The Institute of Mathematics has developed a formula ‑ this may seem complex but there’s a general formula for almost everything. It is known as the “conversion of mass equation”.

This stands for: “The change in density of some stuff in some place is equal to the amount of stuff coming in to said place minus the amount of stuff going out of said place.”

The Institute of Mathematics used a spatial variable with a temporal variable and considered an arbitrary volume.

In short, this equation can be applied to anything ‑ even the perfect cup of tea.

How to stir 

Let’s be honest, you want your cup of tea’s flavour to be evenly distributed. There’s nothing worse than a poorly stirred drink, or even worse, a cuppa with bubbles from stirring too quickly. 

In mathematical terms, two steps are involved in making your cup of tea. Letting the tea seep out of its bag (an uncontrollable random movement) and advection caused by the mixture velocity (how much you stir it).

There are three equations behind stirring your cup of tea perfectly. 

The first equation is: J = ‑D ▽ ρ + ρ v

In this case, the ‘D’ represents the diffusion, which you are then adding to the ‘PV’ which is the advection (the act of stirring the tea).

The Institute of Mathematics found in their research that the ‘advection’ will dominate the ‘diffusion’ when it comes to mixing.

In conclusion, mixing will only be done quickly if you mix hard enough, and that makes a better brew, so don’t mess about.

What about the temperature?

Some prefer it lukewarm so they can sip it, others prepare it piping hot so it almost burns their tongue. But according to maths, what exactly is the perfect temperature for your brew? 

Several factors need to be considered including the size of the mug, material of the mug and heat of the water from the kettle. 

“Newton’s law of cooling” is used to estimate the change in temperature of the tea, without considering the spatial variation. Solving this equation as an ordinary differential equation gives:

T(t) = T1 + Ae-ct

Therefore, the cooling coefficient of tea is quite small; an approximation of c = 0.0005 should be fine for most mugs.

If you want your tea at a specific temperature and want to know how long it will take for your cuppa to be reade, there’s an equation for that too:

T = 20001n (80) ÷ (T2 – 293.15)

In this equation, you need to know your ideal tea temperature and square that, which will tell you how many minutes.