Rotational flow and irrotational flow

Question

I can understand the difference between the definition of rotational and irrotational flow if the flow is in a straight pipe. But in case of circular flow, that confuses me.

Let's consider a flow that is rotating about the origin point $O$.

In both figures, the fluid particle is shown by green color with some axes attached to it.

In the left, the particle moves with flow but it seems that it doesn't rotate about its own axes. In the right figure, the particle is rotating but its axes are always aligned with the polar axes (normal and tangential axes).

Is it true to say that the flow in the left figure is irrotational and the flow in the right is rotational?

Answer 1

The difference between a rotational and an irrotational fluid is that if you place a (infinitesimally) small wheel in the fluids, then the wheel will start to rotate in the rotational fluid, while it won't in the irrotational fluid. What this means is that in the rotational fluids, there is a net "push" by the fluid on the wheel. Therefore, even if you have a fluid where all the particles flow horizontally to the right, then it might still be a rotational fluid,so long as the particles flow faster the higher up you go. That's because a small wheel will feel a bigger push on the top than on the bottom, meaning a net push on the wheel, causing it to rotate clockwise.

Using this idea, it is pretty straightforward to see that the fluid is rotational in both pictures (the further you go from O, the faster the fluid flows, leading to a net push on a wheel for the same reason as the example above).

Technically, a fluid is irrotational if the curl of the velocity vector field is 0.

Answer 2

The answer given by @User3141 is simply wrong. In an irrotational flow the fluid particles DO NOT rotate about their own axes (in this case, the flow in the left is irrotational). It is clear that in the right image, the particles rotate about their own axes as they move, so the flow is rotational.