Solution: method of image for a 2D vortex near a wall

We verify if a 2D vortex and its mirror image with respect to a wall located at $y = 0$ satisfies the no flux boundary condition at the wall. First we need to evaluate the component of the velocity vector in the $y$ direction.

$$\frac{\partial \phi}{\partial y} = \frac{\Gamma}{2\pi}\left(\frac{x}{x^{2}+(y-b)^{2}}-\frac{x}{x^{2}+(y+b)^{2}}\right) \notag$$

Next, we set $y = 0$, and

$$\frac{\partial \phi}{\partial y} = \frac{m}{2\pi}\left(\frac{x}{x^{2}+b^{2}}-\frac{x}{x^{2}+b^{2}}\right) = 0 \notag$$

as expected.