# Everyday Science #2: The milk pouring principle

## Everyday Science #2: The milk pouring principle

Has it ever happened to you, when you tried to pour milk into a bowl, that it stops and goes, spilling liquid everywhere? Well, this chaotic flow is based upon a common physical principle that I’ll explain here. An important post to have a safe breakfast!

## Initial situation

We have a carton with a hole the top, located at an extremity. In this carton, we have milk (or orange juice, or any liquid) and air.

There are two configurations in which we can pour the milk out of the carton. In any case, the dynamic of the liquid is then driven by fluid mechanics.

## Bernoulli’s principle

Before explaining what happens, I need to explain Bernoulli’s principle [1] which applies on a fluid (liquid or gas). A common writing of it is the following equation:

$\frac{v^2}{2}+gz+\frac{p}{\rho}=A$

with v the fluid flow’s speed at a specific point, z the that point’s elevation, p the pressure at that point, g the gravity acceleration, rho the density of the fluid and A a constant.

This equation show that when the pressure on the fluid diminishes, the speed increases, and vice versa.

## When the liquid gets out

When putting the opening on the bottom to pour the milk, the system becomes as the following figure:

In this system, the pressure applied by the force of gravity push the milk down.

While the milk comes out, the pressure inside the carton diminishes and so the speed increases. But the carton tries to equalise the pressure with the outside. To do so, it lets the air come in through the same hole the milk comes out.

The air coming in causes the milk flow to stop and go and be so irregular.

## The other way to pour milk

There is another configuration in which the flow is not perturbed. It happens on the following system:

Two elements change, compared to the previous system:

First, the pressure is significantly lower above the milk, due to less amount of liquid.

Then, the air can come in to equalise the pressure without interrupting the milk flow.

In consequence, the milk can be poured with a regular flow, as long as the carton is tilted at a low angle (a higher angle implying higher pressure and the same issues than before).

## Implications for engineering

The Bernoulli’s principle have many applications.

This is, for example, the main principle that allows a plane to fly. Because of the wings’ shape, the air travels faster on top of the wing than on the bottom, causing a difference of pressure. This difference of pressure causes a lift. [2]

Derivations of the Bernoulli’s principle also have applications. This is the case for several devices used in other vehicles such as carburettors (for small engines) with Venturi effect or injectors (for steam locomotives) with Magnus effect.

## References

[1] Batchelor, G.K. (2000). An Introduction to Fluid Dynamics. Cambridge: Cambridge University Press. ISBN 978-0-521-66396-0.

[2] Eastlake, Charles N. (March 2002). “An Aerodynamicist’s View of Lift, Bernoulli, and Newton” (PDF). The Physics Teacher40 (3): 166–173. doi:10.1119/1.1466553