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An actuator disk accelerating a fluid flow from right to left
Inviscid, frictionless fluid
Actuator disc (uniform flow)
Air far above the rotor is at ambient pressure pa and zero velocity, far downstream it's at wake velocity Vw and at ambient pressure pa again (we assume inviscid and frictionless fluid, so the wake and the surrounding air don't mix). Directly under the rotor, the velocity is Vd and the pressure is pd.
The thrust delivered is equal to the change in momentum with time, so (with dotted m meaning the mass flow rate):
(why Vw and not Vd? Because at Vd there's an additional pressure component)
The power delivered by the rotor is
But at the location of the propeller disc, the power is equal to the work done by the thrust:
So
, the velocity under the rotor is half of the ultimate wake velocity.
The mass flow rate is:
We can write the thrust as:
This gives us Vw in function of the thrust
So the power becomes:

\tfrac12\, \rho\, v^2\, +\, p\, =\, </math>

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