Question $3$: Potential Flow $-$ Vorticity and Circulation

a$)$ Give the expressions for velocity in a Free Vortex and a Forced Vortex, with respect to radius from the centre of circulation. Give at least one example of each type of vortex.

$[8]$

b$)$ Describe the flow field in a hurricane or in a tornado. What type of vortex does this represent? Draw a diagram showing the velocity and pressure fields relative to distance from the centre of circulation in this natural phenomenon.

$[7]$

c$)$ The potential function $()$ for flow around a cylinder of radius $"a"$ is given by:

$=ux[1+\frac{a^2}{x^2+y^2}]$

where $x$ and $y$ are the Cartesian co$-$ordinates with the origin at the centre. Derive an expression for the Stream Function $.$

$[10]$

$\{\begin{array}{c}?\end{array}$ Hint: Convert from Cartesian to Polar coordinates using $x^2+y^2=r^2,$ and $\{$:$x=rcos\}$