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I am thinking of a simple ducted fan with a single set of rotating blades, no stator vanes behind. If the effect of the fan is to raise the pressure of the air, how is there flow into the fan against this pressure gradient?

I understand that the rotating fan blades have the effect of increasing the stagnation enthalpy of the flow. For an incompressible flow, this is equivalent to the stagnation pressure increasing via the fan blades imparting kinetic energy into the flow. I know that static pressure and velocity increase across the fan, so how can there be flow against this pressure gradient?

I think the explanation will have something to do with the different frames of reference chosen to analyse the problem.

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how is there flow into the fan against this pressure gradient?

The blades of the fan do work on the air and push it against the gradient near the fan. Long before the air enters the fan, it is pushed by the pressure gradient towards the fan, so pressure falls along the direction of motion. Only when crossing the region where the fan blades move and when entering the duct, the pressure rises on a short length scale. Along the duct, the pressure then falls along the direction of motion again.

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