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Math

Mathematicians Finally Solved Feynman's 'Reverse Sprinkler' Problem (arstechnica.com) 58

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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Mathematicians Finally Solved Feynman's 'Reverse Sprinkler' Problem

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  • by Viol8 ( 599362 ) on Saturday February 03, 2024 @06:17AM (#64210322) Homepage

    A rocket works by expanding gas pushing back on the rocket as it tries to escape from the nozel. Water in a sprinkler or hose isn't expanding as it exits, its simply moving and any force it exerts in theory should be on whatever corners or curves it goes around before it gets to the exit. Or am I missing something?

    • Technically the expansion gives it a benefit. But if your rocket literally weighed only a few grams and you expelled the fuel without igniting it out the back, yes it would also be propelled. This is the 3rd law of motion. The majority of the force in the sprinkler doesn't come from the fluid moving around a corner, it comes from the expelling of a significant mass.

      Incidentally this is precisely why the suck vs blow in this case produced a 50x difference in resulting spin speed, the internal forces of fluid

      • by Viol8 ( 599362 )

        Newtons law assumes there is a reactive force. Where is the force on the pipe when water is simply moving through it? The reactive force should be back at the pump , not at the hose or pipe simply guiding the water.

        • by caseih ( 160668 )

          I've a got a fire hose you're welcome to use to test your theory with!

          It really is like a rocket in my experience. I suspect it's for a similar reason as the rocket engine as well. The restriction of the nozzle as you'd find in a sprinkler or fire hose causes the water to speed up. It's a combination of the mass and speed that give specific impulse to the nozzle itself.

          I would love to get the full explanation of the physics involved here.

    • Yes, a water jet is like a rocket. If you have a spaceship that has a hose that pushes water out, the spaceship will accelerate in the opposite direction. It does not matter whether there is expansion of gas or not in a rocket. All that matters is that momentum is imparted to the expelled matter, so the opposite momentum is imparted to the rocket. A rocket could work with an ion thruster which accelerates gas without an explosion or expansion. Similarly, it does not matter whether water is pushed out o
    • Think of O'l Isaac Newton and his law of preservation of momentum. The sprinkler is tossing water droplets away into space, which causes a reverse thrust.
    • A rocket works by expanding gas pushing back on the rocket as it tries to escape from the nozzle

      Incorrect.

      A rocket works by expelling mass at high speed, exploiting Newton's third law of motion. The expansion effect you note is real and important, but it just serves to increase the speed at which the exhaust mass is expelled. It's a large efficiency enhancement to the basic underlying principle, not the basic principle.

  • by chill ( 34294 )

    I thought one of those sprinklers spun because they contained an arm on a hinge that was impacted by the water jet, causing it to impart spin when in was pushed out and hit a stop. It then was pulled back by a spring and the process repeated.

    Remove the arm and it is just a nozzle and would flop around like every other nozzle if it wasn't fixed to the ground.

  • Isn't this just a matter of fluid dynamics? A ventura causes a stream but only in a certain direction. then there is the shape of the turbine blades. Why would this be such a difficult problem.
  • The inlet to the sprinkler arm will experience reduced pressure. You are changing the shape of the flow, which will cause a pressure differential. That will lead to a low-pressure area at the tip of the arm, and the water pressure on the backside of the arm will push the sprinkler.
    • Though they say they found internal fluid dynamics were responsible for the spinning... I would very much like them to explain how they ruled out the effect you describe as a factor. (Maybe they did, I did not RTFA.)

  • by rossdee ( 243626 )

    Where does the water come from?

  • There's no way Feynman actually meant for you to BUILD a reverse sprinkler, right? It was a math problem wasn't it?
    I mean just the summary is totally wild, that's great work though.

  • Ok, so it spins backwards. Just like one would expect when the water is also travelling backwards from normal operation...

    The math people finally proved what we have seen forever. Great, but can this be applied to anything to improve existing designs of anything else?
  • This is not really surprising. While the forward sprinkler operates on conservation of momentum (mass of water traveling along a velocity vector), the reverse system inflow suction draws water from all directions, canceling out most of that momentum effect.

    However, flow inside the nozzles is (more or less) laminar. Giving it a defined velocity vector and momentum. Also, any change in direction creates additional velocity vectors which need to be resolved.

    I would think that actual pressure distribution ins

  • The liquid based atmosphere is this "proof" is doing all the work. This is no solution to the original question.

Every nonzero finite dimensional inner product space has an orthonormal basis. It makes sense, when you don't think about it.

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