I'm a Mathematics student, working through a homework sheet for a Fluid Mechanics module. The question is given:
Consider the flow described by the complex potential $$w=4z+\frac{8}{z}.$$
- Determine $\psi$, $\phi$, $u$ and $v$ in plane polar coordinates $(r,\theta)$.
- Determine the location of the stagnation points.
- Show that this complex potential describes an inviscid flow around a solid, finite object, What is the shape of the object?
- Sketch the streamlines for the flow outside the object.
My working out so far for the question is:
(1) Let $z=re^{i\theta}$, and therefore
\begin{align} w&=4re^{i\theta}+\frac{8}{r}e^{-i\theta} \\ &=4r(\cos(\theta)+i\sin(\theta))+\frac{8}{r}(\cos(\theta)-i\sin(\theta)) \\ &=(4r+\frac{8}{r})\cos(\theta)+(4r-\frac{8}{r})i\sin(\theta). \end{align}
Using the Cauchy-Riemann equtaions, $w=\phi+i\psi$, we then have that $\phi=(4r+\frac{8}{r})\cos(\theta)$ and $\psi=(4r-\frac{8}{r})\sin(\theta)$.
Also, we have that $u=\frac{\partial\phi}{\partial r}\implies u=(4-8r^{-2})\cos(\theta)$ and $v=\frac{1}{r}\frac{\partial\phi}{\partial\theta} \implies v=-(4r+8r^{-2})\sin(\theta)$.
(2) Stagnation points are given by $u=0$ and $v=0$. So, from $u=0$, we have that $r^2=2$ or $\cos(\theta)=0$. Similarly from $v=0$, we have that $r^2=-2$ and $\sin(\theta)=0$. Therefore, the stagnation points occur at $(r,\theta)=(\sqrt{2},0),(\sqrt{2},\pi)$.
From here (ie (3) onwards), I fall down. I think that I should use that $\textbf{u}\cdot\textbf{n}=0$, but I'm not too sure how to use this information. Should I be using Bernoulli's theorem for pressure? Is there some assumption I am missing?
Any help would be much appreciated!