Flow Around a Cylinder with Linear Vortex and Points of Stagnation

[CptXray]

Homework Statement

Find the flow around a cylinder with radius $a$ generated by linear vortex $\Gamma$ in point $z=b$. Find points of stagnation. Also $b>a$

Homework Equations

Complex potential of vortex: $$\omega_{vortex} = \frac{\Gamma}{2\pi i}\ln{z}$$
Milne-Thomson circle theorem: $$\omega (z) = f(z) + \overline{f(\frac{a^2}{\overline{z}})}$$

The Attempt at a Solution

Here my $f(z)$ is: $$f(z)=\frac{\Gamma}{2\pi i}\ln{(z - b)}$$
Applying circle theorem: $$\omega (z) = f(z) + \overline{f(\frac{a^2}{\overline{z}})} = ... = \frac{\Gamma}{2\pi i}\bigg( \ln{(z-b)} - \ln{(\frac{a^2}{z} - b)} \bigg)$$
which is, I assume, the complex potential with boundary conditions for cylinder. But how am I supposed to find the flow now? I tried rotation of rotation but it's pointless. I'd be really gratful for help or hints.

 

[dRic2]

I have little clue of what you are talking about (my ignorance), but if that is a complex potential then the real part of the function is the velocity potential and to find the velocity field you just take the gradient of the potential. Hope it is what you are looking for.

 

[CptXray]

Yes, but the problem is that logarithms are centered in points other than $z=0$ and I'm wondering if I'm trying to do it wrong, because I can't separate real and imaginary terms in order to get $\Phi$ - velocity potential and $\Psi$ - stream function,

 

[dRic2]

I found this https://math.stackexchange.com/questions/2260028/find-the-real-and-imaginary-parts-of-lnz
Then try some substitutions... For example $z-a = x + iy - (x_a + iy_a) = (x - x_a) + i(y - y_a) = X + iY$ now you can use the formulae given in the link I posted using $X$ and $Y$ instead of $x$ and $y$. Maybe you can work something similar for the other logarithm