Complex Analysis => Fluid Flow

[cepheid]

I'm struggling with this question right now:

Let the complex velocity potential $\Omega(z)$ be defined implicitly by

$$z = \Omega + e^{\Omega}$$

Show that this map corresponds to (some kind of fluid flow, shown in a diagram, not important).

For background,

$$\Omega = \Phi + i\Psi$$

where Phi is the velocity potential:

$$\mathbf{v} = \nabla\Phi$$

and Psi is the harmonic conjugate of Phi (therefore it is the streamfunction of the fluid flow.

My first thought was that I need to find the level curves of the streamfunction in order to find out what kind of flow this is. But before I can do that, I need to solve for Omega explicitly. THAT's where I'm stumped. Any suggestions on a strategy or approach?

 

[AKG]

Well, you can perhaps find a power series expansion for $\Omega$. You have:

$$I = (I + \exp)\circ\Omega$$

$$(I + \exp)^{-1} = \Omega$$

I think the best way to compute the left side is to look at the power series expansion of

$$\frac{1}{1 - x}$$

where x is replaced with -exp. I can't think of a way to get a closed form solution out of this, but at least it will give you something to work with, hopefully enough to prove the other things you need to prove. Note given any thing like Z = X + iY, it's easy to express Y solely in terms if Z, and similarly X only in terms of Z

X = (Z + Z*)/2

Where Z* is the complex conjugate of Z.

 

[cepheid]

Well, you can perhaps find a power series expansion for $\Omega$. You have:

$$I = (I + \exp)\circ\Omega$$

I'm really not sure what's going on here.

 

[AKG]

Sorry, I is the identity function. The left is I, so I(z) = z. The right is (I + exp)oQ, so

[(I + exp)oQ](z)
= (I + exp)(Q(z))
= I(Q(z)) + exp(Q(z))
= Q(z) + exp(Q(z))

I'm using Q for $\Omega$. Anyways, I don't think the thing I suggested will work...