Why don’t all the bubbles go straight up, wondered Leonardo da Vinci. Now there is an answer

The toughest questions are often the easiest to ask. But after five centuries, a bubble problem seems largely solved.

Anyone who has an aquarium has probably noticed this. If not, you may have seen it in your kettle or in a glass of soda. Bubbles do not always rise directly. But why not? And why do abnormal bubbles take a zigzag path towards the surface of the liquid?

The question has also been called by some a Leonardo’s paradox, named after the artist and inventor Leonardo da Vinci, who was already concerned with it in the 16th century. It has to do with size, he noted; the big bubbles make this path in a zigzag, the smaller ones go straight.

Miguel Herrada and Jens Eggers, respectively from the universities of Seville and Bristol, managed to put an exact number on this: the zigzag effect occurs when the bubble has a radius of more than 0.926 millimetres. And above all, they can discuss what is happening, they write in the magazine PNAS.

From this size, the bubble becomes unstable during the ascent. It warps a bit, so the curve of the bubble will be a bit sharper on one side than the other. This side with the pronounced curve rises a little faster and this initiates an interaction with the fluid. It then flows faster past the bubble, so the pressure decreases slightly and the bubble expands on that side. This causes it to return to its original straight path upwards, after which the mechanism begins again.

Why a bubble is so terribly complicated

As simple as it may seem, it was not to calculate. Studying an ascending bubble is a terribly difficult field, agree the authors. Mathematical theory of bubble motion rarely agrees with experimental observations, especially in water. This is partly because the viscosity of water is so low, which means the effects it has on the bubble are so subtle.

It is above all the game that complicates it; the forces exerted by the water cause the bubble to change shape, which affects the flow of water, which changes the shape of the bubble again, and so on until the bubble reaches the surface somewhere. All of this is very difficult to model. Not to mention the deflection you get if there is something in the water that disturbs the pure course of the bubble.

This unpredictability also has its advantages. Bubbles are regularly used to create random numbers, which are used to secure internet connections or to make computer games less predictable.

The two scientists are therefore quick to point out that their model has its limits. It applies to “ultra-pure” water, as used in a series of experiments in the 1990s that produced a workable model for small bubbles. And even then, Herrada and Eggers saw a small gap, albeit less than 2%. For true bubble purists, a bit of mystery remains.

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Jan Beuving is a mathematician and humorist. For years he wrote a column in Trouw in which he played with natural sciences and language.