= Solution
Use the convention $w=\phi+i\psi$, with $dw/dz=u_x-i u_y$. Then
$$
\boxed{w(z)=\frac12z^2,\qquad\phi=\frac12(x^2-y^2),\quad\psi=xy.}
$$
Indeed $w'(z)=z=x+iy$, giving <velocity> components $(x,-y)$. The <streamlines> are the level sets of the <stream function>:
$$
\boxed{xy=c.}
$$
For $c\neq0$ these are rectangular hyperbolas, restricted to $y>0$. The level $c=0$ consists of the vertical axis and the wall $y=0$. Directly on the wall the normal <velocity> is $-y=0$, so it is a streamline. The arrows point down toward the wall and then away from the stagnation point along the wall.
\Image[/past-exam-of-the-options-i-course-of-the-university-of-oxford/2012-stagnation-streamlines.png]
{title=Streamlines and velocity directions above an impermeable wall}
Back to article page
