Physical meaning of curl

[sadegh4137]

hi

whats's the physical meaning of curl?

and why it is a vector?
it's definition is line integral per volume. i can't understand why this is a vector.

 

[HallsofIvy]

First, my usual proviso- there is no one "physical" meaning to a mathematical concept. But there are specific applications and perhaps a "most important" or "most common" application.

In particular, we can apply the curl of a vector to fluid motion- if $v(x, y, z)$ is the velocity vector of water, say, so that v is depends on the position but not time, the $curl v= \nabla\times v$ describes the "rotation" of the fluid. It is a vector because its direction shows the axis about which the fluid rotates while its length is the speed of rotation.

 

[arildno]

If you place tiny paddle wheels in a moving fluid, the rate by which the paddle wheel rotates about its own axis (perpendicular to the wheel's plane) is roughly equal to the local curl there.

 

[K41]

You can work out the rotational velocity of a fluid element inside a fluid by adding a rotational displacement to it and differentiating this displacement. Following through the derivation, you find the angular velocity. Now it just so happens that a large part of the final equation is in the form of curl v where v is a vector field. There is a constant which we take out and therefore we find that twice the angular velocity = curl v for a fluid based on the continuum approximation.

We refer to to twice the angular velocity as "vorticity" and a fluid element which has zero vorticity is said to be irrotational. This means, in very basic terms that if we have a body axis (on the plane of the page) fixed on the fluid element, then the axis does not rotate (about an axis perpendicular to the page) relative to a global reference system.

Note, we can still allow for some distortion in the shape but this is a little harder to explain without diagrams (but its essentially related to having components of vorticity whereby you they still have a finite derivative but the two derivatives cancel when finding the curl since they are the same - this means we have a regular change in shape but zero total angular velocity here).

Regarding curl itself applied to fluids, the components of curl are actually the direction of the axis about which the rotation component is occurring.

 

[Edgardo]

Here's an intuitive explanation of the curl: http://youtu.be/Mt4dpGFVsYc