Divergence of vorticity vector is zero--intuition behind it

[Urmi Roy]

So mathematically I understand that divergence of curl of something is zero.

However, talking specifically about vorticity, this is what it seems to imply to me:

When there is vorticity in a fluid, the tiny particles spin around their own axes, so a net circulation is formed around the surface of a vortex tube (circulation). So circulation is like the vector sum of all the vorticity,the macroscopic effect of tiny particles spinning. Its the resulting 'loop' of velocities.
So zero divergence of vorticity means it doesn't have a source/sink any where in the fluid.

However I don't completely believe it, because all the spinning motion had to start somewhere...so it doesn't make sense that there is no source of vorticity!

Please tell me which part of my understanding is wrong.

Thanks,
Urmi

 

[mfb]

"It had to start somewhere" is a statement about time, not about space. And you cannot have vorticity at a single point only, that does not work. Every tiny vortex you generate will have a (variable) vorticity in some volume, with zero divergence everywhere.

 

[Urmi Roy]

Thanks for the reply mfb. However I still don't understand why the divergence is zero everywhere.
Also, when you say you can't have vorticity at a single point, does that mean that my understanding that vorticity is basically the spin of fluid particles about their own axis (like Earth spinning about its axis) is wrong?

 

[mfb]

Vorticity with individual particles does not work. The concept assumes the material is continuous and all relevant properties are differentiable.

 

[Urmi Roy]

Hmmm, ok I think I'm getting this point.
About what you mentioned regarding the zero divergence at every point in the field, do you mean that AFTER vorticity is created (e.g. due to viscosity) and we have a steady state, then none of the vorticity can vanish on its own?

 

[mfb]

It can vanish in time, but not in the way you would get a divergence of it. This is as impossible as a number being negative and positive at the same time (and no, 0 is neither). There is simply no possible motion that would give a divercence in the vorticity. No matter how the flow looks like.

 

[Urmi Roy]

I'm not sure I understand your argument, and how you're comparing vorticity with a number being +ve and -ve at the same time. However I think the take away is that if we assume no viscous forces are acting on the vortices anymore, any further vorticity can't be generated or destroyed because the basic reason for creation/destruction of vortices i.e. viscosity doesn't act anymore.

 

[mfb]

and how you're comparing vorticity with a number being +ve and -ve at the same time.

I don't compare the values, I compare the impossibility to happen. No matter how hard you try, you can never find a solution for either of the problems.

However I think the take away is that if we assume no viscous forces are acting on the vortices anymore, any further vorticity can't be generated or destroyed because the basic reason for creation/destruction of vortices i.e. viscosity doesn't act anymore.

This is not a question of viscosity. Even if you have perfect control of the flow velocity and direction at every point and can set it to arbitrary values, you still cannot get divergence of vorticity.