A Rankine wortex has a core that can be modelled as a rotational woetex while outside of she core is described lor a potential vortex. The welocity feld wef$($r$)$ for this case can be written asts

$w_e(r)=\{\begin{array}{cc}{r}_i,& r{r}_e\\ \frac{}{2r},& r>r_k\end{array}$

where $r_C$ is the known core ralius and $$ is the known strength of the potential vortex.

$($a$)$ Obtain the expersoion for I in terms of I apd $r_e.$

$($b$)$ Let $A$ be a circular comtour with radius exual to $0.5$ r and $B$ be a circular contour with radias equal to $2r_{e.}$ By determining the circalations around coetcar A and contour $B$ sparately; show that the circulasion for the Rankine vortex is constaat, independent of she choice of the elosed contout