Typical lawn sprinklers “reverse sprinkler”” consist of a revolving wheel that is surrounded by a number of nozzles that are positioned at different angles. When water is pumped into the sprinkler, the nozzles emit jets that force the wheel to rotate. However, what would occur if the water were instead drawn into the sprinkler so that it could be used? In what direction would the wheel turn at that point, or would it turn at all with any degree of rotation? Since the 1940s, physicists such as Richard Feynman and others have been struggling with the “reverse sprinkler” problem. This is the essence of the problem. Now, applied mathematicians at New York University believe that they have solved the puzzle, as stated in a recent research that was published in the journal Physical Review Letters. The answer to the problem defies the conventional thinking that has been held over the subject.
According to Leif Ristroph, a co-author of the paper who works at the Courant Institute at New York University, “Our study solves the problem by combining precision lab experiments with mathematical modeling that explains how a reverse sprinkler operates.” It was discovered by us that the reverse sprinkler rotates in the opposite direction, or in the other direction, while it is taking in water as it does when it is ejecting water, and the reason for this is a subtle and unexpected one.
Ristroph’s laboratory routinely deals with puzzles of this colorful nature that are based in the real world. For example, Ristroph and his colleagues refined the recipe for the ideal bubble in 2018 by conducting trials with soapy thin sheets. This allowed them to achieve exceptional results. (You should blow gently at a steady 6.9 centimeters per second, and you should use a circular wand that has a perimeter equal to 1.5 inches.) A study was conducted by the Ristroph laboratory in 2021 to investigate the formation processes that lie beneath the so-called “stone forests” that are prevalent in some locations of China and Madagascar. These pointed rock formations, such as the well-known Stone Forest in the province of Yunnan in China, are the consequence of solids dissolving into liquids in the presence of gravity, which results in the production of natural convective flows.
It was in the year 2021 when his laboratory constructed a Tesla valve that was functional, in line with the design of the inventor, and monitored the flow of water through the valve in both directions at a variety of pressures. It was discovered that the water flowed approximately twice as slowly in the direction that was not liked. Furthermore, in the year 2022, Ristroph conducted research into the exceedingly complicated aerodynamics of what constitutes a good paper airplane, precisely what is required for smoothly gliding. They came to the conclusion that the aerodynamics of paper airplanes are significantly different from those of conventional aircraft, which are powered by airfoils to generate lift.
Feynman is often credited with popularizing the concept of the reverse sprinkler issue; however, the origins of this problem may be traced back to a chapter in Ernst Mach’s 1883 textbook The Science of Mechanics (Die Mechanik in Ihrer Entwicklung Historisch-Kritisch Dargerstellt). It wasn’t until the 1940s that a group of physicists from Princeton University started discussing Mach’s thought experiment. Prior to that, it had been mostly forgotten about throughout the decades.
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It was around this period that Feynman was a graduate student at that institution. He enthusiastically participated in the discussion, even going so far as to devise an experiment in the cyclotron laboratory to verify his idea. (In a manner that is characteristic of Feynman, the culmination of that experiment was the explosion of a glass carboy that was utilized in the apparatus due to the high-pressure environment inside of it.)
One might have the presumption that a reverse sprinkler would function in the same manner as a standard sprinkler, with the exception that it would be played backwards. However, it turns out that the physics behind it is more intricate. According to what Feynman said in his book titled “Surely You’re Joking, Mr. Feynman” (1985), “The answer is perfectly clear at first sight.” The problem was that some individuals would believe that it was completely obvious [that the rotation would be] in one direction, while other individuals would believe that it was completely obvious in the opposite direction.
Mach argued that a reverse sprinkler would not have any rotation because the reaction force on the nozzle as it draws in water pulls the nozzle in the opposite direction of the clock, while the water that is pouring into the inside of the nozzle pushes it in the opposite direction. In this steady-state scenario, the two forces eliminate one other and cancel each other out. In the experiment that Feynman conducted himself, there was a little tremor that occurred when pressure was initially applied to pump water through the nozzle. After that, the sprinkler went back to its initial position and remained perfectly still.
On the other hand, there were those who proposed that if the friction was sufficiently low and the inflow rate was sufficiently high, a reverse sprinkler would begin to revolve in the opposite direction as a regular sprinkler. This would be due to the formation of a vortex within the dispenser. Philip Ball, writing for the American Physical Society in Physics, stated that after Feynman’s attempts, “some experiments have shown steady reverse rotation, some showed only transient rotation, and some situations led to unsteady rotation that changed direction or proceeded in a direction that depended on the experimental geometry.”
Into the picture comes Leif Ristroph and his colleagues, who constructed their very own bespoke sprinkler. This sprinkler had ultra-low-friction rotary bearings, which allowed the device to rotate without any restrictions. They submerged their sprinkler in water and then utilized a specialized device to either pump water into the sprinkler or take water out of the sprinkler at flow rates that were precisely controlled. A particularly important aspect of the experiment was the fact that the team was able to watch and measure the flow of water inside, outside, and through the device thanks to the bespoke sprinkler that they had created. The addition of dyes and microparticles to the water, together with the use of lasers to illuminate the water, assisted in the capture of the flows on high-speed video. In order to obtain a more accurate representation of the fluid-flow patterns, they conducted their studies for at least several hours at a time.
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.
Physical Review Letters, 2024. DOI: 10.1103/PhysRevLett.132.044003 (About DOIs).
Listing image by NYU’s Applied Mathematics Laboratory
