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More than 500 years ago, the astrophysicist Leonardo da Vinci watched air bubbles moving in water - as probably any of the great Renaissance artists of his time would have done - and noticed that instead of rising straight to the surface of the water, a number of bubbles inexplicably began to zigzag left and right.
Take a glass, fill it with soda and you too will see the movement of the bubbles.
That's exactly what we're talking about.
Since it was probably not urgent for anyone to check how and why the bubbles move in water, for hundreds of years no one has been able to provide a satisfactory explanation for this deviation in the movement of bubbles in water that has since been called "Leonardo's paradox".
And yet, apparently someone did care, because now a couple of scientists are sure that they have found a solution to this long-standing mystery.
They did this by developing new simulations that match high-precision measurements of bubbles, according to research published in the Proceedings of the National Academy of Sciences.
"Leonardo's Paradox" is finally solved (Photo: ShutterStock, Grey82)
Their results indicate that whales can reach a certain radius that pushes them into new and unstable paths due to interactions between the flow of water around them and the subtle deformations of their shapes.
"The movement of bubbles in water plays a central role for a wide range of natural phenomena, from the chemical industry to our environment," said authors Miguel Harada and Jens Eggers, fluid physics researchers at the University of Savija and the University of Bristol respectively.
"The vigorous rise of a single bubble serves as a much-studied paradigm, both experimentally and theoretically."
They added: "However, despite these efforts and despite the ready availability of enormous computing power, experiments could not be reconciled with numerical simulations of the full hydrodynamic equations for a deformable air bubble in water. This is especially true of the intriguing observation then made by Leonardo da Vinci,
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Indeed, bubbles are so common in our daily lives that it is easy to forget that they are dynamically complex and often complex for experimental study as well.
Air bubbles rising in water are affected by a variety of intersecting forces - such as the viscosity of the liquid, surface friction and all the surrounding pollutants - which distort the shapes of the bubbles and change the dynamics of the water flowing around them.
What Da Vinci noted, and other scientists have since confirmed, is air bubbles with spherical radii much smaller than a millimeter tend to rise in a straight path upward through the water, while larger bubbles develop an oscillation that results in cyclic spiral or zigzag trajectories.
Leonardo da Vinci's bubble paradox (photo: screenshot, Proceedings of the National Academy of Sciences)
Harada and Eggers used the Navier-Stokes equations, which are a mathematical framework for describing the motion of liquids, to simulate the complex interplay between the air bubbles and their aqueous medium.
The team was able to pinpoint the spherical radius that activates this tilt—0.926 millimeters, which is about the size of a pencil tip—and describe the possible mechanism behind the tortuous motion.
A bubble that exceeds the critical radius becomes more unstable and creates a tilt that changes the curvature of the bubble.
The change in curvature increases the speed of the water around the surface of the bubble, which drives the rocking motion.
Then the bubble returns to its original position due to the pressure imbalance created by the deformations in its curved shape, and repeats the process in a periodic cycle.
For those who didn't understand, here's another way to explain it:
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In addition to solving a 500-year-old paradox, the new research could shed light on a series of other questions concerning the behavior of bubbles.
"Today we demonstrate a new mechanism, based on the interplay between flow and bubble deformation," Harada and Eggers concluded in the study.
"This opens the door to the study of small infections, which exist in most practical settings."
Leonardo da Vinci