Why is it that when you drop paper behind a fan, it drops, and is not blown/sucked into the fan, whereas if you drop paper in front of a fan, it is blown away?
Answer
There is a YouTube video that visualizes the air flow around a propeller for various configurations. I caught a screen shot of a moment that more or less shows what is going on:
As you can see, this happens at 2:07 into the clip - this happens to be for a dual rotor configuration (two counter rotating blades) but the principle is the same.
Behind the rotor (above, in this picture) the air is moving slowly. Air over a wide range of area is drifting towards the rotor, where it is accelerated.
I will leave it up to others to describe the mathematics behind this contraction - but I thought visualizing the flow would at least confirm your observation that it is indeed slower behind the fan, and faster in front of it. In other words - it pushes, but doesn't suck.
A better image showing the flow lines around the propeller is given at this article about the mechanics of propellers
As the pressure is increased, the flow velocity goes up and the flow lines end up closer together (because of conservation of mass flow). This gives the flow the asymmetry you observed. But it's still more intuitive than rigorous...
AFTERTHOUGHT Hot Licks made an excellent observation in a comment that I would like to expand on. The air being drawn towards the fan is moving in the pressure differential between the atmosphere at rest, and the lower pressure right in front of the fan blades. The pressure gradient is quite small, so the air cannot flow very fast - and it has to be drawn from a wide region to supply the mass flow. After impact with the blade (or at least after "interacting" with the blade), the air has a LOT more momentum that is directed along the axis of the fan (with a bit of swirl...). This higher momentum gives the air downstream of the fan its coherence as can be seen in the diagram.
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