- #1

MichielM

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## Homework Statement

Consider a source/sink in the origin of a coordinate system with a strength equal to 2m and two sources/sinks, each with a strength of −m, i.e. with a opposite sign with respect to the source/sink in the origin. The two sources/sinks are positioned along the x-axis at x = δx and x = −δx, respectively. Consider the limit for δx → 0 under the condition that

[tex]\lim_{\delta x \rightarrow 0} \delta x^2 m = \mu_2[/tex]

with [tex]\mu_2[/tex] finite. Consider this problem in three dimensions and sketch the resulting streamline patterns. Is there a closed streamline, which does not pass through the origin.

## Homework Equations

The potential [tex]\Phi_e[/tex] for a single source/sink of strength m in 3D is:

[tex]\Phi_e=-\frac{m}{4\pi r}[/tex]

[tex]\bar{u}_e=\nabla\Phi_e=-\frac{m \bar{r}}{4\pi r^3}[/tex]

with [tex]\bar{r}=\bar{x}-\bar{x}_0 [/tex] where x = (x, y, z) and [tex] r = |\bar{x}-\bar{x_0}| = \sqrt{x^2 + y^2 + z^2}[/tex]

## The Attempt at a Solution

Since the poisson equation which leads to the potential solution above is linear, I can add the 2 sinks and 1 source together to get 1 potential:

[tex]\Phi_e=-\frac{m}{4 \pi}\left(\frac{-1}{|\bar{x}-\delta\bar{x}|}+\frac{2}{|\bar{x}|}+\frac{-1}{|\bar{x}+\delta\bar{x}|}\right)[/tex]

I have to take the limit to 0 for δx and I want to introduce [tex]\mu_2[/tex] so I take:

[tex]\Phi_e=\lim_{\delta x \rightarrow 0} -\frac{m \delta x^2}{4 \pi}\frac{1}{\delta x^2}\left(\frac{-1}{|\bar{x}-\delta\bar{x}|}+\frac{2}{|\bar{x}|}+\frac{-1}{|\bar{x}+\delta\bar{x}|}\right)[/tex]

*From this point on I'm highly unsure about my method.*

Taking the limit it follows that [tex]\lim_{\delta x \rightarrow 0} \delta x^2 m = \mu_2[/tex] and [tex]\frac{1}{\delta x^2}\left(\frac{-1}{|\bar{x}-\delta\bar{x}|}+\frac{2}{|\bar{x}|}+\frac{-1}{|\bar{x}+\delta\bar{x}|}\right)=\frac{\partial}{\partial x^2}\left(\frac{3}{|\bar{x}|}\right)[/tex]

So my potential becomes:

[tex]\Phi_e=\frac{\mu_2}{4\pi}\frac{\partial}{\partial x^2}\left(\frac{3}{|\bar{x}|}\right)[/tex]

From here I could proceed to calculate the velocity, but honestly I don't trust my answer enough to take the effort so I first wanted to ask whether anybody could run through my calculation and comment on any mistakes/errors!

Thanks in advance!

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