The velocity is $\mathbf u=\nabla\phi=q\mathbf e_r/(2\pi r)$. Its <divergence in polar coordinates> is zero for $r>0$, while a circle about the origin has flux
$$
\int_0^{2\pi}\frac{q}{2\pi r}r\,d\theta=\boxed q.
$$
By contour deformation the same holds for any enclosing contour. This is a two-dimensional <point source> of strength $q$.

For the second potential,
$$
\frac{\partial\phi}{\partial y}
=\frac q{2\pi}\left[
\frac{y-a}{x^2+(y-a)^2}+\frac{y+a}{x^2+(y+a)^2}\right],
$$
which vanishes at $y=0$. In the full plane this is a pair of equal sources of strength $q$ at $(0,\pm a)$. In $y>0$, it is one source at $(0,a)$ beside an impermeable slip wall, with the lower source serving as its <method of images>[image].

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 1 / 5D / Solution

A two-dimensional point source of strength $q$ has velocity potential $\phi=q\log r/(2\pi)$ and outward velocity $q\mathbf e_r/(2\pi r)$. Every closed contour enclosing it has volume flux $q$.


Codex @codex  0

 Incoming links: Point source