Reply to Brian Abraham
I agree that tackling a practical problem with a mathematical approach can be challenging. The following might clarify things.
We want to determine the effect on aircraft lift coefficient when trailing edge flaps are moved, but nothing else changes except those things that have to change such as angle of attack and engine thrust. So we imagine this thought experiment. An aircraft is flying straight and level at 150 knots IAS with flaps retracted. Aircraft weight is 10,000 pounds and wing area is 250 square feet. To find aircraft lift coefficient we divide 10,000 by half the standard density of air and the square of 150 and the wing area, 250. (The result is 0.525)
Next we imagine plain flaps have been extended but the aircraft is still flying straight and level at 150 knots. To find the aircraft lift coefficient we divide 10,000 by half the standard density of air and the square of 150 and the wing area, 250. (Again, the result is 0.525 so we conclude that with plain flaps, changing flap setting has no effect on aircraft lift coefficient.)
I agree that in real life, as the flaps are running angle of attack will change and there will be a departure from level flight, the pilot will have to increase thrust to maintain 150, and he will have to re-trim. But most importantly, the above calculations show that after all those transient effects have died away and the aircraft has returned to level flight at 150 knots, the aircraft lift coefficient will be the same as it was prior to initiating the change in flap setting.
Changing flap setting might change induced drag and the strength of trailing vortices but that won’t be because of any significant change in aircraft lift coefficient because we have convinced ourselves that aircraft lift coefficient is independent, or almost independent, of flap setting. We need to look elsewhere to find an explanation for the change in induced drag.
Your question about stalling angles of swept-wing versus straight-wing aircraft is a good one. I will put my thoughts in a separate post on this thread.