squire636
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Homework Statement
a. Determine the complex potential for two equal counter-rotating vortices with strength \Gamma, 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\theta)
W(z) = \Phi + i\Psi
where \Phi is the potential function and \Psi 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\Gamma/(2*\pi) * ln(z)
Therefore, for the two vortices, we will have:
W(z) = i\Gamma/(2*\pi) * 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\theta)) = ln(r) + i\theta
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!