What happens to the pressure differential of airflow under and over the wing as the critical angle of attack is reached?
While this may seem pretty easy, it's not for a 12th-grade student like me.


It definitely does not seem easy. It takes solid background in numeric integration, good working knowledge of a numeric library and some serious programming time.

There is a good reason everybody—and I mean aircraft engineers and researchers—just uses XFoil. And that's just the basic analysis—3D calculations require even more complex software packages—that cost hefty sums of money (XFoil is free).


  We all know that the pressure differential is 1/2 ρ〖(v2-v1)〗^2, where v2 is the airflow velocity over the wing and v1 the airflow velocity under the wing.


There is no single $v_1$ and single $v_2$. There is just $\vec{v}(x, y)$ (limiting to 2D analysis like XFoil does for simplicity) that is different at each point. You can't just take $\alpha$ and divine two speeds from it, because the situation is more complex than that.

Remember that:


The air flowing over the wing reaches the trailing edge in significantly shorter time than the air flowing below it!
A thin flat plate is not a particularly good wing, but it does generate some lift.
At the Reynolds numbers involved for full-scale aircraft the flow is rarely laminar beyond 20–30% chord. Turbulence increases the velocity and therefore decreases pressure and increases lift.
In viscous fluid, the velocity must be continuous in all coordinates and that includes the boundary, so the layer directly touching the surface is not moving relative to it. The velocity quickly increases in the boundary layer. The properties of this boundary layer is what governs whether the stream will remain attached or not.
There is no lift in inviscid fluid (= superfluid) and neither would there be in massless fluid. That is, you need to take both inertia and viscosity into account to get any result.


Basically the only way to calculate this is by evaluating the Navier–Stokes equations, which has to be done numerically and in fairly fine grid to achieve any useful precision.

Before numeric integration, there were some simpler analytical methods like the thin airfoil theory, but those didn't work from scratch—some coefficients must be measured experimentally.


  Whenever the critical AoA (AoA at which the aircraft stalls) is reached, how do we calculate pressure differential?


Just like before—by integrating the Navier–Stokes equations.