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I'm working through a homework sheet for a Fluid Mechanics module. The question is given:

Consider the flow described by the complex potential $$w=4z+frac8z.$$

1. Determine $psi$, $phi$, $u$ and $v$ in plane polar coordinates $(r,theta)$.


2. Determine the location of the stagnation points.


3. Show that this complex potential describes an inviscid flow around a solid object, What is the shape of the object?


4. Sketch the streamlines for the flow outside the object.

My working out so far for the question is:

1. Let $z=re^itheta$, and therefore
   beginalign
   w&=4re^itheta+frac8re^-itheta \
   &=4r(cos(theta)+isin(theta))+frac8r(cos(theta)-isin(theta)) \
   &=(4r+frac8r)cos(theta)+(4r-frac8r)isin(theta).
   endalign
   Using the Cauchy-Riemann equations, write $w=phi+ipsi$, $phi=(4r+frac8r)cos(theta)$ and $psi=(4r-frac8r)sin(theta)$. Also, we have that $$u=fracpartialphipartial rimplies u=(4-8r^-2)cos(theta)$$ and $$v=frac1rfracpartialphipartialtheta implies v=-(r+8r^-2)sin(theta).$$
   


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 $sin(theta)=0$. Since $rinmathbbR$ (as it is a distance), we have that $r^2=2$ (from $u=0$) and $sin(theta))=0$ (from $v=0$). Therefore, the stagnation points occur at $(r,theta)=(sqrt2,0),(sqrt2,pi)$.

From here (ie 3 onwards), I fall down. I think that I should use that $textbfucdottextbfn=0$, 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!

To determine the shape of the obstacle, you just look at the streamlines (they're everywhere tangent to the flow). Since you have a potential flow, it holds that $Delta u = 0$, where $u$ is the (real) velocity. So, the term $muDelta u$ in the Navier-Stokes equation will vanish. Obviously, this doesn't itself imply $mu = 0$ so the fluid itsf may not be inviscid, but the flow should be.