Solving the complex velocity of a clockwise vortex gives a counter clockwise rotation?

Question

I know that the complex potential of a clockwise vortex is $Ω = ik\ln{z}$

If I solve its complex velocity by $V = \bar{\frac{dΩ}{dz}}$

$\frac{dΩ}{dz}=\frac{ik}{z}$

$z=re^{i\theta}$

$\frac{dΩ}{dz}=\frac{ik}{r}e^{-i\theta}$

$V=\bar{\frac{dΩ}{dz}}=\frac{ik}{r}e^{i\theta}$

$V=\frac{ik}{r}cos\theta+\frac{i^2k}{r}sin\theta$

$V=-\frac{k}{r}sin\theta+i[\frac{k}{r}cos\theta]$

Therefore,

$V_x = -\frac{k}{r}sin\theta$

$V_y = \frac{k}{r}cos\theta$

Plugging in theta values seems to give me a counter clockwise rotation. Where did I go wrong?

If I do $Ω = -ik\ln{z}$, which is the complex potential for a counter clockwise vortex, it gives me a clockwise rotation.

Answer 1

Going from line 3 to line 4 (first appearance of $V$) you forgot to conjugate the leading constant factor $i$. That sign error reverses the sign of the flow field