In the pump-out case, the water adds positive angular momentum when it flies out, so the sprinkler body must add negative angular momentum (constant torque) to have conservation.
In the Feynman suck-in case, the water drains from the base of the sprinkler without any angular momentum, so the sprinkler head does not have to add an opposite angular momentum (feel a constant torque).
Imagine the S is straight, but a vane attached to the side of the pipe blocks suction from one side.
I tried to illustrate this in ascii art, but it appears HN has an algorithm to destroy ascii art.
That vane will now have a slight vacuum on the side of the pipe, and it seems logical that it should want to move in that direction.
Now imagine that vane curved around the pipe so it forms the end of the S-bend. Same thing.
hello
world ===============
Pipe <---Suck
===============
-----------vane-----------IMO, that's the crucial asymmetry -- fans "blow out" air in a straight line, but "suck in" air from the entire surroundings.
In response to your first question, the water doesn’t add negative angular momentum because the whole system we are conserving is the sprinkler body and the moving water together. The sum must just be zero in both cases. We can track a single drop of water in case two (Feynman) to see how the conservation works. At the start the drop is at zero angular momentum since it is stationary in the tank. It then passes through the whole Sprinkler mechanism (and whatever s-curves there are) and ends up at the suction drain at the center of the sprinkler also at zero angular momentum. Therefore no matter what happened it could not have applied a net torque to the sprinkler, no matter the path it took to get to the drain.
Contrast with case 1, the regular sprinkler, where the water drop starts at zero angular momentum (pumped from the center of the sprinkler) and ends with positive angular momentum (spinning away from the center) therefore it must have applied an opposing net torque at some point in its journey to the body of the sprinkler, which spins.
When you’re forcing the water in, you’ve got a pipe with pressure greater than the water. Pressure is a measurement of force per unit area. The pressure of the water presses evenly on the tube walls, but there is no tube walls at the opening. So, there is a spot where there is no pressure on the tube. This means a net force because that lack of pushing on the opening is not canceled out. Since the pressure differential is in the direction of the rotational axis you get rotation.
In the reverse, there is also more pressure in the tubes than in the water. However, the spot where there is a net difference in pressure is the drain. The drain has an opening putting less pressure, so we get a net force. However the direction of the net force is perpendicular to the rotational axis—so we get no rotation.