A 150-Year-Old Mathematical Theory Has Now Been Tested And It Seems It Was Wrong

The isotropic helicoid printed by the team to test Lord Kelvin's theory. Image credit: Greg Voth/Wesleyan University

Suck it, Lord Kelvin. You may have been right about thermodynamics, fluid dynamics, electronics, countless breakthroughs in engineering, the evolution and life cycle of stars, asymptotic analysis of functions, and kinetic energy, but when it comes to tiny little doodads that spin around in water, you don’t know jack.

In 1871, Lord Kelvin, whose real name was William Thompson, proposed the existence of a certain shape – an “isotropic helicoid” – which should naturally rotate when dropped into a fluid. He described what this shape should look like, and for 150 years his idea was held up as an elegant illustration of the power of symmetry analysis. In fact, the idea was so convincing that, apparently, nobody actually ever went ahead and tested it.

Well, a lot of old Victorian mathematicians are about to feel really embarrassed. A team of physicists has finally put Kelvin’s conjecture to the test – and it looks like he got this one wrong.

“Although symmetry analysis indicates that the particle should start to rotate as it settles, we did not detect any translation-rotation coupling in our experiments,” concludes the team’s paper, published this month in Physical Review Fluids. “This raises the question [of] whether Lord Kelvin’s original argument is flawed.”

Following Lord Kelvin’s instructions, the team 3D printed five small isotropic helicoids – spheres with “fins” placed strategically across the surface at 90- and 45-degree angles to the center circle. Key to the shape is the property that it looks the same from any angle – this is the “isotropic” part of “isotropic helicoid”. The team varied the size and shape of the fins for each of the five trials, but all of them produced the same result: nothing.

According to Kelvin’s original hypothesis, what ought to have happened when the helicoid was dropped into liquid was that it would start spinning, as the peculiar shape interacted with the dynamics of the fluid surrounding it. And the deeper it sunk, the faster it should fall.

What actually happened when the helicoid was dropped into liquid – specifically, silicone oil – was that it fell to the bottom without spinning at all.

In fact, the researchers suspect that the anticlimactic nature of their experiment may be the reason we’ve yet to see others attempt it.

“In Kelvin's manuscript, he explicitly describes how to fabricate an isotropic helicoid, including materials to use, suggesting that he created one,” study lead Greg Voth told Live Science. “I personally suspect that Kelvin and others since have fabricated isotropic helicoids and observed that the measured translation-rotation coupling is determined by limits on the quality of the fabrication, and therefore, they didn't publish their measurements.”

According to the researchers, the problem may be that the “translation-rotation coupling”, which refers to the interaction between the liquid and the shape, was just too small to see. Using mathematical modeling, they worked out that most of the torque – rotational force – created by the fins was getting canceled out over the helicoid. That meant that overall, only a tiny amount of torque developed, and it looked like Kelvin’s hypothesis was a bust.

But with a few modifications, the team thinks, Lord Kelvin’s reputation may be saved. They are now working on optimizing the design of the helicoid to make its spin measurable.

“The coupling is tiny,” Voth told New Scientist, “but it still exists.”

 This Week in IFLScience

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