Quiver and Stream Plots

In this section, you will learn how to build quiver and stream plots using matplotlib

Quiver Plots

A quiver plot is a type of 2D plot that shows vector lines as arrows. Quiver plots are useful in electrical engineering to visualize electrical potential and useful in mechanical engineering to show stress gradients.

To build a quiver plot, first import matplotlib. Again, by convention - the alias plt is used. If using a Jupyter notebook include the line %matplotlib inline. For some of the quiver plots in this section, numpy will be needed as well.

Quiver plot with one arrow

Let's buid a simple quiver plot that contains one arrow to see how matplotlib's ax.quiver() method works. The ax.quiver() method takes four positional arguments:

ax.quiver(x_pos, y_pos, x_direct, y_direct)

Where x_pos and y_pos are the arrow starting positions and x_direct, y_direct are the arrow directions.

Let's build our first plot which contains one quiver arrow at the starting point x_pos = 0, y_pos = 0. We'll define the quiver arrow's direction as pointing up and to the right x_direct = 1, y_direct = 1.

We see one arrow pointing up and to the right.

Quiver plot with two arrows

Now let's add a second arrow to the quiver plot by passing in two starting points and two arrow directions.

We'll keep our original arrow starting position at the origin 0,0 and pointing up and to the right, direction 1,1. We'll define a second arrow with a starting position of -0.5,0.5 which points straight down (in the 0,-1 direction).

An additional keyword argument to add the the ax.quiver() method is scale=5. Including scale=5 scales the arrow lengths so the arrows look longer and show up better on the quiver plot.

To see the start and end of both arrows, we'll set the axis limits between -1.5 and 1.5 using the ax.axis() method and passing in a list of axis limits in the form [xmin, xmax, ymin, ymax].

We can see two arrows. One arrow points to the upper right and the other arrow points straight down.

Quiver plot using a meshgrid

Two arrows is great, but to create a whole 2D surface worth of arrows, we'll utilize numpy's meshgrid() function.

We need to build a set of arrays that denote the x and y starting positions of each quiver arrow on the plot. We will call our quiver arrow starting position arrays X and Y.

We can use the x,y arrow starting positions to define the x and y components of each quiver arrow direction. We will call the quiver arrow direction arrays u and v. On this quiver plot, we will define the quiver arrow direction based upon the quiver arrow starting point using:

$$ x_{direction} = cos(x_{starting \ point}) $$$$ y_{direction} = sin(y_{starting \ point}) $$

Now we will build the quiver plot using matplotlib's ax.quiver() method. Again, the method call takes four positional arguments:

ax.quiver(x_pos, y_pos, x_direct, y_direct)

This time x_pos and y_pos are 2D arrays which contain the starting positions of the arrows and x_direct, y_direct are 2D arrays which contain the arrow directions.

The commands ax.xaxis.set_ticks([]) and ax.yaxis.set_ticks([]) removes the tick marks from the axis and ax.set_aspect('equal') sets the aspect ratio of the plot to 1:1.

Now let's build another quiver plot with the $\hat{i}$ and $\hat{j}$ components of the arrows, $\vec{F}$ are dependant upon the arrow starting point $x,y$ according to the function:

$$ \vec{F} = \frac{x}{5} \ \hat{i} - \frac{y}{5} \ \hat{j} $$

Again we can use numpy's np.meshgrid() function to build the arrow starting position arrays, then apply our function $\vec{F}$ to the $x$ and $y$ arrow starting point arrays.

Quiver plot containing a gradient

Next let's build another quiver plot using the gradient function. The gradient function will have the form:

$$ z = xe^{-x^2-y^2} $$

We can use numpy's np.gradient() function to apply the gradient function to every arrow's x,y starting position.

Quiver plot with four vortices

Now let's build a quiver plot containing four vortices. The function $\vec{F}$ which describes the 2D field is:

$$ \vec{F} = sin(x)cos(y) \ \hat{i} -cos(x)sin(y) \ \hat{j} $$

Again we can build these arrays using numpy and plot them with matplotlib.

Quiver plots with color

Now let's add some color to the quiver plots. The ax.quiver() method has an optional fifth positional argument that specifies the quiver arrow color. The quiver arrow color argument needs to have the same dimensions as the position and direction arrays.

Using matplotlib subplots, we can build a figure which contains 3 quiver plots each in color

Stream Plots

A stream plot is a type of plot used to show fluid flow and 2D field gradiants.

The basic method to build a stream plot in matplotlib is:

ax.streamplot(x_grid,y_grid,x_vec,y_vec, density=spacing)
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Where x_grid and y_grid are arrays of x,y points. The arrays x_vec and y_vec denote the stream velocity at each point on the grid. The keyword argument density=spacing specifies how close together to draw the stream lines.

A simple stream plot

Let's build a simple plot of stream lines on a 10 x 10 grid where all the stream lines are parallel and point to the right.

The plot contains parallel streamlines all pointing to the right.

Stream plot of a field

We can build a stream plot which shows field lines based on a defined 2D vector field.

Stream plot of two point charges

We can build a stream plot showing the electric field due to two point charages. The electric field at any point is dependant upon the position and distance relative to the point charges.

Stream plot showing fluid flow around an object

Stream plots can also be used to show how fluid flows around a stationary object. In this example, we will consider a 2D circle as the statinary object and model the fluid flow around the circle.