'Eyeballing' non-zero divergence and curl from vector field diagrams

[mindarson]

Homework Statement

Explain whether the divergence and curl of each of the vector fields
shown below are zero throught the entire region shown. Justify your answer.https://sphotos-a-ord.xx.fbcdn.net/hphotos-prn2/1185774_4956047513788_517908639_n.jpg

Homework Equations

N/A

The Attempt at a Solution

Ok, here's my attempt at answering for each field pictured.

A: Curl: The curl is not zero everywhere. Using the 'tiny paddlewheel test', there are points in the field (ie in the right half of the region shown) where the torque from the fluid on the left side (clockwise) is greater than the torque from the other side (counterclockwise), which would spin the paddlewheel, giving a non-zero curl. Divergence: It is zero everywhere. The reason is that there is no point at which there is a net inflow or net outflow of the field vectors, no point they 'spread away from' or 'converge upon.'

B: Curl: It is not zero everywhere in this region. Since the field vectors are longer (greater magnitude) nearer the 'center of the vortex' (not pictured), the torques on the near and far sides (relative to the center) are unequal, resulting in rotation of the tiny paddlewheel, i.e. nonzero curl. Divergence: The divergence is zero everywhere, since there are no 'sources' or 'sinks' of the field.

C: Curl: Zero everywhere, since there is no point that would cause a paddlewheel to rotate. Divergence: Not zero everywhere. In fact, the divergence is positive everywhere, since at any point the net outflow of the field exceeds the net inflow (the vectors get larger as we go farther along in the direction of the flow).

D: Curl: Zero everywhere. At every point, the clockwise torque equals the counterclockwise torque on the imaginary tiny paddlewheel, so it wouldn't rotate, therefore no curl (note this would not be the case at the unpictured center, where the wheel would rotate). Divergence: Zero everywhere, since for every point, each component of inward flow is canceled by an equal component of outward flow.

E: Curl: Not zero everywhere. A paddlewheel inserted into the center would spin clockwise. Divergence: Not zero everywhere, since at a point in the bottom left-hand corner, there is net inflow, giving negative divergence.

F: Curl: The curl is zero everywhere, since every clockwise torque on a paddlewheel would be canceled by a counterclockwise torque. This is easy to see in the center, but it appears to hold for every point in the region. Divergence: Not zero everywhere, since it appears to be negative at the center, since net inflow to a point located there exceeds net outflow.

Alright, that's my best shot. I'm not entirely confident in some of these, particularly the divergences. Can anyone spot any mistakes in my answers and, more importantly, in my way of thinking about this? I don't want to rely too much on the paddlewheel or 'source/sink' analogy, but since these are just pictures without numbers or anything, I feel like it gets me pretty far.

Thanks!

 

[TSny]

These can be tricky! For the curls I would not rely solely on the paddle wheel picture. It's also a good idea to imagine some little closed loops and see if it looks like the line integral of the field around the loop is zero.

For a couple of the cases where you said the curl is zero, I think you can argue that the curl is nonzero.

For a couple of the cases where you said the curl is nonzero, I think it is possible that the curl could in fact be zero. In these cases you can't say for certain that the curl is zero without knowing more precisely how the field varies.

For the divergences, I think there is one case where you said the divergence is nonzero but it is possible that it is zero. Again, you can't say for sure for this case without having more information.

For now, I will let you think about them some more to see if you want to change any of your answers.

 

[mindarson]

It does seem that without more precise quantitative information about the fields, there is always [EDIT: not always, but in some cases] room for 'controversy' about the divergence and curl. Eyeballing only goes so far.

My problem right now is with B and D. (They appear to be subtly different, because in B the vectors get bigger away from the 'center'). Just looking at the region, I at first wanted to say that there is a curl in the region because the whole thing just seems to 'rotate'; so if you were to toss a rubber duck into the field (interpreted as velocities of water), the duck would then describe the arc of a circle.

But curl is not really a macroscopic quantity, is it? That's why I second-guessed and tried to use the paddle-wheel idea. The loop idea is a good one that I did not use, but it seems to make sense in light of Stokes' theorem.


Also reminds me of Feynman's heuristic description of circulation. Something like imagining you had a cylindrical loop in the midst of fluid flow, and if you suddenly freeze all the fluid except what is in the cylindrical hose. So I will reexamine my answers in light of the loop idea.

 

[mindarson]

For example, looking at D, it now looks like there would be a circulation, because if I were to draw a loop and then integrate the tangential component of the vector field all around the loop, I would get a nonzero number. Since there is a circulation, by Stokes' theorem, there would be a curl.

Then again, since all the vectors are the same size, maybe every tangential component of the field would be canceled by another going in the opposite direction when I loop around? I think that's kind of what I was thinking in my first argument.

Whereas in the case of B, the vectors (and thus the vectors tangent to a loop) are bigger in some places than in others, i.e. different size going in one direction than in the other direction, giving a net circulation and therefore, by Stokes' theorem, a nonzero curl.

?

 

[TSny]

For B and D consider integrating around loops made of 2 radial lines perpendicular to the field and 2 arcs tangent to the field as shown.

 

[mindarson]

Alright, here's what I'm thinking for B. The circulations along the sides perpendicular to the field direction are zero, because of the perpendicularity. But the circulation along one of the tangential sides is greater than along the other (the one farther from the center being greater) due to the fact that the circulation is an integral over a greater distance.

This would leave a non-zero circulation around the closed loop and therefore a nonzero curl.

Is this correct?

EDIT: Actually, this reasoning applies better to D, since the vectors are all the same magnitude. For B, the vectors farther out (the ones integrated over the longer distance) are actually smaller. Maybe this would mean the circulation is in fact zero over the loop you've drawn in B? Since although we integrate over longer distance, we're integrating smaller vectors.

EDIT: But I suppose, for B, you could only know for sure if you knew the field decreased as 1/r. Then the vectors would shrink by the same factor as the angular path I'm integrating over grew, thus canceling out.

 

[TSny]

Alright, here's what I'm thinking for B. The circulations along the sides perpendicular to the field direction are zero, because of the perpendicularity. But the circulation along one of the tangential sides is greater than along the other (the one farther from the center being greater) due to the fact that the circulation is an integral over a greater distance.

This would leave a non-zero circulation around the closed loop and therefore a nonzero curl.

Is this correct?

EDIT: Actually, this reasoning applies better to D, since the vectors are all the same magnitude. For B, the vectors farther out (the ones integrated over the longer distance) are actually smaller. Maybe this would mean the circulation is in fact zero over the loop you've drawn in B? Since although we integrate over longer distance, we're integrating smaller vectors.

EDIT: But I suppose, for B, you could only know for sure if you knew the field decreased as 1/r. Then the vectors would shrink by the same factor as the angular path I'm integrating over grew, thus canceling out.

Yes! Everything you said here is spot-on. The curl for D is nonzero and the curl for B is zero if the field decreases as 1/r.