A doublet along the x$-$axis is present at the origin of a $2$D xy$-$plane. Additionally, there is

a uniform flow of velocity V in the x$-$direction. Use potential flow theory and superposition

of elementary planar irrotational flows.

$($a$)$ Derive the stream function for the combined flow. $[30]$

$($b$)$ The streamline $=0$ forms a circle, indicating that the combined flow represents the

flow over a cylinder. Find the radius of this cylinder $($i$.$e$.$ r for which $=0).[30]$

$($c$)$ Use the definition of the stream function to derive the velocity on the surface of the

cylinder. $[30]$

$($d$)$ Use Bernoulli$$s equation to find the pressure distribution over the cylinder surface. $[10]$