Solved: Da Vinci's 500-year-old mystery about bubbles

Solved: Da Vinci’s 500-year-old mystery about bubbles

Pour water – or another tasty, bubbly liquid – into a clear glass. Watch the bubbles carefully as they nucleate then float in the glass: you will notice that some of them rise differently than others. The smaller bubbles push upwards, while the larger bubbles bounce rhythmically, slowing their journey. If you’ve ever wondered why this is happening, you’re not alone. No less a natural philosopher than Leonardo da Vinci was confounded by it.

Bubble, bubble, toil and trouble

Children are fascinated by bubbles, and apparently some scientists too, including one describe in this way: “Bubbles are the void, non-liquid, a tiny cloud protecting a mathematical singularity. Born of chance, of a violent and brief life leading to union with the (almost) infinite.

In his famous sketchbooks (like the one depicting a helicopter that finally flew in 2022), da Vinci drew and described the mysterious sparkling phenomenon. Armed with modern theories, the unresolved question caught the attention of 20th century scientists. Their attempts to solve it by handand in the following decades by computer, were only partially successful. None of them are quite right.

But now, a mathematical and conceptual answer may finally have been found. A new paper in the review Proceedings of the National Academy of Sciences describes the solution.

Why Bubbles Flicker

Like all good theoretical work, the paper begins by looking at hard data. Skillful experimental scientists have produced a beautiful database on which to test theories. Their device emitted air bubbles of precisely determined size in hyper-pure water. Bubbles below a certain radius – about 0.91 mm, or just over 1/32 inch – rose straight up in the water. Above this size, the bubbles began to waver or spiral.

Armed with this data, the authors of the new paper built a model to predict the behavior of bubbles. Water and air flow smoothly over each other. When squeezed, these fluids move laterally rather than shrink. The flow patterns of these incompressible fluids are described by the Navier–Stokes equations, a set of rules established in the language of vector calculus. The equations are notoriously unsolved: there is a $1 million prize for all those who “simply do[s] substantial progress” on them.

When faced with impossible equations, researchers have found clever ways to simplify the calculations enough to construct approximately correct solutions with a computer. The details (which involve terms like non-reflecting boundary conditions, eigenfunctionsAnd Hopf bifurcation) are far too technical to explain. Suffice it to say, we can use the computer model to intuitively explain why the biggest bubbles wobble.

As a spherical bubble rises, it flattens somewhat, taking on an oval shape with a flat top and rounded bottom. If its spherical diameter is 0.926 mm or more, it is just large enough that a tiny vortex begins to form under its rounded bottom surface. The low pressure from the swirling vortex destabilizes the bubble, causing it to tip to one side.

A diagram of fluid flows around a bubble

Credit: MA Herrada and JG Eggers, PNAS2023

The upward sloping side of the bubble begins to curve more, accelerating the movement of water over the surface of the bubble on that side. Faster sinking water is more easily pushed aside, causing that side of the bubble to rise faster. The rapid airflow from the rising side of the bubble lowers the pressure there, causing the water outside to push it sideways, creating the zig.

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It is essentially a demonstration of the Bernoulli’s Principle: Higher flow velocity creates lower pressure. (You can test this for yourself by placing a very light piece of paper in your palm and blowing on it. The rapid flow of your breath over the top lessens the pressure above the paper, sucking it up. )

However, the bubble is not moving away; it goes back. The side zig bends the far side of the bubble. Now that side begins to rise and suck in air, creating a new area of ​​low pressure where the water will push back, sending the bubble back in the direction it came from.

What’s the point?

A mathematical computer model to explain the rise of water bubbles is obscure. At the same time, it is another case of scientific progress in the face of the impossible Navier-Stokes equations. Fluid mechanics is the sum of many such small victories. Raise your glass to progress, measured in centuries.

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