Why does the air at the top of the wing move down? I think it's because of a negative pressure field. Some people think it's the viscosity of the fluid. Why does the air at the bottom of the wing move down? I think it's because of a positive pressure field. But others don't think so. What is the reason?
Lift is generated by conservation of momentum. The wing causes air to be directed more downward, to gain momentum downward, and that is balanced by the upward momentum of the aircraft.
It is a classic theorem that no lift is generated from a non viscous fluid. And so in this sense, the viscosity is required to get this additional downward motion. But, on the other hand - even if lift is not being generated, the air has to split, some of it being pushed upward to go over the wing, and that has to be pushed back again. This push back down is effectively due to the pressure gradient. It is an important thing that in a non viscous fluid all these pushes and pulls do really add to zero.
But, also ...
One can get a good approximation to the performance of the wing by using the equations for non viscous fluid plus boundary layer theory. That is, it is only in the interaction of the wing with the air right on the wings surface that viscosity comes into play. So, in this sense - the viscosity cannot be significant in the overall flow and bending of the stream of air around the wing. Viscosity is a bit like tripping someone up to make them fall down the stairs.
The bulk motion is not affected by viscosity, it is driven by the pressure gradient, but the boundary conditions implicitly give the viscosity a big impact.