# Blood pressure and Bernoulli

I first saw this demonstration as a grad student and it has taken me years to get around to building my own demonstration rig. The demonstration illustrates the same Bernoulli behavior as the Ping-Pong ball in a funnel demonstration but requires a lot more equipment.

Equipment

1. A bucket full of water
2. A sump pump
3. A bike pump to pressurize the cylinder
4. A sealed cylinder with:
1. A flexible rubber hose passing down the middle of the cylinder with a connection to the sump pump at one end and an outlet return to the bucket at the other end.
2. A valve to connect to the bike pump to pressurize the cylinder

Equipment

Details on the connections, air seal, inner tube, and valve

The demonstration rig fully assembled

Demonstration

The demonstration is very straight forward. I often try and get a bunch of students involved, one to pressurize the cylinder with the bike pump and one to hold the drain tube. I have a student hold the drain tube because you sometimes get quite a reaction from them when it suddenly starts to pulse.

1. Ensure that the cylinder is not pressurized
2. Connect the sump pump to the rubber inner tube and hold the outlet so that it drains back into the bucket
3. Turn on the sump pump. You should get a steady flow through the inner tube.
4. Connect the bike pump to the valve and start to pressurize the cylinder.  Eventually the pressure in the cylinder will be large enough that the flow through the inner tube starts to pulse. If you have a large sump pump, the pulsing can be quite violent.
5. Release the pressure on the cylinder and the pulsing should stop.

Analysis

The analysis is fairly qualitative. Draw a schematic diagram of the setup as shown below. When the cylinder is not pressurized then the tube has a constant cross sectional area and the flow is steady.

When you pressurize the cylinder you begin to compress the rubber tube. Therefore, there is a section of tube where the cross sectional area is reduced. Conservation of volume tells you that the velocity in the contraction must be higher than in the uncompressed section of the tube. Bernoulli tells you that the pressure in the contraction must be lower than the pressure in the uncompressed section of the tube because of the increased velocity (the tube is horizontal so the z terms cancel). Therefore, if a section of tube begins to compress, it will continue to compress as the water pressure in the tube drops in the contraction.

Eventually, the contracted section will completely collapse blocking the flow. At this point the pressure in the contraction is the pump shut-off head. This pressure is much higher than the cylinder pressure so the blockage is forced open again. This cycle continues as long as the pressure remains high enough in the cylinder. As such, you get a fairly dramatic pulsing flow out of the end of the tube.

This is somewhat analogous (though not a direct analogy) to the measurement of blood pressure. The cuff that is placed on your arm compresses the veins. You can sometimes feel the pulsing while the cuff is pressurized.

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.