Feynman Sprinkler - Solution

Solution

The behavior of the reverse sprinkler is qualitatively quite distinct from that of the ordinary sprinkler, and one does not behave like the other "played backwards." Most of the published theoretical and experimental treatments of this problem have claimed (as did Mach and Gleick) that a sprinkler will not turn when made to suck in the surrounding fluid. It is now understood, however, that an ideal reverse sprinkler (i.e., one which can turn without friction and is surrounded by an ideal fluid) will accelerate towards the incoming fluid as the suction is being switched on, and come to a stop as the suction is switched off. The ideal reverse sprinkler will not experience any torque in its steady state. This behavior may be understood in terms of conservation of angular momentum: in its steady state, the amount of angular momentum carried by the incoming fluid is constant, which implies that there is no torque on the sprinkler itself.

Most experimental setups fail to detect any turning of the reverse sprinkler because the transient torque is not large enough to overcome the friction of the sprinkler's bearing. On the other hand, experiments with very low-friction bearings find a small torque on the reverse sprinkler even in its steady state. This is now understood to be a consequence of the viscosity of the fluid being sucked into the sprinkler, which leads to the dissipation of some of the energy of the incoming fluid and diffuses some of its angular momentum to the surrounding tank. This torque, induced by the viscosity, causes the reverse sprinkler to turn weakly towards the incoming fluid (i.e., in the direction contrary to the motion of a regular sprinkler expelling water).

The smallness of the torque on a reverse sprinkler is closely analogous to the propulsion of the so-called "pop pop boat," a toy boat that moves forward as it alternately expels and then sucks in water through a pipe connected to a small internal tank heated by a candle.