In a jet engine though, you have a nozzle on the back, so the input area is greater than the output area. Heating the air also causes it to take up much more volume. These put together, mean that the output velocity is much greater than the input velocity, so there is a momentum transfer from the atmosphere to the plane.
Without the T, the pipe moved as you'd expect. With the T directing the airflow in equal proportions perpendicular to the axis of the pipe, the pipe stayed still as if the fan wasn't on at all.
Here's my caveman f=ma thought experiment:
1. make it 2-d.
2. replace the fan with a person sitting on a chair on a frictionless surface.
3. instead of air it's an endless field cinder blocks ahead of him.
the person reaches out, and pulls in a cinder block. f=ma says they each move toward the other while the center of mass of the combination of them does not move.
Now, if he throws the cinderblock behind him, he moves further forward - this would be analogous to an airplane propeller. or a fan in an open pipe.
If he, um, splits the cinderblock in two and places each half directly off to his sides there is no net force exerted on him by this. This is the fan in a T-shaped pipe.
the fan+pipe grabs air from ahead, moves this mass backward and then sets it aside. it's not a jet-engine, but it is moving the air mass toward itself and must be moved equally and oppositely.
I don't think it's essential to worry about how the air/blocks rearrange themselves after this - but if the blocks surround and jostle, that's just another effect layered in super-position over this one, and if we don't agree so far then it will only make things more confusing
With your concrete block example, as I start pulling the block towards me I experience an impulse forward. But when it approaches my body I slow it down to zero speed, creating an opposite impulse. So although I might have moved forward a few inches during the motion, my momentum is zero at the end. When you scale this up to large numbers of air molecules, the result is the same.
if the demo was mainly to show that it's the jet of air expelled out the back that's providing thrust, then, fine it does that and its a valuable lesson. I guess I'm hung up on the technicality that there is actually a real movement of air mass even without that rearward jet and that has to be felt by the apparatus - I guess it's just unnoticeably small in the real world demo.
anyhow, my confidence in physics intuition has been shaken. thanks bunches.
The exiting flows out the sides neatly cancel. So what's happening from front-to-back? There is air flowing in at some velocity x cross-sectional area x air density. this momentum has to be balanced completely for the pipe to stay still - but there is no source of momentum in the other direction so the pipe will feel this force and move.
if the pipe was open at the back, there would be momentum exiting the pipe balancing the incoming momentum - or even over-balancing (as in a jet engine)
here's a video going through the math on a similar problem which is a little more complex in detail, but the same in principle: https://www.youtube.com/watch?v=hXApWf1r0Eo
https://upload.wikimedia.org/wikipedia/commons/d/df/F-GTAR_A...
I think the important thing to consider is the relative velocities of the intake and exhaust air (the latter being much larger).
here is a thermal image of a modern jet with thrust reverses on : http://www.rusaviainsider.com/russias-mc-21-tested-for-rever...
The force on the plane is in the opposite direction.