Finally, mathematicians have solved Feynman’s “reverse sprinkler” problem.

Finally, mathematicians have solved Feynman’s “reverse sprinkler” problem.

Microscopic light-scattering particles reveal the reverse-mode (absorption) flow pattern of the spray, illustrating the vortices and complex flow patterns that form within the central chamber. Credit: K. Wang et al., 2024

A typical lawn sprinkler features different nozzles arranged at angles on a rotating wheel; When water is pumped, it releases jets that cause the wheel to spin. But what would happen if the water was sucked into the sprinkler instead? In which direction will the wheel turn then, or will it turn at all? This is the essence of the “backspray” problem that physicists like Richard Feynman, among others, have faced since the 1940s. Now, applied mathematicians at New York University think they’ve solved that mystery, according to a recent study published in the journal Physical Review Letters — and the answer challenges conventional wisdom on the issue.

“Our study solves the problem by combining careful laboratory experiments with mathematical modeling that explains how backwash works,” said co-author Lev Ristrov of New York University’s Courant Institute. “We have found that the reverse sprinkler rotates in the opposite or opposite direction when drawing in water as it does when releasing it, and the reason is subtle and surprising.”

Ristrov’s lab frequently tackles these kinds of colorful real-world puzzles. For example, in 2018, Ristroff and colleagues fine-tuned the ideal bubble recipe based on experiments conducted on thin films of soap. (You want a circular wand about 1.5 inches in circumference, and you should blow gently at a constant speed of 6.9 cm/s.) In 2021, Ristroff’s lab investigated the formation processes behind so-called “stone forests” common in certain regions. China and Madagascar. These pointed rock formations, like the famous Stone Forest in Yunnan Province of China, are the result of solids melting into liquids in the presence of gravity, resulting in natural convection flows.

In 2021, his lab built a working Tesla valve, according to the inventor’s design, and measured water flow through the valve in both directions at different pressures. They found that the water flows about twice as slowly in the non-preferred direction. In 2022, Ristrov studied the very complex aerodynamics that make a good kite, specifically what is required for smooth flight. They found that the aerodynamics of a kite differ significantly from conventional aircraft, which rely on ailerons to generate lift.

رسم توضيحي لـ أ "Reaction wheel" From a book <em>Mechanics</em> By Ernst Mach (1883).” src=”×692.jpg” width=”640″ height=”692″ srcset=”https 2x”/><figcaption class=
Zoom in / Illustration of the “reaction wheel” by Ernst Mach Mechanics (1883).

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The backscatter problem is associated with Feynman because he popularized the concept, but it actually dates back to a chapter in Ernst Mach’s 1883 textbook Mechanics (The mechanisms in their development are presented historically and critically). Mach’s thought experiment remained in relative obscurity until a group of physicists at Princeton University began discussing the issue in the 1940s.

Feynman was a graduate student there at the time, and he threw himself into the debate with great enthusiasm, even devising an experiment in the cyclotron laboratory to test his hypothesis. (In true Feynman fashion, this experiment culminated in the device’s glass carriage exploding due to the high internal pressure.)

One might think that a reverse sprinkler would work just like a regular sprinkler, only run backwards, so to speak. But the physics turns out to be more complicated. “The answer is quite obvious at first glance,” Feynman wrote You must be joking, Mr. Feynman (1985). “The problem is that some people might think it’s pretty obvious (that the rotation will be) one way, and someone else might think it’s pretty obvious the other way.”

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