Chaotic Advection and Blinking Vortices
Students Ryan Schilt, Joe Adams, Anna Lieb
Dates of Involvement 2008-2009
Faculty Advisor Jim Meiss
Graduate Student Mentor Krissy Snyder
Background
Hassan Aref, in his paper "Stirring by Chaotic Advection", outlined an analytical method to observe chaos in two dimensional stokes flow within a disk. All fluid motion within the disk is the result of “blinking vortices”, one vortex is on and will rotate the fluid while the other is off.
Chaotic behavior is seen with at least three dimensional systems; all three spatial, or two spatial with one in time. The two dimensional disk has only two spatial dimensions therefore has to be time dependent in order to display chaotic behavior.
a circular boundary, black unit circle, to satisfy a “no-flux” boundary condition.
The black dot is the rotating vortex within the disk and the red vortex is the
“hypothetical” image vortex. The resultant velocity is then tangent to the
boundary, noted by the blue vector.
Using the method of images (Fig.1), Aref was able to find an analytic solution with one rotating vortex:
Where a is the radius of the disk containing the fluid, b is the location of the vortex, 
is the location of a specific particle of fluid, 
is the strength of the vortex and 
. The above differential equation can then be solved:
Which is an arc of a circle in the complex plane with center at 
, radius 
, and angle 
. Choatic behavior can then be observed by blinking the vortices for a certain amount of time.
Simulation
Below is a link to a flash-simulation of Aref’s blinking vortex setup. You can experiment with different blinking periods to determine when chaotic behavior can be seen. Hint: start with a lower period, say 0.10.
To view simulation, follow the link below, then click on “Assorted Simulations” on the left side-panel, then on the red “Stirring Simulation” link. Then follow the instructions below to run the simulation.
Simulation Link: http://appmsaga.colorado.edu/
Simulation Instructions:
1) Select an option from Initial Points. This will color certain fluid particles so we can see how they move. The "User Defined" option allows you to click inside the circle to color 30 particles around the selected point.
2) Enter in the desired number of particles to color, a good value is around 2500.
3) Click the "Reset" button to show the particles you choose to color.
4) Enter a value for the period, such that it is greater than zero and ends in zero (EX: 0.1250, or 23.1250)
5) Finally use the Enter key to update the position of the fluid particles to a time, t = 0.5*period, in the future.
Experimental Setup
The experimental apparatus is relatively simple. There is a cylindrical tank on top of a rectangular base. Two steel rods,acting as vortices, travel out of the base and through the tank. Each rod is attached to a motor in the base. There is some circuitry required to make the motors blink on and off, thus simulating the Aref model. This includes: speed controllers, relays, and a microcontroller. The microcontroller can easily be controlled by a laptop with the proper code to make the motors blink.
into the tank. These rods act as the vortices by spinning. (right) Front view of
the structure. The two rods attach to the motors which are controlled by the
other circuits in the base.
Fig 3: This is a close-up of the base from the side. The blue circuit on the left
is an Arduino microcontroller. The black box at the bottom is a power supply.
The brown circuit above it is one of the relays. The green circuit on the right is
a speed controller.
About the Mixing Group:
Ryan Schilt – Ryan is a senior in Applied Mathematics with and emphasis in Planetary Sciences. Ryan is also participating in the five year bachelor-master program. Future plans for MCTP include investigations into methods of characterizing chaotic behavior within Aref’s model and other three dimensional mixing models. Aside from working for MCTP he programs in actionscript/flash for the Applied Math SAGA program.
Joe Adams – Joe is a fourth-year student in Applied Mathematics with a focus in Mechanical Engineering. He is working towards a BS/MS degree. Some of his future work with MCTP will be extending Aref’s blinking vortex model into three dimensions. He also works with the Applied Math SAGA program using actionscript/flash.
Anna Lieb – Anna is currently a junior Applied Mathematics major. She is interested in physics (especially quantum mechanics) and biology. She hopes someday she can teach people that math is not scary -- its beautiful!
