Vorticity is equivalent to angular momentum?

[hanson]

Hi all.
In Fluid dynamics, is vorticity equiavlent to angular momentum?
It seems that vorticity is twice of the rate of rotation of a fluid element at a point, and angular momentum is the density times the rate of rotation of a fluid element, so they just differ by some constant?

But it is a bit strange to me...
I know the the rate of rotation of a fluid element is proportional to shear stress in fluid dynamics. So shear stress gives rise to rate of rotation of a fluid element and hence angular momentum of the element. But from physics, the presence of shear stress (think of force) shall give rise to "rate of change of angular momentum", isn't it?

So...which is the correct one? Shear stress just give angular momentum or rate of change of angular momentum?

Please kindly explain.

 

[hanson]

Can someone help?

 

[Cyrus]

Vorticity is just twice the angular velocity.

 

[olgranpappy]

Vorticity is just twice the angular velocity.

In general the "vorticity" (\vec \omega) is given by
<br /> \vec \omega = \nabla \times \vec v<br />
where v is the fluid velocity.

*If* the fluid velocity happens to be given by the expression
<br /> \vec v = \vec \Omega \times \vec v<br />
for some *constant* Omega, then
<br /> \vec \omega = 2\vec \Omega<br />
(cf. Landau and Lif****z volume 6 Eq. 8.2 and text following.)

 

[Cyrus]

Vorticity is defined as twice the rotation vector. I don't understand your post.

 

[olgranpappy]

did you cf. the reference I gave?

 

[olgranpappy]

Vorticity is defined as twice the rotation vector. I don't understand your post.

Furthermore, you seem to be posting about a rigid body for which there can be a single "rotation vector." The OP asked about a fluid (for which rigid rotation is a very special case.)

 

[Cyrus]

the rotation vector is given by omega. \omega = \frac{1}{2} \nabla \times V

A fluid element has a rotation vector associated with it. I am not making any claims to rigid bodies. I don't know what you mean by c.f. your references.

 

[hanson]

thanks for all the replies.
I know that vorticity is twice the rate of rotation of infinitesimal fluid element at a point.
But I would like to know if the vorticity is related to the angular momentum of that fluid element as well. Since angular momentum is related to angular velocity of the element by the moment of inertia.

If vorticity is related to angular momentum, then vorticity shall be conserved when there is no net torque acting on the fluid element, right?

So, here I consider a Couette Flow (a viscous shear flow). Picking an element from the flow, there would be shear stress of equal mgnitude but different direction acting the upper and lower surfaces of the element. So, there shall be a net torque acting on the element, right? So, the angular momentum of the element shall not be conserved in this case?

However, in a Couette flow, the vorticity is constant everywhere. If we consider the vorticity of an element, it is still constant as it flows. So, why the shear stresses on the upper and lower surface of the element do not change the angular momentum (or vorticity) of the fluid element?

 

[olgranpappy]

the rotation vector is given by omega. \omega = \frac{1}{2} \nabla \times V

um... okay. So, you have defined the rotation vector as half the vorticity. Fair enough.

 

[olgranpappy]

thanks for all the replies.
I know that vorticity is twice the rate of rotation of infinitesimal fluid element at a point.
But I would like to know if the vorticity is related to the angular momentum of that fluid element as well. Since angular momentum is related to angular velocity of the element by the moment of inertia.

If vorticity is related to angular momentum, then vorticity shall be conserved when there is no net torque acting on the fluid element, right?

I don't know about that... maybe not. I don't think that vorticity is necessarily proportional to angular momentum... but really I can't remember.

I do seem to recall that the line integral of the velocity along a closed loop which itself travels along streamlines is conserved... so for a small loop in this sense the vorticity remains constant as the fluid particle travels along a streamline... but perhaps this is a red herring.

Maybe Landau has something useful. Section 18 deal with Couette flow, but just glancing at it I did not find anything too helpful.

Good luck.

 

[Cyrus]

um... okay. So, you have defined the rotation vector as half the vorticity. Fair enough.

Not I, my fluid mechanics text and aerodynamics text have.