# Radial flow, Bernoulli, and levitating an index card

This experiment was suggested by Dr. John Foss from Michigan State University. He presents a more detailed write up, along with a discussion of Reynolds number effects in Experiments in Fluid Mechanics, R.A. Granger, Ed. Holt, Rienhart and Winston, (1988). The basic idea is that a radial outflow between parallel plates has a velocity that decreases with increasing radial distance. Therefore, if a high Reynolds number situation exists such that the shearing effects are small and a nominal balance exists between the pressure gradient and the acceleration, the pressure will increase with radial distance. If the outflow boundary condition is atmospheric pressure then the pressure between the parallel plates must be a vacuum pressure. This is surprisingly easy to do with very simple equipment.

Equipment:

There are a couple of methods for doing this. Two are described below. The index card version is the easiest to do in large numbers.

• Foam Plate Version
• 1 Foam plate
• 1 Straight pin
• A can of compressed air
• Tape
• Index Card Version
• 1 straw
• 1 3”x5” index card
• 1 straight pin
• Tape
• Adhesive putty (optional, may help hold the straw in the spool)

Procedure:

• Foam Plate Version
1. Place the straight pin through approximately the center of the foam plate. Tape the pin to the bottom of plate to stabilize it and to keep it perpendicular to the plate.
2. Have someone hold the plate so that the pin is horizontal
3. Hold the spool of thread to where it covers most of the pin but is not touching the plate
4. Take the can of compressed air and aim it through the spool of thread. The nozzle does not need to be placed directly in the spool. Just aim it so that the air will flow through the spool.
5. When the air is released from the can the plate should move towards the spool of thread and should stay there without support as long as the compressed air can is blowing

*CAUTION:  Do not aim the compressed air can downward. The air will become very cold and could possibly burn someone

• Index Card Version
1. Place the straight pin through approximately the center of the index card. Tape the pin to the bottom of the card to stabilize it and keep it perpendicular to the card
2. Place the index card on the table with the pin pointing straight up
3. Hold the spool of thread directly above the pin but not touching the index card
4. Insert the straw into the spool. The adhesive putty can be used to attach the straw to the spool so that you only need one hand.
5. Blow through the straw and, if the spool is close enough to the card, you should be able to lift the card off the table. The card should stay as long as there is a steady flow of air

** Trial and error may need to be used to in both experiments to determine the gap distance needed between the spool and card/plate in order to pick the object up.

Foam Plate Version                             Index Card Version

Analysis

Consider an incompressible fluid flowing horizontally and radially out from a point source between two parallel plates separated by a distance T. At any arbitrary radial distance r from the source the area of the flow is

A= 2π r T

(see diagram below). For a constant volume flow rate Q the velocity is given by

U(r)=Q/2 πr T

Writing Bernoulli’s equation from r to the outlet at a radial distance R and taking the outlet pressure to be atmospheric leads to

Pr = (Q/2 π T)2 (R-2-r-2)

Therefore, given that r<R, Pr<0 and the plate/card will be pushed toward the spool. However, for this to work the gap width needs to be small. If T is too large the pressure vacuum pressure over the card will no be enough to overcome the weight of the card.

In reality life is a little more complex and a more detailed analysis of this problem is given by Prof. Foss in Experiments in Fluid Mechanics, R.A. Granger, Ed. Holt, Rienhart and Winston, (1988).

Thanks to John Foss for suggesting the demonstration and helping with the write up. Thanks to Meredith and Alex for testing the procedure, putting together pictures, and making the videos. Videos to follow soon.

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.