Point source derivation

Question

I've just been reading this on a website to derive a point source.

All makes sense, but I am not sure why it is $(2\pi r)v_r$. Thanks!

Answer 1

the volumetric flow rate is the volume of fluid which passes through a given surface per unit time. -- Wikipedia

The article goes on to explain that this is the integral of normal component of $v$ over the surface. But we are in two dimensions, so the "surface" is the circle of radius $r$. The normal component of $v$ is $v_r$, which is constant on the circle. So, the integral of $v_r$ of the circle is $2\pi r v_r$.