A Spanish and a British Researcher Solve Leonardo da Vinci’s Paradox

A Spanish and a British Researcher Solve Leonardo da Vinci's Paradox

Fluid mechanics specialist Miguel Ángel Herrada, from the University of Seville, together with Professor Jens G. Eggers, from the University of Bristol, discovered a mechanism that explains the unstable movement of bubbles that rise in water. The results, published in the journal PNAScan be useful to understand the movement of particles, since their behavior is intermediate between a solid and a gas.

Leonardo da Vinci observed five centuries ago that air bubbles, if large enough, periodically deviate from straight-line motion, whether zigzag or spiral. However, there was still no quantitative description of the phenomenon nor a physical mechanism to explain it.

Air bubbles deviate from a straight line when rising in water if their radius is greater than 0.926 millimeters

The authors of this new study have developed a numerical discretization technique to accurately characterize the bubble’s air-water interface, which allows simulating its motion and studying its stability. Their models are consistent with high-precision measurements of unstable bubble motion.

The deviation occurs if the spherical radius of the bubbles exceeds 0.926 millimeters, a result within 2% of the experimental values ​​obtained with ultrapure water in the 1990s.

Herrada and Eggers propose a mechanism for the instability of the bubble trajectory in which its periodic slope alters the curvature. This affects the rate of ascent and causes the path of the bubble to wobble: the curvature increases and the side of the bubble slopes upwards.

Then, as the fluid moves faster and the fluid pressure drops around the high curvature surface, the pressure imbalance returns the bubble to its original position, restarting the periodic cycle.


Herrada, MA et al. “Instability of the path of an air bubble rising in water”. PNAS (2023)


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