= Logarithmic flow in a sine-mapped half-strip
{title2=$W(z)=\log(a+\sin z),\quad a>1$}
The <conformal map of a vertical half-strip by sine> sends $-\pi/2<\operatorname{Re}z<\pi/2$, $\operatorname{Im}z>0$ 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 $W'=\cos z/(a+\sin z)$, with a boundary singularity at $z=-\pi/2+i\operatorname{arcosh}a$. For $z=x+iy$, define $D=(a+\sin x\cosh y)^2+\cos^2x\sinh^2y$. Then
$$
u=\frac{\cos x(a\cosh y+\sin x)}D,\qquad v=\frac{\sinh y(\cosh y+a\sin x)}D.
$$
The 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 $W$.
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