$2.(\backslash (\backslash$mathbf$\{4\}\backslash )$ points$)$ The stream function for a free vortex shown in Figure $\backslash (\backslash$mathbf$\{2\}\backslash )$ is given by

$\backslash [$

$\backslash$psi$=-\backslash$frac$\{$K$\}\{2\backslash$pi$\}\backslash$ln r

$\backslash ]$

where $\backslash ($ K $\backslash )$ is the strength of the vortex.

a$.$ Find the velocity potential function and determine if the flow is irrotational.

b$.$ What is the vorticity?

c$.$ Assume that the streamline is in a horizontal plane, determine in terms of the circulation an expression for the pressure gradient

i$.$ along a streamline, and

ii$.$ normal to a streamline.

rigure $\backslash (<mtext></mtext><mi>\backslash </mi><mi>)</mi>$