Complex potential of a two-dimensional potential flow ($W=\Phi +i\Psi ,W^{\prime }=u-iv$)

For an incompressible irrotational two-dimensional potential flow, an analytic complex potential of a two-dimensional potential flow combines the velocity potential $\Phi$ and streamfunction $\Psi$. The Cauchy-Riemann equations give $u={\Phi }_x={\Psi }_y$ and $v={\Phi }_y=-{\Psi }_x$, hence $W^{\prime }(z)=u-iv$. Curves of constant $\Psi$ are streamlines. An analytic coordinate change $z=f(\zeta )$ transforms the complex velocity by $dW/dz=(dW/d\zeta )/f^{\prime }(\zeta )$ wherever $f^{\prime }\ne 0$.

The conformal map of a vertical half-strip by sine sends $-\pi /2<\mathrm{Re}z<\pi /2$, $\mathrm{Im}z>0$ to the upper half-plane. The complex potential of a two-dimensional potential flow $W(z)=\log (a+\sin z)$ therefore pulls back the radial logarithmic flow originating at $\zeta =-a$. Its complex velocity is $W^{\prime }=\cos z/(a+\sin z)$, with a boundary singularity at $z=-\pi /2+i\mathrm{arcosh}a$. For $z=x+iy$, define $D=(a+\sin x\cosh y{)}^2+{\cos }^{2}x{\sinh }^{2}y$. ThenThe bottom boundary carries rightward flow, the right boundary upward flow, and the left boundary flow away from the singular point. The velocity tends to $(0,1)$ at large height. The finite bottom corners are stagnation points, not singularities of $W$.