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In an airfoil, a low-pressure region forms in the turbulent wake created when the flow separates. Wouldn't this cause an increase in the net vertical force acting on the airfoil since the pressure at the top decreases?

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This question could get better answers in the Aviation StackExchange.

You are right partially in that at the beginning of the stall the vortices separate from the top of the wing but the wing still has significant lift albeit with a lot of drag.

But in the next stage of the stall, these separated vortices get blown past the trailing edge of the wing and the vacuum let's some of the high-pressure wind from under the wing turn around the trailing edge and acting on top of the wing and causing loss of the lift and stall.

[the wind from under the wing turns around1

watch this slide show. wing stall

http://eng-web1.eng.famu.fsu.edu/~dommelen/research/airfoil/airfoil.html

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To add more to Kamran's answer, Generally for streamlined bodies like aerofoils, pressure drag force << Skin friction drag force << lift force. If the flow is separated, there will a reverse flow on the surface of the aerofoil and this will lead to a drastic increment in skin friction drag. so the efficiency $(C_L/C_D)$ of the airfoil also comes down.

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