Problem $4.(30$pts$)$ Ideal plane flows can be described by the complex potential $F,$ which is a function of $z,$ where $z$ is represented by $z=$rexp$(i).$ Consider uniform flow with speed $U$ in the $+$x $-$direction flowing around a cylinder with radius $r_0.$ The cylinder is also spinning about its axis. This flow case can be constructed using the complex potential as the sum of a uniform stream, a doublet, and a vortex, as given below.

$F=Uz+U\frac{r_o^2}{z}+\frac{i{}_a}{2}ln(\frac{z}{r_0})$

a$)$ Generally, how is the complex potential related to the velocity potential $$ and the streamfunction $($i$.$e$.$ what is the definition of F $)?$

b$)$ Using the definition of $z$ given above, derive the velocity potential $$ and the streamfunction $$ for this flow. Hint: you may need to use the following definitions:

$z=$rexp$(i)=r(cos+isin)$

$ln(\frac{\text{r}\text{e}\text{x}\text{p}(i)}{r_0})=ln(\frac{r}{r_o})+i$

c$)$ Derive expressions for the velocity components $v_r$ and $v_{}$ from your results in $($b$).$ Note: here you should get the same answer whether you start from the velocity potential or the streamfunction.

d$)$ Calculate the point or points on the surface of the cylinder where there is a stagnation point if the circulation constant is ${}_a=2r_{0}U.$ Please provide your answer in terms of $$ in degrees from the $+$x direction $($the direction of the fredstream velocity $U).$