Study says Leonardo da Vinci’s 500-year-old “paradox” has finally been solved

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Leonardo da Vinci statue. Photo: Victor Ovies Arenas via Getty Images

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More than 500 years ago, Leonardo da Vinci was watching air bubbles float in water—as you do when you’re a Renaissance polymath—when he noticed that some of the bubbles inexplicably started bubbling up or zigzagging instead of going straight up to the surface.

For centuries, no one offered a satisfactory explanation for this strange periodic anomaly in the movement of some bubbles through water, which has called Leonardo’s paradox.

Now, a pair of scientists think they may finally have solved the long-running mystery by developing new simulations that match high-resolution measurements of the impact, according to A study published on Tuesday in Proceedings of the National Academy of Sciences.

The results indicate that bubbles can reach a critical radius that pushes them onto new, unstable trajectories due to interactions between the flow of water around them and subtle distortions of their shapes.

said the authors Miguel Herrada and Jens Eggers, researchers in fluid physics at the University of Seville and the University of Bristol, respectively, in the study. “The burgeoning rise of a single bubble serves as a much studied model, both experimental and theoretical.”

“However, despite these efforts, and despite the ready availability of massive computing power, it was not possible to reconcile the experiments with numerical simulations of the complete hydrodynamic equations for a deforming air bubble in water,” the team continued. “This is especially true of the interesting observation, already made by Leonardo da Vinci, that air bubbles large enough perform a periodic motion, rather than rising along a straight line.”

Illustration of Leonardo da Vinci's bubble from Codex Leicester.

Indeed, bubbles are so ubiquitous in our daily lives that it is easy to forget that they are dynamically complex and often difficult to study experimentally. Air bubbles rising in water are affected by a combination of intersecting forces—such as fluid viscosity, surface friction, and any surrounding contaminants—that twist the shapes of the bubbles and change the dynamics of the water flowing around them.

What da Vinci noticed, and has since been confirmed by other scientists, is that air bubbles with spherical radii much smaller than a millimeter tend to follow a direct upward path through the water, while larger bubbles oscillate causing a cyclic or zigzag vortex. tracks.

Hirada and Egger used the Navier-Stokes equations, a mathematical framework for describing the motion of viscous fluids, to simulate the complex interaction between air bubbles and their aqueous medium. The team was able to determine the spherical radius that causes this tilt — 0.926 millimeters, which is about the size of a pencil tip — and describe a possible mechanism behind the zigzag motion.

A bubble that has exceeded the critical radius becomes unstable, which results in a tendency that alters the curvature of the bubble. The shift in curvature causes the water to speed up around the bubble’s surface, which then sets off the oscillating motion. The bubble then returns to its original position due to a pressure imbalance caused by deformations in its curved shape, and the process repeats in a cyclic cycle.

In addition to solving a 500-year-old paradox, the new study could shed light on a host of other questions about the mercurial behavior of bubbles, and other things that defy easy classification.

“While it was previously thought that bubble wakes become unstable, we now demonstrate a new mechanism, which is based on the interaction between flow and bubble deformation,” Hirada and Eggers concluded in the study. “This opens the door to studying small contaminations that are present under most practical conditions, simulating particles somewhere between a solid and a gas.”

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