If Bernoulli's Principle can only be applied to a streamline, how does it justify the pressure difference that causes lift According to the Bernoulli's principle, an increase in the speed of a fluid causes a decrease in pressure.
In the case of an aerofoil or aircraft wing, the speed of the fluid (air) above the wing is greater than the speed below it.
But since the Bernoulli's Principle can be applied ONLY along points along the SAME streamline, how does the speed difference justify the pressure difference above and below the wing?
 A: Pressure on a surface is always related to the vertical velocity component of molecules striking it, so a change of the parallel velocity can not possibly change the pressure on the surface. In this respect, Bernoulli's law is not a valid explanation.
The reason for the pressure difference is that the underside of the airfoil is facing largely into the airstream (increasing the pressure there) whereas the topside is largely in the shadow of the airstream (reducing the pressure there). This is due to a combination of the shape (camber) of the airfoil and the fact that the wing has a certain inclination in flight (also called angle of attack ).
The fact that the air flows faster above the airfoil is just a side effect caused by the pressure difference between the front and the back due to the effects mentioned above, in combination with the viscosity of the air. It is not the cause of the pressure difference (as Bernoulli's law suggests), but rather the other way around.
A: $\def\¿{\small}$Yes, Bernoulli's theorem can be applied only along one streamline. Therefore the following arises the common misunderstanding that the theorem is applied considering points C and D, those are not along the same streamline.

$$\¿ P_{\¿C}+\frac12\rho v_{\¿ C}^2=P_{\¿ D}+\frac12\rho v_{\¿ D}^2$$(neglecting the term $\¿ \rho gh$)
But technically we are not applying Bernoulli's equation directly into different streamlines. Consider the figure below.

We apply the equation to streamlines AC and BD seperately. This gives,
$$\¿ P_{\¿A}+\frac12\rho v_{\¿ A}^2=P_{\¿ C}+\frac12\rho v_{\¿ C}^2\tag{1}$$
$$\¿ P_{\¿B}+\frac12\rho v_{\¿ B}^2=P_{\¿ D}+\frac12\rho v_{\¿ D}^2\tag{2}$$
When we define points A and B to be far away from the airfoil, we can assume that $\¿ v_{\¿A}=v_{\¿B}$ and $\¿ P_{\¿A}=P_{\¿B}$. So the equations (1) and (2) give the initial equation.
Thus you can see Bernoulli's theorem is still valid along the same streamline.
 The above is a basic description about Bernoulli's theorem and airfoil. But airfoil's physics is more complicated. See this for more info. 
A: An intuitive way to imagine what's happening is to remember pressure is actually the result of fluid (e.g. air) molecules bouncing randomly around and hitting a surface.
Imagine on the side with slower air movement, the molecules are bouncing off the airfoil surface on a certain angle on average (they're going in random directions, but consider the average). On the fast-moving side the air molecules will bounce similarly but on average on a shallower angle because they're moving faster in the parallel direction to the surface. That means they bounce with less perpendicular force.
In the extreme of a very very very fast air movement parallel to the surface, you can imagine most molecules just flying by without having time to hit the surface, and those that do simply graze the surface and bounce off at a very shallow angle. There would be very little perpendicular force from the bouncing molecules.
A: It really doesn't, or at least it is widely misunderstood, to the point where I would almost call it a misconception to say a wing flies because of Bernoulli.
It's true that Bernoulli's principle applies to fluid streamlines above and below the wing, as it does to any incompressible flow.  But the main reason lift occurs is much simpler: an angled surface (red) moves forward and pushes incoming the air down, which by Newton's 3rd Law pushes the wing up.  That's it.
The air changes direction (blue vectors).  If you subtract the blue vectors (and then take the negative) you get the green vector, which is imparted to the wing.
This video explains it better than I could in a post https://youtu.be/aFO4PBolwFg

