Consider the velocity potential in plane polar coordinates

$$\phi(r, \theta)=U\left(r+\frac{a^{2}}{r}\right) \cos \theta+\frac{\kappa \theta}{2 \pi}$$

Find the velocity field and show that it corresponds to flow past a cylinder $r=a$ with circulation $\kappa$ and uniform flow $U$ at large distances.

Find the distribution of pressure $p$ over the surface of the cylinder. Hence find the $x$ and $y$ components of the force on the cylinder

$$\left(F_{x}, F_{y}\right)=\int(\cos \theta, \sin \theta) p a d \theta .$$