same maths can be useful when you want to open a light plastic bag – like a trash bag – but find it is sticking together. The best trick is to open the bag a tiny amount, and then quickly blow some air into it from a few centimetres away. This quick change in speed, with no change in height, creates a region of low pressure in front of the opening of the bag – which will cause air nearby to rush away from the high pressure and towards the low pressure, filling it up.

This same idea works in any situation involving fluids – where one of the three variables of height, pressure and speed is fixed, the other two will be related. Even the flight of an aeroplane can be modelled in the same way: the curved shape of plane wings is designed so that the air above them moves faster than the air below. This creates a region of low pressure above the wing that produces suction to lift the aeroplane higher into the sky.

So you can thank Bernoulli’s principle next time you pour a drink – maths is there to keep everything flowing smoothly.

Katie Steckles is a mathematician, lecturer, YouTuber and author based in Manchester, UK. She is also adviser for New Scientist’s puzzle column, BrainTwister. Follow her @stecks

For other projects visit newscientist.com/maker