Each summer, lawns are marked by a familiar addition: "silly sprinklers," whose loops and spirals spew water in creative ways. While seemingly frivolous in their construction, a team of mathematicians has used their design to address a long-standing mystery surrounding the laws of physics.
For decades scientists have been trying to solve Feynman's Sprinkler Problem: How does a sprinkler running in reverse—in which the water flows into the device rather than out of it—work? Through a series of experiments on custom-designed sprinklers with different shapes, the researchers arrived at a clear answer and, more generally, have determined how flowing fluids exert forces and move structures.
"This work provides the experimental answer for Feynman's Sprinkler Problem by showing, across several sprinkler types, how the angular momentum of water flows drives sprinklers' rotation," explains Leif Ristroph, an associate professor at New York University's Courant Institute School of Mathematics, Computing, and Data Science and the senior author of the paper, which appears in the journal Proceedings of the National Academy of Sciences.
The paper's authors note that the work's significance stretches beyond resolving a long-standing physics problem.
"Our findings provide a firmer understanding of how components respond to fluid flows—knowledge that can guide future engineering and technological advances for devices, such as turbines, that convert these flows into energy," notes Brennan Sprinkle, an assistant professor at Colorado School of Mines and one of the paper's co-authors.
In 2024, the research team reported its first investigations into Feynman's Sprinkler Problem, which was made famous in the 1980s by physicist Richard Feynman's account of his failed experiments.
In an earlier study, the team found that a reverse sprinkler rotates much more slowly than does a conventional one—about 50 times slower—even though the mechanisms are fundamentally similar. A conventional forward sprinkler acts like a rotating version of a rocket powered by water jetting out of the arms. By contrast, a reverse sprinkler acts as an "inside-out rocket," with its jets shooting inside the chamber where the arms meet. The researchers discovered that the two internal jets collide but they do not meet exactly head on—a subtle effect which produces forces that rotate the sprinkler in reverse.
Ristroph, Sprinkle, and their colleagues labeled this the momentum flux theory as it relates to how swirling flows move through the device.
However, this 2024 study examined only conventional sprinklers, which bear S-shaped arms, leaving open the possibility that more complicated ones—such as the silly sprinklers that spread water through differently curved and loopy tubes—might yield a different answer. Also, their earlier work was not able to disprove other leading theories.
In the new study, the authors created their own set of silly sprinklers of varying contours and then tested them in forward and reverse modes: in the forward mode, water is sprayed out like a regular sprinkler; in the reverse mode, the water is sucked in. The distinctly shaped sprinklers allowed the researchers to measure the sprinklers' rotational motions, the flows outside and inside the devices, and the torque or twisting force on the sprinklers when held in place.
The researchers used the results to test their theory as well as two other long-standing theories. One, put forth in the 1880s by physicist Ernst Mach, posits that fluid swirls in one direction and the sprinkler in the other. However, this theory could not account for reverse rotations and torques that Ristroph and his colleagues observed in their experiments. A second, attributed to Feynman and those following up on his famous study, centers on water flows occurring at the very outside of the sprinklers' arms. However, the new experiments showed the outer portions of the arms and the flows there had no effect on the sprinkler motions and torques.
By contrast, the authors found strong support for their momentum flux theory, which they generalized and found to apply equally well for both reverse and forward modes and for all the differently shaped sprinklers. Their work also shows that the shapes of the arms can control the jet flows, which could be useful in applications.
"By showing that momentum flux is the answer to Feynman's Sprinkler Problem, our findings address a long-standing open problem in flow physics and provide useful knowledge about how these devices work and their effectiveness," concludes Ristroph.
The work was supported by grants from the National Science Foundation (DMS-2407787 and DMS-2407788).
The paper's other authors were NYU graduate students Jesse Smith and Mingxuan Zuo, as well as Will Kuhlke, an NYU undergraduate.
