Imagine we have a plane that is already moving at a speed $v_{plane}$. At a certain time $t=0$, a motor starts moving a propeller whose blades consist on symmetrical airfoils with $90^\circ$ of pitching (that is, their chords are perfectly parallel to the fuselage of the plane). Under which conditions (if any) would this propeller produce thrust?
Intuitively, one may think that, with $90^\circ$ of pitching, this will just be impossible. Note however that we start from the condition that the airplane is already moving.
Imagine that the rotating speed of the propeller was $\omega_{prop}$. If we analysed the forces generated by a blade in a cross section at radius $R$, they should look like the following (blue for the velocity of air and red for forces produced by the airfoil):
Since the plane is moving and the propeller is rotating, from the perspective of the blade the incoming air will have a nonzero apparent angle of attack. This will produce lift and, if $\frac{c_L}{c_D}$ is sufficiently large, the direction of the resulting force vector ($F_{TOTAL}$ in the diagram) will point slightly to the direction of movement, hence generating thrust.
Is this correct? If so, would a propeller with long blades spinning sufficiently fast generate thrust under these constraints?