If a fluid has the complex potential $w(z)=\frac{-\Gamma i}{2 \pi}\operatorname{log}z$ what are its radial and transverse velocity components?

Question

If a fluid has the complex potential $$w(z)=\frac{-\Gamma i}{2 \pi}\operatorname{log}z$$ Can anyone show me how to find its radial and transverse velocity components in polar coordinates?

They are meant to be $u_r=0$ and $u_\theta=\frac{\Gamma}{2r \pi}$

Answer 1

Oh, I've forgot, what is the complex potenial. So,

$$w = \varphi + i \psi$$

where $\varphi$ is a potential and $\psi$ is a stream function.

Thus, $\boldsymbol v = \text{grad} \varphi \;$:

$$v_r = \frac{\partial \varphi}{\partial r} = \text{Re} \left[ \frac{\partial \, w(r e^{i \varphi})}{\partial r} \right] $$

$$v_{\theta} = \frac{1}{r} \frac{\partial \varphi}{\partial \theta} = \text{Re} \left[ \frac{1}{r} \frac{\partial \, w(r e^{i \varphi})}{\partial \theta} \right] $$

You could use stream function $\psi$ instead of potential though.

I've used FriCAS to evaluate things to the answer:

(15) -> D(real(-G*%i/(2*%pi)*log(r*exp(%i*phi))),r)

   (15)  0

(16) -> D(1/r * real(-G*%i/(2*%pi)*log(r*exp(%i*phi))),phi)

            G
   (16)  ──────
         2%pi r