Surface Pressure Coefficient Distribution of a Doublet in a Uniform Flow

[WhiteWolf98]

Homework Statement

A flow is defined by the complex potential:

$F=U_{\infty}(z+ \frac {R^2} {z})$

Show that the surface pressure coefficient distribution is given by:

${C_p}=1-4\sin^{2}\theta$

Relevant Equations

$z=x+iy=re^{i\theta}$
$F=\phi+i\psi$
$\frac {dF} {dz}=u-iv$

So, it's a long way to the solution, but I'm finding it difficult to find a starting point. I'm going to say that as a first step, I should find what the value of the stream function $\psi$ is, at the surface. In order to do this, I need to use the following equation:

$F=\phi+i\psi$

If I can decompose $F$ into its real and imaginary parts, then I can find what $\psi$ is ($\psi=Im(F)$). I would just like to add that for a solid body, $\psi=0$. In the case of a doublet in a uniform flow, you end up getting a flow around a circle. As no fluid passes through into this boundary, nor out of it (and it's a closed streamline), it can be see as a solid body and thus $\psi$ must be $0$. I want to prove it anyway as this might be the case here, but perhaps not in a different case. My problem with decomposing $F(z)$ is the $z$ in the denominator. So you'd end up with:

$F(z)=U_{\infty}(z)+\frac {U_{\infty}R^2} {z}=U_{\infty}(x+iy)+\frac {U_{\infty}R^2} {x+iy}$

It's quite easy to know what to do with the first part $(U_{\infty}x+iU_{\infty}y$), we quite nicely have a real and imaginary part there. But I've no clue what to do with the second part, where we have $x+iy$ in the denominator. Ultimately if I want pressure, I need to know what $u$ and $v$ are at the boundary (I'm assuming anywhere along the circle as no specific point has been given), and to know those I need to know what $\psi$ is.

 

[vela]

Does this help?
$$\frac 1{x+iy} \cdot \frac{x-iy}{x-iy} = \frac{x-iy}{x^2+y^2}$$

 

[WhiteWolf98]

Does this help?
$$\frac 1{x+iy} \cdot \frac{x-iy}{x-iy} = \frac{x-iy}{x^2+y^2}$$

I... think so, yes! So skipping all the algebra, the expression I end up with is:

$F(z)=U_{\infty}x+\frac {U_{\infty}R^2x} {x^2+y^2}+(U_{\infty}y-\frac {U_{\infty}R^2y} {x^2+y^2})i$​

As:

$\psi=Im(F)$
​

Then it would follow that:

$\psi=U_{\infty}y-\frac {U_{\infty}R^2y} {x^2+y^2}$
​

Now knowing what $\psi$ is, I'm able to find what its value (should be) at the stagnation points.

If:

$F(z)=U_{\infty}(z)+\frac {U_{\infty}R^2}{z}$​

Then:

$\frac {dF}{dZ}=U_{\infty}-\frac {U_{\infty}R^2}{z^2}$
​

At stagnation points:

$\frac {dF}{dZ}=0$​

Hence:

$U_{\infty}-\frac {U_{\infty}R^2}{z^2}=0$
​

Solving for $z$:

$z=\pm R$​

If $x_{st}+iy_{st}=\pm R$, then $x_{st}=\pm R$ and $y_{st}=0$. Until this point, can I correctly say that there are two stagnation points at $x=-R$ and $x=R$? I've done a little more, but it's probably best to break it into pieces rather than putting it down all in one go.