Image vortex at a plane wall ($U=\frac{\Gamma }{4\pi b}$)

The method of images enforces impermeability at a plane wall by reflecting a line vortex across it and reversing its circulation. The two logarithmic stream functions cancel on the wall, which is consequently a streamline. For a vortex of circulation $\Gamma$ at $(0,b)$ above $y=0$, the image of circulation $-\Gamma$ at $(0,-b)$ induces velocity $(\Gamma /(4\pi b),0)$ at the real vortex. The signed drift parallel to the wall is therefore $\Gamma /(4\pi b)$.

In the quadrant $x,y>0$, a line vortex of circulation $\Gamma$ at $(x,y)$ has three images: circulations $-\Gamma$ at $(-x,y)$ and $(x,-y)$, and $+\Gamma$ at $(-x,-y)$. Their induced velocity isIn polar coordinates, these equations give $\dot{r}=\Gamma \cot (2\theta )/(2\pi r)$ and $\dot{\theta }=-\Gamma /(4\pi r^2)$. Direct differentiation yields $\frac{d}{dt}(r\sin 2\theta )=0$, so the trajectory obeys $r\sin 2\theta =\text{constant}$.