Understanding the Nature of Free Vortices

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In summary, a free vortex is a circular flow of fluid in which the angular momentum of the water causes a centrifugal force that pushes the water outwards, creating a cavity in the middle. The velocity times the radius is a constant, resulting in an increase in centrifugal force towards the center. The shape of the surface can be determined using differential equations, with the speed of the water increasing as it closes in towards the center of the vortex.
  • #1
powerball
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Iam having some trouble understanding the nature of free vortexs, with the internet and library being extremely scarce in answers. My understanding is that the angular momentum of the water causes a centripetal force to push the water outwards leaving a cavity in the middle. I understand that at any point the angular momentum times the radius is a constant. Can anyone give me a more indepth explanation on how vortices work? I need equations to work out circulation, vorticity and possibly the height and radius of a vortex.
 
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  • #2
can anyone answer this instead? why is there greater centripetal force at the bottom of the beaker than the top
 
  • #3
Originally posted by powerball
My understanding is that the angular momentum of the water causes a centripetal force to push the water outwards
It's rather a centrifugal force. From latin fugare=to flee. Because the water flees from the center.
I understand that at any point the angular momentum times the radius is a constant.
No. The velocity times the radius is a constant. In other words, the angular momentum is a constant.
Can anyone give me a more indepth explanation on how vortices work?
Yes.
The idea of a water surface is, that at any point on the surface the total force on a particle is normal to the surface. With no motion, there's just gravity (vertical), so the surface is horizontal. With rotation, you get an extra centrifugal force [tex]F_{cf}=\frac{mv^2}{r}[/tex]acting horizontally. The surface will be normal to the vector sum of these 2 forces. Now, since [tex]vr = c[/tex] you get [tex]v=\frac{c}{r}[/tex] and thus [tex]F_{cf}=\frac{mc^2}{r^3}[/tex]. Thus, the centrifugal force will increase strongly towards the center, making the surface very steep.

Note: This holds only for values of r above a certain limit. Below that, angular momentum and energy will be consumed by turbulence, the water starts behaving erratic, and <poetry on> your unlucky ship is destroyed in the great Maelstrom <poetry off>.

The next step calls for differential equations to find the shape of the surface. Is that what you want to do?
 
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  • #4
if you could guide me on the differentiation needed to work out the shape of the surface that would be helpful
 
  • #5
Well if a point on the surface has coordinates (r,h) then the tangent vector is (dr,dh). The force vector is (Fcf,-mg).

Force being normal to tangent means
[tex]
dr \cdot F_{cf} - dh \cdot mg = 0.
[/tex]
Thus,
[tex]
dr \frac{mc^2}{r^3} - dh \cdot mg = 0
[/tex]
or
[tex]
\frac{dh}{dr}=\frac{c^2}{gr^3}.
[/tex]
Can you integrate that to get h(r)?
 
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  • #6
Thanks very much i can integrate from there. Just wondering in a vortex is the speed of the water rotating the same at any depth in the vortex?
 
  • #7
No. The idea is that it speeds up as it closes in. Did you read my post? I said vr = c. Yielding
[tex]
v = \frac{c}{r}
[/tex]
 

FAQ: Understanding the Nature of Free Vortices

1. What is a free vortex?

A free vortex is a type of fluid motion in which the fluid particles rotate around a central axis without any external forces acting on it. In other words, it is a self-sustaining rotational flow.

2. How is a free vortex different from other types of vortices?

A free vortex is different from other types of vortices because it does not require any external forces to maintain its rotation. Other types of vortices, such as forced vortices, are created and maintained by external forces.

3. What factors affect the behavior and stability of free vortices?

The behavior and stability of free vortices can be affected by factors such as the shape and size of the vortex, the density and viscosity of the fluid, and the presence of any external forces or disturbances.

4. How are free vortices used in real-world applications?

Free vortices have various applications in engineering and science, such as in fluid dynamics, aerodynamics, and meteorology. They are also used in the design of turbines and pumps, and in studying the behavior of ocean currents and weather patterns.

5. Can free vortices exist in all types of fluids?

Yes, free vortices can exist in all types of fluids, including liquids, gases, and even plasmas. The behavior and characteristics of the vortex may vary depending on the properties of the fluid, but the basic concept remains the same.

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