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# 7.5: Stream Functions

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In everything we did above poor old $$\psi$$ just tagged along as the harmonic conjugate of the potential function $$\phi$$. Let’s turn our attention to it and see why it’s called the stream function.

##### Theorem $$\PageIndex{1}$$

Suppose that

$\Phi = \phi + i\psi$

is the complex potential for a velocity field $$F$$. Then the fluid flows along the level curves of $$\psi$$. That is, the $$F$$ is everywhere tangent to the level curves of $$\psi$$. The level curves of $$\psi$$ are called streamlines and $$\psi$$ is called the stream function.

Proof

Again we have already done most of the heavy lifting to prove this. Since $$F$$ is the velocity of the flow at each point, the flow is always tangent to $$F$$. You also need to remember that $$\nabla \phi$$ is perpendicular to the level curves of $$\phi$$. So we have:

1. The flow is parallel to $$F$$.
2. $$F = \nabla \phi$$, so the flow is orthogonal to the level curves of $$\phi$$.
3. Since $$\phi$$ and $$\psi$$ are harmonic conjugates, the level curves of $$\psi$$ are orthogonal to the level curves of $$\phi$$.

Combining 2 and 3 we see that the flow must be along the level curves of $$\psi$$.

## Examples

We’ll illustrate the streamlines in a series of examples that start by defining the complex potential for a vector field.

##### Example $$\PageIndex{1}$$: Uniform flow

Let

$\Phi (z) = z. \nonumber$

Find $$F$$ and draw a plot of the streamlines. Indicate the direction of the flow.

Solution

Write

$\Phi = x + iy. \nonumber$

So

$\phi = x \text{ and } F = \nabla \phi = (1, 0), \nonumber$

which says the flow has uniform velocity and points to the right. We also have

$\psi = y, \nonumber$

so the streamlines are the horizontal lines $$y =$$ constant (Figure $$\PageIndex{1}$$). Figure $$\PageIndex{1}$$: Uniform flow to the right. (CC BY-NC; Ümit Kaya)

Note that another way to see that the flow is to the right is to check the direction that the potential $$\phi$$ increases. The Topic 5 notes show pictures of this complex potential which show both the streamlines and the equipotential lines.

##### Example $$\PageIndex{2}$$: Linear Source

Let

$\Phi (z) = \log (z). \nonumber$

Find $$F$$ and draw a plot of the streamlines. Indicate the direction of the flow.

Solution

Write

$\Phi = \log (r) + i \theta.\nonumber$

So

$\phi = \log(r) \text{ and } F = \nabla \phi = (x/r^2, y/r^2),\nonumber$

which says the flow is radial and decreases in speed as it gets farther from the origin. The field is not defined at $$z = 0$$. We also have

$\psi = \theta,\nonumber$

so the streamlines are rays from the origin (Figure $$\PageIndex{2}$$). Figure $$\PageIndex{2}$$: Linear source: radial flow from the origin. (CC BY-NC; Ümit Kaya)

## Stagnation points

A stagnation point is one where the velocity field is 0.

##### Definition: Stagnation Points

If $$\Phi$$ is the complex potential for a field $$F$$ then the stagnation points $$F = 0$$ are exactly the points $$z$$ where $$\Phi '(z) = 0$$.

##### Definition: Proof

This is clear since $$F = (\phi_x, \phi_y)$$ and $$\Phi ' = \phi_x - i \phi_y$$.

##### Example $$\PageIndex{3}$$: Stagnation Points

Draw the streamlines and identify the stagnation points for the potential $$\Phi (z) = z^2$$.

Solution

(We drew the level curves for this in Topic 5.) We have

$\Phi = (x^2 - y^2) + i2xy.\nonumber$

So the streamlines are the hyperbolas: $$2xy =$$ constant. Since $$\phi = x^2 - y^2$$ increases as $$|x|$$ increases and decreases as $$|y|$$ increases, the arrows, which point in the direction of increasing $$\phi$$, are as shown in Figure $$\PageIndex{3}$$. Figure $$\PageIndex{3}$$: Stagnation flow: stagnation point at $$z = 0$$. (CC BY-NC; Ümit Kaya)

The stagnation points are the zeros of

$\Phi '(z) = 2z, \nonumber$

i.e. the only stagnation point is at the $$z = 0$$.

The stagnation points are also called the critical points of a vector field.

This page titled 7.5: Stream Functions is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Jeremy Orloff (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.