What is the mathematical expression for fluid rotation in 3D?

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In summary, the conversation discusses rigid and non-rigid rotations in n dimensions and how they apply to a fluid sphere. The concept of non-rigid rotation is clarified and potential scenarios for non-rigid rotations are discussed. The possibility of a single expression with a viscosity term and the role of forces such as gravity and surface tension are also mentioned. The conversation then shifts to discussing a specific example involving non-interacting massive particles and the idea of combining position and momentum vectors for a six-dimensional rotation.
  • #1
ImaLooser
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I can figure out how to do a rigid rotation in n dimensions. Next I want to look at non-rigid rotations.

Lets says that we have a rotation sphere of fluid in 3D. How mathematically does that rotate?

It is important how viscous the fluid is. If it is infinite viscous them it would essentially be a solid, right? So it seems that there could be a single expression with a viscosity term.

A fluid sphere with zero viscosity would be a superfluid, I think, and hence excluded. Or perhaps not.
 
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  • #2
Higher dimensional rotation of

Rotations usually take place in a coordinate system with the center of the sphere at the origin, to avoid unnecessary mess.

Now consider a rotation in four dimensions. Such a rotation has two axis. What if one of the axis does not go through the origin? I think that a sphere would be non-physical and the result would be some sort of ellipsoid instead, but at this point I don't even know what questions to ask. I would guess that a rigid rotation would be impossible, but a fluid rotation would be.

I guess it would depend on the forces that hold the ellipsoid together. I think that gravity alone would not tend to make such a structure, nor would surface tension. It is hard to thnk of what would naturally do this. Maybe a combination of two forces, like gravity and an extremely strong magnetic field (like 10^14 gauss, maybe.) But by this point I'm walking on air like Wile E. Coyote. Anybody have any ideas.
 
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  • #3
What do you mean by "non-rigid rotation"? Any motion (without translation so the center of the object does not move) can be interpreted as a rotation and stretches and so as a "non-rigid rotation".
 
  • #4
HallsofIvy said:
What do you mean by "non-rigid rotation"? Any motion (without translation so the center of the object does not move) can be interpreted as a rotation and stretches and so as a "non-rigid rotation".
Quite so. So there are quite a few situations where it would arise. Let's look at a specific simple example. Let's assume that we have CNIMP (completely non-interacting massive particles). Initially they are evenly distributed through the galaxy. They will be attracted by massive bodies and pass through in cometary or elliptical orbits. It seems that there should be some simple matrix for the case of a single star, though I'm too dumb to figure it out. My best guess is that there is some way to combine position and momentum vectors to get a six-dimensional sphere, then have a rigid rotation of that. Or something like that.
 
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  • #5
I don't know any fluid dynamics but mathematically a vector field that is everywhere orthogonal to the lines from the origin would integrate to a flow that is an instantaneous rotation at every point.
 

1. What is the mathematical expression for fluid rotation in 3D?

The mathematical expression for fluid rotation in 3D is given by the curl of the velocity vector field, which is represented by the symbol ∇ x u, where ∇ is the del operator and u is the velocity vector field.

2. How is fluid rotation related to the concept of vorticity?

Fluid rotation is directly related to the concept of vorticity, which is a measure of the local spinning motion of fluid particles. The vorticity vector is equal to twice the curl of the velocity vector field, or 2∇ x u.

3. What is the physical significance of the curl of the velocity vector field?

The curl of the velocity vector field represents the tendency of fluid particles to rotate around a point in the fluid. It is a measure of the local angular velocity of the fluid and can help determine the presence and strength of vortices in the flow.

4. How does the mathematical expression for fluid rotation differ from 2D to 3D?

In 2D, the mathematical expression for fluid rotation can be represented by a single scalar value known as the vorticity. However, in 3D, the expression becomes a vector quantity due to the added complexity of fluid motion in three dimensions.

5. Can the mathematical expression for fluid rotation be applied to both laminar and turbulent flows?

Yes, the mathematical expression for fluid rotation can be applied to both laminar and turbulent flows. In laminar flows, the vorticity is typically concentrated in specific regions, while in turbulent flows, it is distributed throughout the flow field.

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