Mathematicians Finally Solved Feynman's 'Reverse Sprinkler' Problem

Jennifer Ouellette reports via Ars Technica: A typical lawn sprinkler features various nozzles arranged at angles on a rotating wheel; when water is pumped in, they release jets that cause the wheel to rotate. But what would happen if the water were sucked into the sprinkler instead? In which direction would the wheel turn then, or would it even turn at all? That's the essence of the 'reverse sprinkler' problem that physicists like Richard Feynman, among others, have grappled with since the 1940s. Now, applied mathematicians at New York University think they've cracked the conundrum, per a recent paper published in the journal Physical Review Letters -- and the answer challenges conventional wisdom on the matter. 'Our study solves the problem by combining precision lab experiments with mathematical modeling that explains how a reverse sprinkler operates,' said co-author Leif Ristroph of NYU's Courant Institute. 'We found that the reverse sprinkler spins in the 'reverse' or opposite direction when taking in water as it does when ejecting it, and the cause is subtle and surprising.'

Enter Leif Ristroph and colleagues, who built their own custom sprinkler that incorporated ultra-low-friction rotary bearings so their device could spin freely. They immersed their sprinkler in water and used a special apparatus to either pump water in or pull it out at carefully controlled flow rates. Particularly key to the experiment was the fact that their custom sprinkler let the team observe and measure how water flowed inside, outside, and through the device. Adding dyes and microparticles to the water and illuminating them with lasers helped capture the flows on high-speed video. They ran their experiments for several hours at a time, the better to precisely map the fluid-flow patterns.

Ristroph et al. found that the reverse sprinkler rotates a good 50 times slower than a regular sprinkler, but it operates along similar mechanisms, which is surprising. 'The regular or 'forward' sprinkler is similar to a rocket, since it propels itself by shooting out jets,' said Ristroph. 'But the reverse sprinkler is mysterious since the water being sucked in doesn't look at all like jets. We discovered that the secret is hidden inside the sprinkler, where there are indeed jets that explain the observed motions.' A reverse sprinkler acts like an 'inside-out rocket,' per Ristroph, and although the internal jets collide, they don't do so head-on. 'The jets aren't directed exactly at the center because of distortion of the flow as it passes through the curved arm,' Ball wrote. 'As the water flows around the bends in the arms, it is slung outward by centrifugal force, which gives rise to asymmetric flow profiles.' It's admittedly a subtle effect, but their experimentally observed flow patterns are in excellent agreement with the group's mathematical models.

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