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March 22nd, 2016, 07:43 AM   #1
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Divergence & flow integral

Hi! I got this task:

S = the mantle area of the cylinder.
F = the vector field.

$\displaystyle x^2+y^2=4 $

$\displaystyle -2<= z <=2$

$\displaystyle F(x,y,z) = (x, 2y, 1-3z)$

As div(F) = 0,
shouldn't the flow through the mantle area S also 0 ?

Because in one solution I got, which I don't really believe in, the take:
flow(total cylinder) - flow(top part cylinder) - flow(bottom part cylinder) = flow(mantle cylinder)

I get why they subtract like that, but shouldnt it be zero as div(F) = 0 ?

Last edited by 9robban6; March 22nd, 2016 at 07:47 AM.
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March 26th, 2016, 07:53 PM   #2
jks
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Hi 9robban6, and welcome to the forum.

Quote:
As div(F) = 0,
shouldn't the flow through the mantle area S also 0 ?
No, I don't think so. I think that div(F)=0 means that the flow through the whole cylinder is zero (this follows from the Divergence Theorem).

Quote:
Because in one solution I got, which I don't really believe in, the take:
flow(total cylinder) - flow(top part cylinder) - flow(bottom part cylinder) = flow(mantle cylinder)
Let's calculate each term.

-----

First, we know that the total flow through the cylinder is 0 as mentioned above.

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Next, for the flow through the top of the cylinder, the x and the y components of F do not play a role since they are in a direction perpendicular to the normal of the surface. The z component is obviously parallel to the normal so in the dot product, cos(angle) = 1 (you probably know all of this but I thought I would state it to be sure).

So on the top of the cylinder z=2 (constant) and F(z)=-5. The area is $\pi \left(2^2 \right) \ = \ 4\pi$. The normal is in the +z direction so the flow is:

$\displaystyle -20 \pi$

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For the bottom part of the cylinder, again the x and y components do not play a role. Here, z=-2 (constant), so F(z)=7. The area is $4\pi$ and the normal is in the -z direction. So the flow is:

$\displaystyle -28\pi$

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For the mantle of the cylinder, the z component of F does not play a role. Also, I found it easier to convert to cylindrical components using $\rho$ and $\phi$ (at least I think the following is correct; someone may correct me if my procedure is wrong).

$\text{x} \ = \ \rho \cos(\phi)$ and $\text{y} \ = \ \rho \sin(\phi)$, and the normal is in the $+\phi$ direction, so the angle between the x component and the normal is $\cos(\phi)$ and the angle between the y component and the normal is $\sin(\phi)$. With $A$ as the area this gives the flow as:

$\displaystyle \text{Flow(x)}=\rho \cos^2(\phi) \ dA \qquad \text{and} \qquad \text{Flow(y)}=2\rho \sin^2(\phi) \ dA$

$\rho \ = \ 2$ (constant), so the flow is given by:

$\displaystyle \text{Flow} \ = \int_0^{2\pi}2\cos^2(\phi) \ dA + \int_0^{2\pi}4\sin^2(\phi) \ dA$ where

$\displaystyle dA \ = \Delta z \ \rho \ d \phi = 8 d \phi$ giving

$\displaystyle \text{Flow} \ = \int_0^{2\pi}16\cos^2(\phi) \ d \phi + \int_0^{2\pi}32\sin^2(\phi) \ d \phi$

Since

$\displaystyle \int_0^{2\pi} \cos^2(\phi) \ d \phi \ = \ \int_0^{2\pi} \sin^2(\phi) \ d \phi \ = \ \pi$ the flow is

$\displaystyle \text{Flow} \ = \ +48 \pi$

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So the flow equation becomes:

$\displaystyle 0-(-20 \pi)-(-28 \pi) \ = \ 48 \pi$ which holds per the Divergence Theorem.

In the solution that you referenced, was the flow $48 \pi$ for the mantle area?
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