The explanation based on Newton's laws of motion is more to the effect that the wing interacts with the air in such a way as to accelerate some of the air towards the ground. The reaction force is upwards.
The Navier-Stokes equations merely model fluid flows. Understanding them provides no understanding of the behaviour of such flows. That behaviour is emergent from the interaction of a great many particles.
>The explanation based on Newton's laws of motion is more to the effect that the wing interacts with the air in such a way as to accelerate some of the air towards the ground. The reaction force is upwards.
But that doesn't have any explanatory power at all. If we assume Newton's laws hold, then obviously if there's a force upward on the airfoil then there's a reaction force downward on the air.
It'd be like explaining the combustion engine by saying "the drive shaft from the engine rotates this way, and the reaction force - because the engine is more-or-less rigidly mounted to the frame - is resisted through the suspension by the wheels being in contact with the ground". OK, sure, but I still don't know how the engine actually works.
I dunno. If I look at even a very simple diagram of the flow of air around a wing I see air deflected downward on the bottom and air accelerated around a curve on the top. Both would be expected to produce a downward reaction force.
Added: Or more Newtonish (no action at a distance), there is more upward vertical force contributed by the particles in both cases than downward force.
Yes, if someone tells you how the air flows around a wing you can immediately deduce that it's producing lift, since the air is deflected downwards. The real task is explaining why the air flows the way it does.
The Navier-Stokes equations merely model fluid flows. Understanding them provides no understanding of the behaviour of such flows. That behaviour is emergent from the interaction of a great many particles.