Zero curl but nonzero circulation

In summary, the vector field \vec{F} = <\frac{-y}{x^2 + y^2},\frac{x}{x^2 + y^2},0> has a zero curl, but the circulation around a unit circle on the xy plane is not zero, instead it is equal to (+/-)2\pi. This is because the domain of \vec{F} is not simply connected, as it is undefined at (0,0,z), and therefore the surface used in Stoke's theorem does not exist.
  • #1
MHD93
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The vector field [itex]\vec{F} = <\frac{-y}{x^2 + y^2},\frac{x}{x^2 + y^2},0>[/itex] has a zero curl, which means its circulation is zero. However

[itex]\int \vec{F}.d\vec{s}[/itex] around a unit circle on the xy plane is equal to [itex](+/-)2\pi[/itex] and not zero

Is it because F is undefined at (0,0)? No, because Stoke's theorem allows me to choose an arbitrary surface not including the origin(0, 0, 0)?
 
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  • #2
Is it because F is undefined at (0,0)?
Right - more general, it is undefined at (0,0,z). As result, the domain of F is not simply connected - and the surface you want to consider does not exist.
 

What is "zero curl but nonzero circulation"?

"Zero curl but nonzero circulation" refers to a situation in fluid dynamics where the curl of the velocity field is equal to zero, but the circulation (the integral of the velocity around a closed path) is not equal to zero. This means that the flow is irrotational, but still has a non-zero swirl or rotation around certain paths.

How is "zero curl but nonzero circulation" different from other flow patterns?

"Zero curl but nonzero circulation" is different from other flow patterns because it is a combination of irrotational and rotational flow. In most cases, a fluid flow is either completely irrotational or completely rotational, but this situation has elements of both.

What are some real-world examples of "zero curl but nonzero circulation"?

A common example of "zero curl but nonzero circulation" is the flow of air around a spinning ball or cylinder. The flow around the sides of the object is rotational, while the flow around the top and bottom is irrotational. Another example is the flow of water around a boat propeller, where the flow at the tips of the blades is rotational, but the overall flow around the propeller is irrotational.

What are the implications of "zero curl but nonzero circulation" in fluid dynamics?

The presence of "zero curl but nonzero circulation" in a fluid flow can have important implications for the behavior and properties of the flow. For example, it can affect the pressure distribution, drag, and lift on objects in the flow. It can also impact the mixing and diffusion of substances within the flow.

How is "zero curl but nonzero circulation" related to the concept of vorticity?

Vorticity is a measure of the rotation or spin of a fluid element. In "zero curl but nonzero circulation" flow, there is a non-zero vorticity and therefore a non-zero rotation at certain points, but overall the flow is considered irrotational. Vorticity is closely related to circulation, which is why the presence of nonzero circulation in this flow pattern is significant.

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