Laminar (Un)mixing

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"Unmixing" a mixture is impossible, right? So what's going on here?

Big Ideas: 
  • What is laminar flow?
  • How can we tell if a fluid flow is laminar or turbulent?
  • What is Reynolds Number?

In the video above, it appears that we are mixing together and then separating a uniform mixture of liquids.  According to the second law of thermodynamics (and common sense), this should be impossible -- so, what's going on?
 
Setting up the Demonstration
 
Our "mixing vessel" consists of one stationary outer cylinder, and one rotating inner cylinder.  A high-viscosity liquid (we used glycerol) sits in the space between the two cylinders.  The coloured dots are also glycerol, mixed with food colouring and then injected into the cylinder using a syringe.  In order to "mix", we slowly rotate the inner cylinder for 2-3 full rotations (720 - 1080 degrees), so that the colour appears uniform around the entire cylinder.  To "unmix", we rotate the inner cylinder in the reverse direction back to its starting point.  
 
How does it work?
The reason why our uniform mixture appears to unmix is because it was never mixed (homogeneously) in the first place!  Because of the high viscosity of the glycerin (or corn syrup) and the cylinder's slow rate of rotation, the flow of the liquid is very laminar -- the liquid flows in orderly, parallel layers that slide past each other, with little disruption in between.  This means that the displacement of each layer of liquid depends only on its distance from the moving inner cylinder.  The liquid closest to the rotating cylinder moves the most, while the liquid closest to the stationary cylinder doesn't move at all.  
 
 
 
This laminar flow causes the coloured drops to shear into sheets that wrap all around the cylinder, giving the appearance of a uniform mixture from the side view:
 
Visualization, side view
 
If you look from the top of the cylinder, you can see that our "mixture" is not uniform at all:
 
 Visualization, top view
 
Reversing the rotation causes the sheets of dye to come back together into concentrated drops.
 
How can we tell if flow is laminar?  (What does viscosity have to do with anything?)
One way we can characterize a fluid flow is using Reynolds number, which measures the ratio of inertial forces to viscous forces for a given flow.  Laminar flow occurs at low Reynolds numbers -- viscous forces are dominant, and flow occurs in a smooth, constant motion.  At high Reynolds numbers, inertial forces dominate, which produces unstable, chaotic flow that encourages mixing.  Reynolds number is defined as: 
\textup{Re} = \dfrac{\rho \textup{v}L}{\mu}
 
where \rho is the density of the fluid, v is its velocity, \mu is its viscosity and L is the characteristic dimension -- in our case, the thickness of the ring of fluid. We can compare the Re computed for our system using both glycerol and water:
 
Notes:
  • By convention, fluid velocity was defined as the container radius (.1m) multipled by the angular velocity of the mixer (estimated to be 90°/s)
  • Properties for water and glycerol were taken at STP.

From the table above, we can see that the high viscosity of glycerol (over 1586 times that of water) is the biggest reason why its flow is much more laminar than water -- while glycerol is also denser than water, the difference in density is not nearly as dramatic.  The greater turbulence in the water flow is the reason why this demo doesn't work if you replace the glycerol with water.

Make your own laminar (un)mixer!
The apparatus in the video above was manufactured professionally, and allows us to repeat and reverse our movements very precisely in order to maximize the "unmixing" effect.  However, it is certainly possible to recreate this effect using materials found in the household. Below, we have provided instructions on how to make your very own homemade laminar (un)mixer.
 
Materials:
  • 1 aluminum pop can
  • 1 clear, straight-sided 2L plastic bottle
  • 2 popsicle sticks
  • 1 styrofoam plate
  • 500 - 750 mL glycerin or corn syrup*
  • Sand (optional)
  • Food colouring
  • Opaque, light-coloured tape (white duct tape or electrical tape works well)
  • Scissors/craft knife
  • Pen or pencil
  • Hot glue gun
  • Eyedroppers or pipets (1 for each colour)
*Corn syrup is less expensive and more readily available than glycerin, but will spoil if left out.  Glycerin can be found in small quantities at most pharmacies and is more expensive, but can be decanted and reused.
 
Directions:
 
 
  1. Wrap the sides of the pop can with tape. Cut the top off the plastic bottle, leaving an open cylindrical container 15cm high.
  2. Cut out a ring shape from the styrofoam plate, with the outer diameter the same as the plastic bottle (~10.5cm) and the inner diameter the same as the pop can (~6cm).
  3. Trim both popsicle sticks down to the diameter of the bottle, and glue together to form a "+" shape.
  4. Slide the styrofoam ring onto the pop can and position it 1 inch from the bottom.  Check that it fits in the bottle and seal with hot glue. Glue the popsicle sticks onto the top of the pop can.  Check to make the entire piece still fits in the bottle and can freely rotate (trim pieces if necessary).
  5. Pour the glycerin or corn syrup into the plastic bottle.  It should reach approximately 1/3 to half-way to the top.
  6. Fill the pop can with water or sand to weigh it down, and carefully place it into the liquid, styrofoam end first.  Make sure the can is not tilted to one side, and add more liquid if desired.
  7. For the coloured liquid, mix 2 drops of food colouring for every 1 tbsp of glycerin or corn syrup, and "inject" it into the container using a pipet or eyedropper.

Notes:

  • For a good result, ensure that the can is properly aligned with the bottle at all times while mixing.
  • You may need to experiment to find an optimal turning rate for the inner cylinder.  The key is to turn slowly enough to take care in aligning the can, but not too slowly, otherwise the food colouring will begin to diffuse into the surrounding glycerol, which produces an less dramatic final effect.  (You may have noticed that the latter occurred in our video above.) 
  • If left out for a long enough time (several days) the food colouring will eventually diffuse completely into the surrounding liquid and leave you with a lightly coloured container (a real uniform mixture) of glycerin or corn syrup.
  • If you want to immediately repeat the demonstration using the same liquid, remove the pop can from the container and stir the liquid vigorously -- this creates some turbulence and speeds up the diffusion process.
 
Top view of homemade "(un)mixer" with blue glycerin, at (L-R) 0°, 360°, 720°, 900°, 0°
 

 Red, green and blue glycerin "sheets"

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