How Does Vorticity Relate to Blood Flow Dynamics?

[ELLE_AW]

Homework Statement

What is the difference between vorticity vs angular velocity? I can see the equations, but conceptually I still don't really understand the difference. Also, what is curl of velocity and how does one obtain that?

Relevant Equations

angular velocity = 2 pie rad/s,
vorticity = curl of velocity (nabla x velocity)

This is not homework. I'm studying fluid mechanics/dynamics in the heart/blood vessels and I just want to understand this, so I can have a better appreciation for it's clinical relevance. I'm more of biology/biochem type of person so this has been a bit of challenge. I have basic physics course in undergrad, but it's been a while and we certainly did not discuss curl of velocity and/or vorticity at that time.

Thank you in advance for your help.

 

[Steve4Physics]

Homework Statement:: What is the difference between vorticity vs angular velocity? I can see the equations, but conceptually I still don't really understand the difference. Also, what is curl of velocity and how does one obtain that?
Relevant Equations:: angular velocity = 2 pie rad/s,
vorticity = curl of velocity (nabla x velocity)

Well, I’ll give you my interpretation, based on the assumption that you don’t need to know the maths (vector-calculus) and just want the underlying concepts. I expect other contributors will address any inadequacies in my explanation!

Drop a leaf in (moving) water. The leaf moves with the water. A leaf (at instantaneous position P) could be moving along, rotating, or both. Another leaf (at a different instantaneous position Q) could be moving differently.

Each point on the water-surface has a value of ‘vorticity’. P has a vorticity, Q has a (probably different) value of vorticity. The value of vorticity is (for mathematical reasons) twice the angular velocity of the leaf at that point (i.e. twice the angular velocity of the water about the point)

We really need to think in 3D, but the principle is the same. Imagine watching a tiny sphere, with markings so you can see it rotate, carried along in, say, blood. The vorticity at a point in the blood is twice the angular velocity of the blood at that point.

If the flow-pattern has reached a steady-state, vorticity at all points will be constant. Otherwise (e.g. during turbulent flow) the vorticity at a point will not be constant over time.

Notes:

Your statement: “angular velocity = 2 pie rad/s” is (very) incorrect!
It would mean angular velocity is always about 6.28 rad/s, which makes no sense.
Also, ‘pie’ should be ‘pi’ (or even better, π).

If you want a simplified definition:
Angular speed = number of radians rotated per second (= angle/time)
If the period of rotation is T seconds, then angular speed = 2π/T rad/s.

Reference to angular velocity requires additional information about the direction of the axis of rotation, because velocities are vectors.

‘Curl’ is a mathematical operation involving partial derivatives and vectors It let's you calculate the vorticity at a point, providing you have a formula for the velocity-vector as a function of position. You can ask for further details if you are familiar with partial derivatives and vectors expressed in component-form; otherwise I don’t know how to explain it.

 

[Charles Link]

I can add something to the above that is hopefully kind of simple : By Stokes' theorem $\int \nabla \times \vec{v} \cdot dS=\oint \vec{v} \cdot dl$, so that if the curl is some constant, the velocity vector integrated around a loop or circle will be proportional to that constant, (and also proportional to the circumference of the loop). This says that when the curl is non-zero, you have a swirling going on around any loop about that region.