Past exam of the fluids and waves course of the University of Oxford / 2023 / 2 / b

The complex potential of a two-dimensional potential flow givesAt a stagnation point, $q=0$. For $m\ne 0$, taking the two square-root alternatives givesThereforeThe denominators are nonzero because $b>a$. Moreoverso both stagnation points satisfy $|z_{\sigma }|=a$. They are distinct: equality would require $2e^{-i\alpha }(a^2-b^2)=0$. Neither is a singularity of $q$. Conversely, the numerator of $q$ is a quadratic polynomial with nonzero leading coefficient, so these are all its finite zeros. Thus there are exactly two boundary stagnation points and none in the open fluid region. If $m=0$, the flow instead vanishes everywhere, which is the degenerate exception to the assertion.