Before you read this, I suggest you read post 17.15.
We will think about this question by considering a plane whose wings have the cross-sectional shape shown in the picture below.
In this picture, the engines of the plane move it in the direction shown by the arrow. Now let’s think about the movement of the surrounding air relative to the wing (post 16.4). Let’s suppose that the air flow is laminar (post 17.15) and consider a streamline moving from A to B. The streamline moving over the wing has a greater distance to travel than the streamline moving under the wing. If the two streamlines leave A at the same time and arrive at B at the same time, the one that moves over the wing must have a higher speed, vover, than the speed, vunder, of the streamline that moves under the wing. We can write this as vover > vunder.
According to Bernoulli’s equation the quantity
p/ρ + v2/2 + Ψ
is conserved in laminar flow of a fluid with negligible viscosity (post 17.15). In this equation p is the pressure in the flowing air, ρ is its density and v is its speed; Ψ is a measure of the potential energy which is the same at A and B whether the air has flowed over or under the wing. So, when we apply Bernoulli’s equation to air flowing over and under the wing, we get the result that
pover/ρ + v2over/2 = punder/ρ + v2under/2.
Since vover > vunder,it then follows that pover < punder. In other words, there is a net pressure under the wing of
Δp = punder – pover.
If the total wing area of the plane is A, there is an upward force on the wings of AΔp (post 17.5).
At the same time, there is a gravitational force of mg, where m is the mass of the plane and g is the magnitude of the gravitational field (post 16.16), acting downwards on the plane.
If AΔp > mg, the plane rises. If AΔp = mg, the plane stays in level flight. If AΔp < mg, the plane falls.
So the upward force, called lift, on the plane depends on its speed, since Δp depends on speed.
This is the explanation for why planes fly that you will find in most textbooks. It’s what I was taught when I was a student. And the idea that difference in air speed under and over the wing causes lift explains why planes fly. But if we apply the arguments exactly as I have done above, you don’t get the right answer for the lift. I think the reason is that air can flow from A to B by the two paths (over and under the wing) without arriving at exactly the same time, while still preserving laminar flow. More details are given at https://www.grc.nasa.gov/www/k-12/airplane/bernnew.html.
And remember that all this applies only to laminar flow. If the plane encounters turbulent air flow (post 17.15), the pattern of forces acting on the wings is much more complicated – leading to the irregular motion that is familiar to all passengers on planes!