$($b$)$ For $2D$ steady, inviscid and incompressible flow past a rotating cylinder subjected to uniform free$-$stream velocity $U_{},$ the stream$-$function $$ is given as

$=U_{}sin(r-\frac{a^2}{r})+a^{2}ln(\frac{r}{a})$

where $a$ is the radius of the cylinder and $$ is the angular speed of the rotating cylinder. Obtain the expression for the coefficient of pressure $c_p=\frac{(p-p_{})}{\frac{1}{2}{}_{}{U}_{}^2}$ on the surface of the cylinder. Also obtain an expression for the lift coefficient $c_L$ of this cylinder. $($Clarification: Lift is the component of the aerodynamic force which is perpendicular to the free$-$stream flow direction$)$

$[12$ Marks$]$