Every summer, lawns are filled with colorful “silly sprinklers” whose looped and twisted tubes spray water in unusual patterns. Their designs may seem playful, but researchers are now using these backyard devices to investigate serious, decades-old questions in physics.
This puzzle is known as Feynman’s sprinkler problem. Ask what happens when a sprinkler operates in reverse, drawing water into its arms instead of pushing it outward. A team of mathematicians came up with a clear experimental answer by building and testing different shapes of sprinklers. Their results also provide broader insight into how moving fluids push, twist and rotate physical structures.
“This study provides an experimental answer to Feynman’s sprinkler problem by showing how the angular momentum of the water flow drives sprinkler rotation across several types of sprinklers,” explains Leif Listlov, an associate professor in the New York University Courant Institute’s Department of Mathematics, Computing, and Data Science, and senior author of the paper published in the same journal. Proceedings of the National Academy of Sciences.
Why is the sprinkler issue important?
Researchers say the discovery helps do more than solve a famous scientific puzzle. Understanding how objects react to moving fluids can help engineers improve machines that capture or convert energy from flowing fluids.
“Our findings provide a more robust understanding of how components respond to fluid flows, knowledge that can guide future engineering and technological advances in devices such as turbines that convert these flows into energy,” said Brennan Sprinkle, assistant professor at the Colorado School of Mines and one of the paper’s co-authors.
The research team began studying Feynman’s sprinkler problem in a study published in 2024. The problem became widely known in the 1980s after physicist Richard Feynman described his unsuccessful attempts to investigate it experimentally.
This early research shows that reverse sprinklers rotate about 50 times more slowly than regular sprinklers, even though both rely on closely related physical mechanisms.
Traditional sprinklers operate like rotating rockets. Water is ejected from the arm, creating a force that rotates the device. Reverse sprinklers work like an “inverted rocket” because the water jet moves inward and enters the central chamber where the arms are connected.
Inside that chamber, the two incoming jets collide. However, they do not clash completely head on. This slight misalignment creates a force that rotates the sprinkler in the opposite direction.
Ristroph, Sprinkle, and their colleagues described this explanation as momentum flux theory. This focuses on how swirling water transfers momentum through sprinklers.
Testing sprinklers with twists and loops
The 2024 experiment focused only on standard sprinklers with S-shaped arms. That leaves open the possibility that sprinklers with more complex shapes, such as the curved, looped tubes found in silly sprinklers, may behave differently.
Previous studies also did not completely rule out other major explanations for sprinkler movement.
For the new study, the team built a collection of silly sprinklers with different contours. Each device was tested in two configurations. In forward mode, water is sprayed outward like a regular lawn sprinkler. In reverse mode, water was drawn into the sprinkler.
This unusual shape allowed researchers to examine several features at once. They recorded how the sprinklers rotated, observed water moving inside and outside the device, and measured the torque and twisting forces created when the sprinklers were prevented from rotating.
Competing physical theories put to the test
Scientists compared momentum flux theory to two other explanations that have been proposed over the years.
The first one dates back to the 1880s and was introduced by physicist Ernst Mach. This suggests that the fluid rotates in one direction and the sprinkler rotates in the opposite direction. However, Mach’s explanation failed to account for the reverse rotation and torque measured during the new experiment.
The second theory, associated with Feynman and subsequent researchers, focuses on water flowing near the outer end of the sprinkler arm. New tests show that neither the outer portion of the arm nor the water moving around the arm affects sprinkler movement or torque.
Instead, the results strongly supported momentum flux theory. The researchers extended the theory and found that it accurately describes both forward and reverse behavior across all sprinkler shapes tested.
This experiment also revealed that the water jet can be changed and controlled by changing the shape of the arm. This ability could prove useful when designing real fluid-based devices.
“By showing that momentum flux is the answer to Feynman’s sprinkler problem, our findings address a long-standing open question in flow physics and provide useful knowledge about how these devices work and their effectiveness,” Ristoroff concluded.
Other authors on the paper were New York University graduate students Jesse Smith and Mingxuan Zuo, and New York University undergraduate Will Kuhlke.
This research was supported by grants from the National Science Foundation (DMS-2407787 and DMS-2407788).
