Curl Vector: What exactly is rotating?

Question

I am a little bit confused over the exact conceptual meaning of the curl vector. So I am familiar with the paddle wheel interpretation, but I don't think I am satisfied with that analogy because it could give different meanings. There are two ways which I am imagining the curl. To make this explanation easier, I will talk about a fluid on a two dimensional plane where the vector field is the velocity vector field given by F= M(x,y)i+ N(x,y)j, as usual.

So when we talk about some fluid, say water, and we say there is a curl vector field on the plane, are we saying that it is the water molecules themselves which are rotating about their axis due to this field? (suppose that these water molecules are perfect spheres and it's axis is parallel to the z-axis). In other words, is it the water molecule itself that is located at (x_o,y_o) rotating about it's axis which is parallel to the z-axis?

Or, are we saying that these water molecules (perhaps some bunch) are "rotating" in the sense that they follow a circular path which is centered at (x_o, y_o) and so it is not actually the molecules itself rotating about their own axis.

I am thinking it is the first one. Thanks in advance!

P.S. if anyone would like the edit my question using Latex, please do so! Thank you.

Answer 1

In the water surface analogy (the vector field representing local velocity), think of placing a little paper ship on it. In equilibrum, the ship will travel with the same speed as the water, so the vector field also describes the ships velocity. But the ship will also start rotating because even seen from the ship, the water is not completely still: It may flow backwards on starboard and forward on port side, thus trying to rotate the ship.

Answer 2

In physics ${\rm curl}$ is usually applied to force fields, not to fluid fields. It measures the "nonconservativity" of such a field.

But we can apply ${\rm curl}$ to a velocity (or fluid) field ${\bf v}=(P,Q)$ as well and try to understand what it means. The field $${\bf v}:=\left({-y\over x^2+y^2}, \ {x\over x^2+y^2}\right)$$ describes the rotation of a taifun around its center. Drawing the little arrows you get instantly the feeling of it. Nevertheless for this field $${\rm curl}\>{\bf v}= 0$$ at all points where it is defined, i.e., in the full punctured plane. The reason for this is that ${\rm curl}$ measures a purely local property of a field, and this is another matter. In particular ${\rm curl}$ deals with the subtle changes of the field when we move from a given point ${\bf z}_0$ to a very ("infinitesimally") nearby point ${\bf z}_0+{\bf Z}$, $\>|{\bf Z}|\ll1$. In order to make these changes of the field better visible I have subtracted in the following figure the constant field ${\bf v}({\bf z}_0)$ (whose ${\rm curl}$ is zero anyway) from ${\bf v}$. Now you can see that in some cases there is a hint of rotation of the "field increment" $$\Delta{\bf v}({\bf Z}):={\bf v}({\bf z}_0+{\bf Z})-{\bf v}({\bf z}_0)$$ around ${\bf z}_0$, counterclockwise or clockwise, and in other cases the field changes in the various directions of the increment vector ${\bf Z}$ do not generate a rotational effect.