I've been struggling a bit to find a satisfying explanation for why the component of net fluid force along a 2D wing's chord cannot cause a torque. At the moment (no pun intended), I'm only considering a symmetrical wing, not an airfoil. Say, a rectangle.
I have two explanations:
The cross-product formula gives zero torque, since the component of net fluid force along the wing's chord is parallel to the moment arm, and so the sine of the angle between them is zero.
The component of net fluid force along the wing's chord cannot generate a torque; it can only translate the wing, with a push or a pull.
The component of net fluid force along the wing's chord cannot generate a torque because such a force is generated from fluid pressure around the wing that cancels out, and so only the perpendicular component of net fluid force is associated with a pressure imbalance around the wing, which can cause a torque, namely if the aerodynamic center is not at the center of mass. I feel this is not quite right ...
The above explanations are unsatisfactory to me, for some reason.
What point am I missing that could help me feel more confident about the idea? Preferably, something that's physically intuitive.
Should I instead not overthink it, and simply treat the cross-product formula as a definition of the torque force?