Circulation Definition: Understanding Positive, Null, and Negative Circulation

[Jhenrique]

1st which is the math definition for circulation ($\Gamma = \int_s \vec{f}\cdot d\vec{s}$)? And 2nd, what means positive, null and negative circulation?

 

[HallsofIvy]

The "math definition" for circulation is exactly what you give: \int \vec{f}\cdot d\vec{s}.

I think you mean a geometric or physical "interpretation" of the circulation. In that case, "circulation" of a vector function is the total of the components of \vec{f} that are tangent to the circle over which you are integrating. Physically, it is the total motion around the axis (the line through the center of the circle and perpendicular to it).

"positive circulation" means the circulation is in the counterclockwise direction, "negative circulation" is in the clockwise direction, and "null circulation" means there no net rotational motion.

 

[Jhenrique]

Well, I can to compute the flux through of a closed or open surface, so, can I compute the circulation through of a closed or open curve too?

 

[Matterwave]

You can certainly compute the line integral of a vector field over a closed or open curve...but in the case of an open curve, I don't think you'd call it "circulation" anymore.

 

[chogg]

Physically, it is the total motion around the axis (the line through the center of the circle and perpendicular to it).

What circle?

Doesn't the expression make sense for arbitrary paths? Or, for that matter, arbitrary dimensions, so that the concept of "axis" won't necessarily make sense?

 

[Matterwave]

What circle?

Doesn't the expression make sense for arbitrary paths? Or, for that matter, arbitrary dimensions, so that the concept of "axis" won't necessarily make sense?

I believe they are talking about the circulation of a fluid field, defined as the line integral of the velocity vector field around an (arbitrary) closed circle.

See: http://en.wikipedia.org/wiki/Circulation_(fluid_dynamics)

 

[chogg]

Maybe I'm being too pedantic. By "circle", you just mean an arbitrary closed path, right?

If so, I do not see what value is added by introducing the concept of an "axis" in this situation.

Also in the "chogg being too pedantic" category:

"positive circulation" means the circulation is in the counterclockwise direction, "negative circulation" is in the clockwise direction, and "null circulation" means there no net rotational motion.

Wouldn't this be exactly backwards if the path is clockwise? Doesn't "positive" simply mean the net rotation is along the path, whichever direction the path takes?

Seems to me like it's more natural to express it relative to the path, instead of having to define what one means by (counter-)clockwise.