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Below is an image for a Fowler Flap: enter image description here

Notice that as the flap is deployed, both the camber and the area increase. This change in wing shape modifies the lift curve as follows:

enter image description here

(source: Development of a Fowler Flap System for High Performance General Aviation - Wentz (NASA))

This behavior can be isolated and viewed in the simpler figure below: enter image description here

Note that the lift curve slope has increased.

Here is my analysis: If Fowler Flap is deployed, the planform area increases. If the planform area increases, the aspect ratio decreases (since wing span is unchanged). If AR decreases, then the lift curve slope should decrease (see How does the aspect ratio of a wing impact its lift? for details).

So where in my analysis do I go wrong? I know the lift curve slope should increase, then why does my analysis indicate otherwise?

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Note that your plots show lift coefficient over angle of attack. This lift coefficient is referenced to a reference area which must be defined somehow. It is customary to use the projection of the clean wing area in the x-y plane for all flap settings in order to keep coefficients comparable. The lift curve slope increases because the real wing area increases while the reference area remains unchanged. The dashed line in the lower plot shows the isolated effect of camber increase while the upper solid line combines both effects (area and camber increase).

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