Would the Casimir effect make this perpetually rotating engine possible? If I understand the Casimir effect right, two microscopic plates approach each other in a vacuum because the space between them is limited allowing only shorter wavelengths to arise from quantum fluctuations.
As this is due to the space between two plates, I imagine that the same effect could affect a "V" shaped structure, as the space inside the structure is less than the space available outside. So the structure would move towards the opening of the "V".

If this proves to be true, we could make a perpetual rotation engine using two of these "V"s:

Can this simplistic idea really work?
 A: The Casimir force is just an electromagnetic force between charged particles in the conducting plates. When the plates are parallel planes, there is a funny trick to calculate it by attributing it to vacuum fluctuations. The result is less accurate than a correct calculation, and it doesn't work with other plate geometries. It isn't even how Casimir originally derived the force. See The Casimir Effect and the Quantum Vacuum by R. L. Jaffe for more.
It's a conservative force, so you can't build a reactionless drive or perpetual-motion machine out of it. You can imagine that every configuration of your system is associated with a potential energy, and the force tries to minimize the potential. The potential will be lower if the vertex angle of the V is smaller, so there is an inward force on the plates in each V (exerting no net torque), but translational and rotational symmetry imply that the potential is not any different when the V is in a different place, so there's no rightward force in your first diagram or torque in your second diagram.
A: 
From here
Just a technicality. As gap is increased x3, force is increased x100. So at least you need to make sure there is no difference greater than x3 or so in your gap.
Also someone already invented this idea somewhere around 2012, here
And someone else filed a patent in 2005, here
So this idea gets reinvented every 10 years or so.
Its hard to test it with casimir effect. Lets consider a version with magnets. If magnets are nearby and at an angle, like V, and attracting they should also move in the same direction, towards the V opening. Like in your example. This is how half of the perpetuum mobile devices with magnets are made.
If we try to 'simulate' it by taking several points, it becomes clear. For simplicity take V with sharp corner.
Lets take 4 points, 2 in V corner, and 2 in V edges. Forces edge-to-corner are within the plane, and act against the material. Forces corner-to-corner act symmetrically, and arent interesting.
Or 6 points, 2 in corners, 2 in the middle of the line, 2 in the edges. Corner-middle are within the line, acts against the material, so are corner-edge. Middle-middle and corner-corner are not interesting. Middle - corner do have force vectors at an angle, so probably it will work. Middle1 -> corner2 we can split into forces within the line thatbare cancelled out by the material, and perpendicular to it. As we consider one side, there is a side force. But it is cancelled exactly as we consider corner2 -> middle1 force. This is similar how it would react if you would connect a tensioned spring acros an angle, from a middle of one side to the corner of another side.
I've invented the magnetic version with angled magnets too when I was younger. And even built it. It didnt work, sadly.
All of these perpetuum mobile machines can be analyzed this way, by calculating forces betwen points. All of the confusion comes when people try to generalize the force and give it a direction, to the larger body than a point.
But I think it is safe to say that ibteraction happens between point-like objects, or spheres. And calculating forces through them helps in such cases. Even if there is just 4-6 points, it can show the situation differently.
