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In this configuration the fluid is not going around in a circle but there is still an acceleration around a focal point. I have looked for math behind how to calculate this as there are certainly factors different from a normal gyroscope set up but I can't find anything (perhaps because there is no effect).

Joe
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I assume the fluid is flowing through a conduit (a pipe of some sort).

There is a form of mass flow metering that uses a curved pipe, the pipe is constructed such that it can be driven to a particular vibration frequency, and sensors are in place that measure how the response of the pipe (to being driven) is different when fluid is flowing through the pipe.

The animations in the wikipedia article about Mass flow metering were created by me.


The precise shape of the pipe is not important. (Mass flow meters do have a characteristic design, but the design is engineering driven, not physics constrained.)

The design of the curved pipe is such that fluid flows towards a section of pipe that is free to vibrate, and then the pipe turns back.

Since the pipe turns back the fluid is moving in a plane, around some central point. For simplicity you can think of that as motion around the center of mass of that fluid. Again, the shape of the motion in that plane is not particularly important, the important feature is that the fluid is being turned back.

With the fluid in the pipe not flowing:
Then the situation is symmetric, and the vibration will be the same for the section of pipe away from the hinge point and the section towards the hinge point.

With the fluid in the pipe flowing:
Now the situation is not symmetric. In one section of the pipe fluid is flowing away from the hinge point, and in the return section fluid is flowing towards the hinge point. The fluid-flowing-away-from-the-hinge section has to push the fluid to make the fluid co-accelerate with the driven vibration. So that section will tend to lag behind. The fluid-flowing-back-to-the-hinge has to decelerate the fluid that is coming into that section. So that section will tend to pull ahead of the overall vibration.

The combined effect is a twisting effect.

So the overall operating principle of a mass flow meter is as follows.

(1) There is a starting rotating motion
(2) A second motion, around a perpendicular axis, is added
(3) The combination leads to motion around a third axis that is perpendicular to (1) and (2)

Connection with gyroscopic precession:
This effect is key to gyroscopic precession.

There is a 2012 answer by me on explanation of gyroscopic precession, using the above as the basis of the explanation.


So: no connection with parabola.

There is a connection though:
Thinking about the operating principle of Mass flow metering helps to understand gyroscopic precession.

Cleonis
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