I think I got it. A real puzzler.
First, arrange the flaps around the cowl such that the xyz axes are oriented uniformly so that a rotation in one axis will move the flap in the correct direction. Move the Hot Points for all of the flaps to the leading edge of the flaps. This can be done all at once by selecting all of the flaps at the same time and while the Caps Lock key is active, move the Hot Points.
Use the Universal Manipulator to view the axes relative to the flap, rather than relative to the workspace. (See image)
Select each flap and apply a “Ball Joint” Constraint. In the case of this model the rotation of each flap occurs around the y axis, so the constraints for each axis was x-Locked, y-Limited, z-Locked. The “Rotation Order” needs to be changed to YZX before it will work correctly.
(FULL DISCLOSURE: I have little knowledge about what “Rotation Order” does. I found out by trial and error that I needed this setting to make all of the flaps rotate the correct amount in the correct direction.)
Select one flap (I used the top one) and give it a unique name, such as “control flap”.
Select each of the other flaps in turn and apply a “Track” Modifier. In the modifier properties check “Track Rotation” and uncheck X, Y, and Z. In the “Towards” parameter choose the flap that you renamed “control flap”.
After the “Track” Modifier has been applied to all of the flaps, rotating the “control flap” around the Y axis should cause all of the other flaps to follow suit.
EDIT: For some reason my animated GIF isn’t displaying properly when posted here, but you can still kind of see what’s happening. I’ll fix it if I can.
EDIT2: If you click on the animation it will open in another window where it will display properly.
Click thumbnail to see full-size image