Past exam of the perturbation methods and applied complex variables course of the University of Oxford / 2008 / B / 5 / iii

A uniform horizontal velocity potential in the upper $Z$-plane is a real constant times $Z$. Since $dz/dZ=(2/\pi )R/Z$, chooseup to an additive constant. Its complex velocity isAs $Z\to \infty$ this tends to one, giving physical velocity $(1,0)$. As $Z\to 0$, $R\to i$ and $Z/R\to 0$, giving zero velocity far down the channel. The imaginary part of $w$ is constant on the real $Z$-axis, so the physical boundary is a streamline and the impermeability condition holds. The usual complex velocity convention is $dw/dz=u-iv$.

To supply nonzero channel flux while letting the speed decay in the large upper region, take a logarithmic velocity potential:Use the logarithm analytic on the upper half-plane, with argument between zero and $\pi$. Its complex velocity isAt upper infinity it tends to zero. At the channel end $Z\to 0$, it tends to $1/i=-i$, which means $(u,v)=(0,1)$, as required. The streamfunction is constant on each real boundary interval, and differs by two between the two channel walls. That flux equals the channel width two times the unit upward speed, providing an independent normalization check.