The Student Room Group

Fluid dynamics

I'm not sure the best way to start this problem:

I have a model of a 2D flow given by the potential
w(z)=12πln(z+a2z)w(z) = \frac{-1}{2\pi}ln(z+\frac{a^2}{z})
and need to find the velocity components in the x and y directions.

I know that dwdz=uiv\frac{dw}{dz} = u - iv but when I differentiate w and then try and split into real and imaginary components, I get horribly long fractions which doesn't seem right.
I've also tried writing w=Φ+iΨw = \Phi + i\Psi and then evaluating the partial derivatives of phi and psi to get the velocity components, but again that leads to some very messy algebra.

Am I missing a slick method, or do I have to try and force my way through the algebra?
Any pointers in the right direction are welcome.
Reply 1
IIRC you want u=ϕ\nabla \phi for ϕ\phi a standard non-complex potential. I can't recall how it works for a complex potential.
Are you simply trying to find dwdz\frac{dw}{dz} and then taking the real and imaginary parts as components for u?