Explaining Lawn Sprinkler's Movement: P> & P< Pa

[vladimir69]

the picture attached resembles a lawn sprinkler but it works on air not water.

what i am trying to find is

1. if P> Pa where Pa is the atmospheric pressure, does the rotor turn and, if so, in which direction?

2. if P< Pa where Pa is defined as above, does the rotor turn and, if so, which direction.

for 1 i am pretty sure about which way it turns but i have been unable to come up with a substantial reason as to why it turns in that direction. i tried 2 explanations which weren't satisfactory, namely stating Newtons 3rd law and tried using the rocket equation(?) to explain its movement (ie m dv + u dm = 0) where u is the effective exhaust velocity. i also suggested that the kink in the sprinkler being the part that makes it turn but couldn't explain why

can anyone come up with a "sufficient" argument to explain the movement of the rotor for both parts 1 and 2?

i think it mat have something to do with conservation of angular momentum but i am not sure about the details

 

[Clausius2]

Very good question boy!

1)if P>Pa, it turns clockwise, because the stored air tends to escape. As it flows out, the two jets creates a torque (you are right about rockets). It is only due to the momentum conservation of the air that passes thorugh the rotor. Without employing any equation, you should know any deflection of a fluid due to a reflecting surface creates a force in this surface. The rotor is only absorbing the deflection because of the proper flow. Surely, the torque would not be constant in time due to the continuous decreasing of the internal pressure.

2)if P<Pa, it turns also clockwise. The same reason is valid, due to the same stream deflection. I suppose that vessel is larger enough to absorb some amount of external air.

3) third case, I will formule this time. P<Pa but Mach number in the outflow section is 1. The pipe has sonic flow everywhere.