Video of “Compressible vs incompressible flow and conservation of mass”

Below are GIFs of the compressible and incompressible versions of the “Compressible vs incompressible flow and conservation of mass” demonstration. The full videos are linked from the GIF headings.

Compressible flow (air)

Incompressible flow (water)


An index of all the demonstrations posted on this blog can be found here. Don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (nbkaye@clemson.edu). I also welcome comments (through the comments section or via email) on improving the demonstrations.

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Compressible vs incompressible flow and conservation of mass

This is a really simple demonstration of how conservation of volume can be used for incompressible fluids but not for compressible fluids. The demonstration was suggested by Dr. Baburaj of IIT Madras. I teach in a civil engineering department where practically everything is incomopressible and we mostly talk about conservation of volume. The demonstration below is so simple yet so clear.

Equipment

You will need:

  1. Two identical syringes,
  2. A few feet of clear tubing that fits tightly over the end of each syringe,
  3. Some water, and
  4. Food dye (optional)

Demonstration

Compressible flow

  1. Have one syringe (A) with the plunger fully pushed in and the second plunger (B) fully pulled out.
  2. Connect each end of the tube to the syringes
  3. Slowly press the plunger on syringe (B)

Assuming that the syringe plunger’s are a little stiff you should be able to push the plunger on (B) all the way in before the plunger on (A) is pushed all the way out. Mass is conserved because there are no leaks but volume is not conserved as the plungers move different distances on identical syringes. This works better with stiffer syringe plungers.

Incompressible flow

  1. Have one syringe (A) with the plunger fully pushed in and the second plunger (B) fully pulled out and the syringe full of water.
  2. Fill the tube with water (food dye can help with visualization) and connect the tubes in the same way as for the previous version. This is tricky as you want to ensure that there are no air bubbles in the lines.
  3. Slowly push in the plunger on syringe (B). The plunger in syringe (A) should move out at exactly the same speed. you can show this clearly by having the syringes pointing away from each other with the plunger ends next to each other. As you push one in the other should move right next to it.

Analysis

There is no analysis for this demonstration. The gas is compressible so volume is not conserved whereas the liquid is incompressible so volume is conserved. Analysis of the change in pressure in the compressible case and resulting motion of the plungers is complex as you need to know about the friction in the syringe.

Thanks again to  Dr. Baburaj for suggesting the demonstration.


An index of all the demonstrations posted on this blog can be found here. Don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (nbkaye@clemson.edu). I also welcome comments (through the comments section or via email) on improving the demonstrations.

Video of “‘Rotational buoyancy’ – Hydrostatic pressure in solid body rotation”

Here is a video from the “‘Rotational buoyancy’ – Hydrostatic pressure in solid body rotation” demonstration. The GIF is a little hazy but clearly shows the cork staying centrally located under rotation and the marble being pushed to the end of the jar (and up next to the lid). The full video is here.

Video Apr 13, 3 04 54 PM 00_00_55-00_01_10


An index of all the demonstrations posted on this blog can be found here. Don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (nbkaye@clemson.edu). I also welcome comments (through the comments section or via email) on improving the demonstrations.

‘Rotational buoyancy’ – Hydrostatic pressure in solid body rotation

Solid body rotation of a fluid about a vertical axis results in a horizontal pressure variation which provides the centripetal force required to rotate the fluid particles. The pressure gradient is, therefore, given by

dp/dr=ρw2r                                              (1)

Where ρ is the fluid density, w is the angular velocity and r is the distance from the center of rotation.

This is easily demonstrated by spinning a cup of water and showing the paraboloid surface that forms with the low point at the center of pressure. However, the pressure gradient exists regardless of the free surface and, just as with the non-rotating hydrostatic pressure variation, an immersed object will experience a net force in toward the region of lower pressure. This can be demonstrated using a sealed container

Equipment

  1. A small cork
  2. A marble
  3. A Lazy Susan or some other cheap turntable
  4. A mason jar
  5. Some Velcro strips or something else to secure the Mason jar to the turntable.

Photo Apr 13, 3 04 44 PM

Demonstration

  1. Place the cork and marble in the jar and fill it with water
  2. Seal the jar so that there are no bubbles (or at least no bubbles that are large compared to the size of the cork) and attach it to the turntable so that its long axis is horizontal and it is centered on the turntable (see figure above).
  3. Shake the jar until the marble and the cork are near the center of the jar (this is so that when the marble moves it is clearly due to the rotation of the jar).
  4. Rapidly spin the turntable. The marble should be pushed to one end of the jar while the cork should remain centered.

Analysis

The horizontal hydrostatic pressure gradient (equation above) means that any submerged object will experience a net pressure force acting toward the center of rotation. For a rectangular object of width s in the radial direction and area A normal to the radial direction located a distance r  from the center of rotation, the net pressure force toward the center of rotation is given by

FpAw2((r+s/2)2-(r-s/2)2)/2                      (2)

See figure below.

rotation

Expanding leads to

Fp=ρAw2rs= ρ∀w2r                                          (3)

where is the volume of the object. Therefore, the net force toward the center of rotation is the angular acceleration multiplied by the mass of water displaced by the object. This is directly analogous to the buoyancy force in a stationary fluid. Therefore, if the object displacing the fluid has a lower density than the fluid then the centripetal pressure force will exceed that needed to maintain the angular velocity of the object and it will be pushed toward the center of rotation (as in the cork). If the object is denser than the fluid the centripetal pressure force will not be enough to maintain rotation and the object will move radially outward (note that this is a somewhat simplified linearized analysis and the integration constant is left out of the pressure equation used in (2) but it gets the appropriate result).


An index of all the demonstrations posted on this blog can be found here. Don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (nbkaye@clemson.edu). I also welcome comments (through the comments section or via email) on improving the demonstrations.

Video of “Buoyancy: Throwing rocks from boats (and weighing coins)”

Here is the video of the “Buoyancy: Throwing rocks from boats (and weighing coins)” demonstration. The two GIFs show the dropping of the coins into the cup and then the dumping of them into the water. The full video is linked here.

Dropping the coins into the cup showing that 9 quarters displaces 50 ml of water.

drop

Dumping the coins out showing that, when submerged, they displace a lot less than 50 ml.

dump


An index of all the demonstrations posted on this blog can be found here. Don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (nbkaye@clemson.edu). I also welcome comments (through the comments section or via email) on improving the demonstrations.