- #1

squire636

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## Homework Statement

a. Determine the complex potential for two equal counter-rotating vortices with strength [itex]\Gamma[/itex], the positive one located at z=-a and the negative one at z=a.

b/ Show the shape of the streamlines for this case.

## Homework Equations

z = x + iy = r*e^(i[itex]\theta[/itex])

W(z) = [itex]\Phi[/itex] + i[itex]\Psi[/itex]

where [itex]\Phi[/itex] is the potential function and [itex]\Psi[/itex] is the stream function

## The Attempt at a Solution

a. This part is relatively easy. I know that the complex potential for a vortex at the origin is

i[itex]\Gamma[/itex]/(2*[itex]\pi[/itex]) * ln(z)

Therefore, for the two vortices, we will have:

W(z) = i[itex]\Gamma[/itex]/(2*[itex]\pi[/itex]) * ln((z+a)/(z-a))

b. This is where I start to have trouble. I need to separate W(z) into the real and imaginary parts, and then I know that the imaginary part is the stream function. However, I don't know how to do this. It is easy for a vortex at the origin, because then I would have

ln(z) = ln(r*e^(i[itex]\theta[/itex])) = ln(r) + i[itex]\theta[/itex]

However, the addition and subtraction of 'a' inside of the log is giving me a lot of trouble. I've tried to separate it every way that I can think of but haven't had any success. Any help would be much appreciated. Thanks!