# Complex Potential Flow - Two Vortices offset from the Origin

• squire636
In summary, the homework statement is that two vortices with the same strength, Gamma, will have a complex potential. The real part of the potential will be the stream function, which will be different for each vortex. It is easier to find the real part of the potential for a vortex at the origin, because then the equation is reduced to ln(r*e^iθ). However, the addition and subtraction of 'a' inside of the log is giving me trouble. I have tried to separate it every way that I can think of, but have not had any success. Any help would be much appreciated.
squire636

## Homework Statement

a. Determine the complex potential for two equal counter-rotating vortices with strength $\Gamma$, the positive one located at z=-a and the negative one at z=a.

b/ Show the shape of the streamlines for this case.

## Homework Equations

z = x + iy = r*e^(i$\theta$)

W(z) = $\Phi$ + i$\Psi$
where $\Phi$ is the potential function and $\Psi$ is the stream function

## The Attempt at a Solution

a. This part is relatively easy. I know that the complex potential for a vortex at the origin is

i$\Gamma$/(2*$\pi$) * ln(z)

Therefore, for the two vortices, we will have:

W(z) = i$\Gamma$/(2*$\pi$) * ln((z+a)/(z-a))

b. This is where I start to have trouble. I need to separate W(z) into the real and imaginary parts, and then I know that the imaginary part is the stream function. However, I don't know how to do this. It is easy for a vortex at the origin, because then I would have

ln(z) = ln(r*e^(i$\theta$)) = ln(r) + i$\theta$

However, the addition and subtraction of 'a' inside of the log is giving me a lot of trouble. I've tried to separate it every way that I can think of but haven't had any success. Any help would be much appreciated. Thanks!

To separate real and imaginary parts, it is easier to write the log as ln(z+a)-ln(z-a). In both cases, this is just a shift by +-a.

I tried that, but still did not make any progress.

ln(z+a) - ln(z-a)
ln(r*e^iθ + a) - ln(r*e^iθ - a)

Now what? I can't split it up into ln(r)+iθ anymore.

What about ##r^+##,##\theta^+## corresponding to the magnitude and phase of z+a?

I'm sorry I don't quite follow you, could you explain in more detail? I'm unfamiliar with that notation. Thanks.

That is a notation I invented for your specific problem.

z+a can be written as ##z+a=r^+ e^{i \theta^+}## with some real values ##\theta^+## and ##r^+## - there are formulas how to convert an arbitrary complex number to that shape.
In the same way, z-a can be written as ##z-a=r^- e^{i \theta^-}## with some real values ##\theta^-## and ##r^-##.

Would I do something along these lines?

ln(z+a)
ln(r*e^iθ + a)
ln(r*cos(θ)+i*r*sin(θ) + a)
ln((r*cos(θ)+a) + i*r*sin(θ))

And then try to find a new r and θ in order to put this back into the form of r*e^iθ ? That doesn't seem to me like it will be possible, and I can't find the formulas that you mentioned. Are they available online somewhere?

That is possible.
and I can't find the formulas that you mentioned. Are they available online somewhere?
The standard formulas how you get r and θ if you have a complex number a+ib? They are everywhere.

I think I figured it out, thanks so much!

## 1. What is complex potential flow?

Complex potential flow is a mathematical model used to describe the flow of fluids, such as air or water, over surfaces. It is based on the concept of a complex potential function, which combines both the velocity potential and the stream function to describe the flow field.

## 2. How are two vortices offset from the origin in complex potential flow?

In complex potential flow, two vortices offset from the origin are represented by two point vortices, each with a specified strength and location. These vortices create a flow field that is a combination of uniform flow and circular motion, resulting in a spiral-like flow pattern.

## 3. What is the significance of offsetting two vortices from the origin in complex potential flow?

Offsetting two vortices from the origin in complex potential flow allows for the creation of more complex and realistic flow patterns. This is particularly useful in studying the behavior of fluids in real-world scenarios, such as the flow around curved surfaces or objects.

## 4. How is the flow field affected by the strength and location of the two vortices in complex potential flow?

The strength and location of the two vortices in complex potential flow directly affect the shape and intensity of the resulting flow field. The closer the vortices are to each other, the stronger the resulting flow will be. Similarly, increasing the strength of the vortices will also result in a stronger flow field.

## 5. What are some applications of studying complex potential flow with two vortices offset from the origin?

Studying complex potential flow with two vortices offset from the origin has many practical applications, such as in the design of aircraft wings and propellers, understanding the flow of air around buildings and structures, and even in predicting weather patterns. It can also be used to model the flow of other fluids, such as water in rivers and oceans.

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