I still keep coming back to the bird in the cage, or a helicopter in a
hanger.
The bird steps off the perch and starts to hover (must be a humming bird).
It pushes one puff or air down, followed by another, and the bird finds
itself in equilibrium. But soon the downdraft will set up a circulation
pattern - up along the sides and back down the middle. So now the bird is
flying in a downdraft (which it created itself), which means it needs to
flap harder, which sets up a bigger downdraft ....
(And all this time, a scale under the cage will read less than the total
weight! The bird an cage may be stationary, but the air is getting
compressed below the bird, so the c.m is still falling, so the normal force
from the scale doesn't have to hold the whole weight.)
Where does it stop? I think the answer is in the pressure. Eventually the
air below the bird gets compressed by all that air getting pushed down. I
suppose you could think of it as flying in ground effect, since the bird
gets enough additional lift from the air molecules that are bouncing back
up to overcome the downward force from the downdraft.
(It is only at this point that equilibrium is truly achieved, and the scale
under the cage will truly read the correct weight.)
AHA! moment (I think). Even level flight is an intrinsically
non-equilibrium state and any explanation of flight must at least recognize
this.
If I might be so bold as to go way out on a limb... Bill's model
corresponds best to the bird just jumping off the perch. The downward puff
holds the bird up, but it is not a truly equilibrium state. (Of course in
real life, the plane simply flies to a new "cage", i.e. an undisturbed
parcel of air and continues its non-equilibrium state.) John's model more
accurately looks at the bird in true equilibrium, where only the
interaction of the air with the ground can ultimately hold the plane and
the air off the ground. (But other than ground effect, this is not really
close to the situation of normal flight.)