Streamlines, Streaklines, Pathlines, and Gridlines

GridStreamStreakPath

The above animation aims to be a slight improvement over one on Wikipedia, which (incidentally) does not correctly describe the velocity field that it is depicting. The Wikipedia image doesn’t show a checkerboard of moving material, nor does it have a nice depiction of streamlines.

Before describing this animation, it might be helpful to look at a simpler motion (a rolling body) in order to review the difference between streamlines, streaklines, and pathlines. Consider a simple rigid body consisting of a disk of small radius (shown in gray below) along which it rolls along a tabletop, along with a larger-radius extension of the body (shown in color below) which can dip down below the table surface (as if there is a slot cut into the table so that part of the body rolls under it).

STREAMLINES: These are tangent to the instantaneous velocity field.  For a rolling rigid body, the motion is always circular about the instantaneous center of rotation at the bottom of the wheel. Accordingly, this image shows the streamlines at various points in time as the disk rolls along:

particleStreamLineRolling

This image of streamlines is drawn not just on the body itself but also on its “virtual extension” in order to emphasize that (for rigid rolling) the instantaneous velocity is circular around the instantaneous center of rotation (bottom of the wheel). A particular set of streamlines is drawn in red. These are the ones that pass through a set of points that are evenly distributed on a spoke of the wheel (shown in black).

STREAKLINES: These are the lines you would see if a magic gremlin were to sit at a given location in space and “spraypaint” the material as it passes by.  Suppose that an assembly line of gremlins (located where you see the dots in the first image) are pointing their spray paint cans at the body while it rolls past. Then they would form the black streaklines shown here at various times:

streaklinesOnBefore

Important: The streaklines are made by gremlins who are sitting still and spraying material as it passes by.

PATHLINES: Are made by gremlins who “ride” with the material, spraying a record of where they have been (as if we were watching the rolling body from behind a window, and those whacky gremlins would spray paint onto the window as they pass by). Accordingly, here are the pathlines for group of gremlins who were initially coincident with the gremlins in the above streakline plot:

particlePathLineRolling

GRIDLINES are any set of lines that are painted on the body like tattoos. Such lines move with the body (like a tattoo).

 

Now let’s get back to the animation,

GridStreamStreakPath

The above animation is a set of progressively overlaid images:

1. In the far background is a yellow-pink-white-gray checkerboard. Imagine that this initially square (perfectly rectilinear) checkerboard grid is spray-painted (tattooed) instantaneously onto moving material at time t=0. The animation of the checkerboard then shows that the material translates to the right while shearing upward with a shear intensity that increases with coordinate x1.   This mapping uses the following continuum mapping function from initial coordinates (X1,X2) to deformed coordinates (x1,x2) at time t:

x1 = t + X1
x2 = 1/3 t (t+X1)^2 + X2

2. Overlaid on top of the checkerboard is a family of green streamline arrows showing velocity directions at a given instant of time. Note that they point in the whatever direction the material (checkerboard) happens to be moving. The local material velocity is

v1=1
v2=1/3 x1 (2 t+x1)

3. The red line is a path line. The part of this line to the left of the origin shows where a particular material point had been located prior to time t=0 before it arrived at the origin at time t=0. The animated part of the red path grows longer during the animation so that the tip of the red line stays always tracking with this same material point, which is clear because the evolving tip of the red line stays always on the same intersection of the checkerboard grid lines. It might seem that the red line itself is moving, but that is an optical illusion. Hold your finger near any point on the red line to see that the red line is stationary. It shows where the particle has been in the past. The key is that the red line shows a locus of points occupied in the past by a particular particle. This curve only grows longer. The point being tracked by the red line in this animation has initial coordinates (0,0) at time t=0. At some \[Tau] in the past, the mapping function was

x1 = \[Tau] + X1
x2 = 1/3 \[Tau] (\[Tau]+X1)^2 + X2

Evaluating this using X1=X2=0 gives

x1 = \[Tau]
x2 = 1/3 \[Tau]^3

These parametrically describe the path line. The parameter \[Tau] varies over times in the past up to the current time t. The part of the red line (to the left of the origin) indicates where this particle had been prior to time t=0, so that part of the path uses t0).

4. The streak line shows what you would see if there were a source of dye at the origin, effectively putting tatoo ink or “spray paint” onto the material as it passes by the origin. At past time \[Tau], the mapping was

x1 = \[Tau] + X1
x2 = 1/3 \[Tau] (\[Tau]+X1)^2 + X2

Suppose that there is a juvenile delinquent standing fixed at the origin spraying material as it passes by. Putting his coordinates (x1,x2) = (0,0) into the above past mapping gives:

0 = \[Tau] + X1
0 = 1/3 \[Tau] (\[Tau]+X1)^2 + X2

Solving this for (X1,X2) then tells us which material particle he was spraying at the past time \[Tau]:

X1 = -\[Tau]
X2 = 0

According to this result, each spraypainted point was originally on the X2=0 grid line in the past, to the left of the orgin by a distance \[Tau]. The equation for the streak line is therefore the image (deformed locations) of these points in the current configuration, found by applying the mapping at time t with X1=-\[Tau] and X2=0:

x1 = t – \[Tau]
x2 = 1/3 t (t-\[Tau])^2

The blue line in the animation is this set of equations plotted parametrically with the parameter \[Tau] ranging from 0 to t.

The Mathematica commands to generate this animation may be downloaded from here.

Just right-click it and “save-as” to get the native notebook file.

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