Show that the complex potential in the complex $\zeta$ plane,

$$w=(U-i V) \zeta+(U+i V) \frac{c^{2}}{\zeta}-\frac{i \kappa}{2 \pi} \log \zeta$$

describes irrotational, inviscid flow past the rigid cylinder $|\zeta|=c$, placed in a uniform flow $(U, V)$ with circulation $\kappa$.

Show that the transformation

$$z=\zeta+\frac{c^{2}}{\zeta}$$

maps the circle $|\zeta|=c$ in the $\zeta$ plane onto the flat plate airfoil $-2 c<x<2 c, y=0$ in the $z$ plane $(z=x+i y)$. Hence, write down an expression for the complex potential, $w_{p}$, of uniform flow past the flat plate, with circulation $\kappa$. Indicate very briefly how the value of $\kappa$ might be chosen to yield a physical solution.

Calculate the turning moment, $M$, exerted on the flat plate by the flow.

(You are given that

$$M=-\frac{1}{2} \rho \operatorname{Re}\left\{\oint\left[\frac{\left(\frac{d w}{d \zeta}\right)^{2}}{\frac{d z}{d \zeta}}\right] z(\zeta) d \zeta\right\} \text {, }$$

where $\rho$ is the fluid density and the integral is to be completed around a contour enclosing the circle $|\zeta|=c$ ).