Problem $9$

Imagine a flow composed of a vortex added to a cylinder of radius $r_0$ in a uniform flow of free stream velocity U:

$=$Ursin$-\frac{r_0^{2}Usin}{r}+\frac{}{2}lnr$

a$)$ Determine expressions for the velocity components. Show detailed work.

Governing Equation$($s$)$

b$)$ Derive an expression for the pressure coefficient $C_p.$ The pressure coefficient is defined as

$Cp=\frac{p-p_{}}{\frac{1}{2}{U}^2}$

Here, Take P as the pressure at the surface. Hint : Use Bernoulli's formula. Show detailed work

c$)$ Derive an expression for the lift force per unit depth into the page. Show detailed work.