Circulation of a stream function

[marklar13]

Homework Statement

The stream function of a 2D incompressible flow is given by:

\Psi=(\Gamma/2pi)*ln(r)

Show that the circulation about a closed path about the origin is \Gamma and is independent of r.

Homework Equations

\Gamma=-\ointV dot ds

The Attempt at a Solution

So far I know that the radial velocity component Ur=(1/r)*d(Psi)/d(theta)=0 because Psi does not depend on theta.
This means that the angular velocity component is the only velocity component: Utheta=-d(Psi)/d(r)=-L/2pi(r)
After this point I am not exactly sure what to do.

 

[marklar13]

Cant really understand the latex stuff... working on it