The inverse sprinkler, worked through

Sample worked problem — the "projects include problems" content type. Replace with problems you've actually chewed on.

Problem. A lawn sprinkler sprays water and spins one way. Run it in reverse — submerge it and make it suck water in through its nozzles. Which way does it rotate, if at all?

This is the famous problem Feynman describes agitating over (and flooding a Princeton cyclotron lab with) in Surely You're Joking. The temptation is to answer by symmetry: reverse the flow, reverse the torque. The temptation is wrong, and seeing exactly why it is wrong is the entire value of the problem.

Setting up the accounting

Work in the frame of the (initially stationary) sprinkler and consider a control volume enclosing one nozzle arm. Steady-state angular momentum balance: the torque on the arm equals the net flux of angular momentum out of the control volume, plus pressure torques on its boundary.

Spraying case. Water leaves the nozzle tangentially with speed $v$ at radius $R$: an angular momentum flux $\dot{m} v R$ out of the volume. The reaction torque spins the sprinkler backwards relative to the jet. Clean and familiar — a rotary rocket.

Sucking case. Naively, water now enters tangentially at the nozzle, carrying angular momentum $\dot{m} v R$ in, so the sprinkler should spin the opposite way with equal magnitude. But the two situations are not mirror images:

An outgoing jet is collimated — all its momentum points one way. An intake draws fluid from a broad region around the nozzle; the incoming fluid's momentum is nearly isotropic far away and only organizes near the opening. Meanwhile, there is a pressure asymmetry: the low pressure just inside the sucking nozzle pulls on the arm.

The two effects cancel — almost

Do the bookkeeping carefully (steady, ideal flow): the angular momentum flux carried in by the intake and the pressure force on the nozzle lip produce torques that cancel exactly in the ideal-fluid, steady-state limit. The ideal inverse sprinkler does not turn at all.

Reality is not ideal, and the corrections pick a direction:

1. Transients. When suction starts, before the flow is established, there is a net torque and the sprinkler twitches toward the incoming water (the reverse of the spraying direction).
2. Viscous dissipation. The intake flow sheds vortices inside the arm; the swallowed angular momentum is partially destroyed instead of transmitted, leaving a small steady torque — again reverse of spraying, and this is what careful experiments (e.g. the 2004 UMD tabletop version and the 2024 NYU work) actually measure: a weak, sometimes unsteady rotation toward the flow.

Answer. Ideally: it doesn't rotate. Really: it creeps weakly in the reverse direction of a spraying sprinkler, driven by the transient and by the asymmetry between a coherent jet and an incoherent sink.

The moral, which generalizes: time-reversing a flow does not time-reverse its momentum accounting when dissipation is anywhere in the loop. Symmetry arguments are only as good as the symmetry.