Scientists solve 500-year-old mystery of Leonardo's paradox

Researchers used a numerical discretization technique.
Nergis Firtina
Leonardo da Vinci.
Leonardo da Vinci.


You might have heard of the 500-year-old Leonardo da Vinci air bubbles paradox. Floating up on the water's surface, he observed bubbles were spiraling and zigzagging rather than going up. 

A method to explain the erratic movement of bubbles rising in water has been found by professors Miguel Ángel Herrada from the University of Seville and Jens G. Eggers from the University of Bristol. The study may be helpful in comprehending the mobility of particles whose behavior is transitional between a solid and a gas, claim the researchers.

Scientists solve 500-year-old mystery of Leonardo's paradox
Leonardo’s sketch showing the spiral motion of an ascending bubble.

A numerical discretization technique

As explained in a press release, the air-water interface of the bubble can be precisely characterized by the authors of this new study using a numerical discretization technique, allowing them to simulate the bubble's motion and investigate its stability.

They demonstrate that bubbles depart from a straight trajectory in water when their spherical radius exceeds 0.926 millimeters, a result that is within two percent of the experimental values obtained with ultrapure water in the 1990s. Their simulations closely match high-precision measurements of unsteady bubble motion.

The researchers suggest a mechanism for the instability of the bubble trajectory in which periodic tilting of the bubble changes its curvature, influencing the upward velocity and resulting in a wobble in the bubble's trajectory, tilting up the side of the bubble whose curvature has increased. The pressure imbalance then causes the bubble to return to its original position, continuing the periodic cycle, as the fluid travels more quickly and the fluid pressure decreases near the high-curvature surface.

As Vice underlines, “The motion of bubbles in water plays a central role for a wide range of natural phenomena, from the chemical industry to the environment,” said authors Miguel Ángel Herrada and Jens Eggers, who are fluid physics researchers at the University of Seville and the University of Bristol. “The buoyant rise of a single bubble serves as a much-studied paradigm, both experimentally and theoretically.”

“Yet, in spite of these efforts, and in spite of the ready availability of enormous computing power, it has not been possible to reconcile experiments with numerical simulations of the full hydrodynamic equations for a deformable air bubble in water,” they also added.

“This is true in particular for the intriguing observation, made already by Leonardo da Vinci, that sufficiently large air bubbles perform a periodic motion, instead of rising along a straight line.”

The study was published in PNAS on January 17.

Study abstract:

It has been documented since the Renaissance that an air bubble rising in water will deviate from its straight, steady path to perform a periodic zigzag or spiral motion once the bubble is above a critical size. Yet, unsteady bubble rise has resisted quantitative description, and the physical mechanism remains in dispute. Using a numerical mapping technique, we for the first time find quantitative agreement with high-precision measurements of the instability. Our linear stability analysis shows that the straight path of an air bubble in water becomes unstable to a periodic perturbation (a Hopf bifurcation) above a critical spherical radius of R = 0.926 mm, within 2% of the experimental value. While it was previously believed that the bubble’s wake becomes unstable, we now demonstrate a new mechanism, based on the interplay between flow and bubble deformation.

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