An incompressible, irrotational two$-$dimensional flow has the complex potential

$w(z)=Uz+\frac{}{2z}$

where $z=x+iy=r{e}^i,$ and $U,$ are real positive constants. Explain why this can represent a two$-$dimensional flow about a circular cylinder of radius

$a=\sqrt[\phantom{2}]{\frac{}{2U}}$

centred at the origin, and deduce further that there are stagnation points at $z=+-a.$

Find the direction of the flow at infinity, and find the speed on the surface of the cylinder as a function of the angle measured from this direction.