Jurgen M said:
Where is this function here?
You are the one asking the question. You tell us.
Watching the video...
We have a hand-drawn airfoil (or cross-section of an airfoil) which is claimed to be symmetric.
The author lays out coordinate axes with x going left to right from leading edge to trailing edge on the horizontal and y vertically at right angles.
The origin appears to be at the midpoint on the leading edge. A chord is drawn clumsily -- visibly curved so that it arches toward the top surface. The author indicates that this is not intentional.
The author draws a vector indicating the flow direction of the free stream. The free stream is angled upward (the air foil has a positive angle of attack). But, as above, we are using coordinates anchored to the air foil so it is at least notionally the free stream that is angled upward and the airfoil that is horizontally aligned.
The author proceeds to parameterize the upper surface so that ##S_u## denotes the path length along the airfoil surface from the leading edge to a particular point on the top surface. Similarly, ##S_l## denotes the path length from the leading edge to a particular point on the bottom surface.
One assumes that this is setting up so that ##S_u## and ##S_l## will be the variables to integrate over.
The author proceeds to talk about pressure as a function of ##S_u## and about the angle between a normal to the surface and the vertical also expressed as a function of ##S_u##. So we have ##P(S_u)## and ##\theta(S_u)##.
The force element on an incremental surface element is now simple to state, though the author has not done so yet. Yes, he is definitely setting up for an integral.
[At this point we are 3:02 into the video]
The author goes on to define a shear force (or shear "distribution") which is tangent to the surface. This is obviously at angle ##\theta(S_u)## with respect to the horizontal. But you do not care about shear, so neither will I.
The author repeats, talking about a point on the lower surface with pressure and shear as functions of ##S_l## in the obvious manner. No surprises there. Just repetition to make sure it sinks in.
[This takes us to 4:29 in the video]
At this point is is blindingly obvious that we can integrate over the upper surface to get the vertical component of the pressure force and also [though perhaps less significantly] the vertical component of the shear force. Similarly, we can integrate over the lower surface to get the vertical component of the pressure force and of the shear force.
The [vector] sum will give us lift.
It is also blindingly obvious that we can do essentially the same integrations to get the net horizontal components and arrive at a vector sum for drag. Just need to swap sines for cosines and vice versa.
The required inputs to this process are a pressure field parameterized in terms of path length along the surfaces and a shear field parameterized the same way.
Without watching the rest of the video, it is not clear whether the author will attempt to derive either input from the flow field. [Skipping forward, it looks like he spends the rest of the video talking about the sines and the cosines and making sure the sign conventions are right]. He's just setting up the integrals in the obvious way.
