# Why don't we talk about angular momentum at all in fluid mechanics?

People usually talk about similar (or maybe not?) things like vorticity or enstrophy in fluid mechanics, but no one talks about angular momentum, why?

• Symmetry of the stress tensor takes care of it, most of the time – Hydro Guy Jul 22 '14 at 21:32
• A quick search for 'fluid angular momentum' suggests that people do talk about the angular momentum of fluid (well, discrete parcels of fluid at least). – Kyle Oman Jul 22 '14 at 21:34
• @Kyle Well, that's true but when you look at fluid textbooks no one talks about angular momentum... – physixfan Jul 22 '14 at 21:39
• Batchelor discusses angular momentum in fluids right on the first chapter, have you read it? – Hydro Guy Jul 22 '14 at 21:45
• @user23873 I just looked it up and it also didn't talk about angular momentum in detail.. – physixfan Jul 22 '14 at 21:52

We don't need to talk about angular momentum because the conservation law is summed up by vorticity. Consider the vorticity equation (in the context of a rotating frame as well): $$\frac{D\boldsymbol\omega}{Dt}=\boldsymbol\omega\cdot\nabla\mathbf u$$ (ignoring all other terms that are normally contained in this term). If we take the coordinate system where $s$ is along the vortex line, then the component of this gives $$\frac{D\omega_s}{Dt}=\omega\frac{\partial u_s}{\partial s}$$ This shows that the vorticity along $s$ changes due to the stretching of the vortex lines, which is principle of angular momentum conservation.

A fluid is modelled as a vector field and therefore we use vorticity to describe its spinning motion. Angular momentum is more often used for a single object or particle, but not so often for a vector field (even though it is still applicable in principle). For a fluid in general, vorticity is twice the mean angular velocity and this fact to me makes it less useful as a quantity when modelling fluids.

• But I can calculate the angular momentum of any subset of the vector field by integrating over it. If I do it for the whole field, I get its total angular momentum. It's not like it's undefined... – Kyle Oman Jul 22 '14 at 21:33
• @Kyle that is true but it seems to me it wouldn't be as useful as vorticity. Where would you use angular momentum for example? – Constandinos Damalas Jul 22 '14 at 21:43
• @Kyle I have made an edit which hopefully answers (at least partially?) your question. – Constandinos Damalas Jul 22 '14 at 21:55
• I wouldn't, I would use vorticity ;) I was more just pointing out that angular momentum is well defined for a vector field (well, you'd also need an associated scalar (density) field). If you desired, you could formulate a constraint on the mechanics of a dissipationless flow based on the conservation of angular momentum. Really I'm just quibbling that your answer reads as "we can't use angular momentum for vector fields" instead of "there is a more useful alternative in this case". – Kyle Oman Jul 22 '14 at 22:19
• To paraphrase something the mods keep saying: comments aren't meant to be permanent, can be and often are deleted, etc. -> make an edit! – Kyle Oman Jul 22 '14 at 22:26

There is a paper that discusses the delicate (and often superficially brushed off) issue of diffusion of angular momentum in the Naiver-Stokes equation, the stress tensor, and its symmetries by Berdahl and Strang titled: The Behavior of a Vorticity-Influenced Asymmetric Stress Tensor in Fluid Flow. Here is a link: http://www.dtic.mil/dtic/tr/fulltext/u2/a181244.pdf .

Consider an incompressible, isotropic, Newtonian, viscous fluid that is truly continuous (i.e. it is not composed of discrete particles that have some non-zero rotational inertia) in a large cylindrical drum (with full slip boundary conditions in the fluid state) and initially frozen solid. Spin the drum at some angular speed along its center z-axis, then, at time t=0, blast the solid with a burst of radiation that instantaneously melts it into its fluid state and raises it to an arbitrarily high temperature that is spatially uniform. The initial condition for this flow has zero shear strain rate – but everywhere has a constant vorticity equal to twice the rotation rate.

Rotational motion does not magically shut off diffusion along radial lines. Since the speed in the theta direction is larger at r + dr than it is at r, this velocity in the theta direction will initially diffuse radially inward and “the band at r + dr” will exert a viscous torque on “the band at r” and cause it to experience an angular acceleration in the theta direction. Angular momentum is conserved as it initially diffuses radially inward – not because the stress tensor is symmetric.

In fact, if the stress tensor were symmetric, this would imply that e.g. a shear flow v = (Cz,0,0) would not only have a shear stress exerted on the x-y plane in the positive x direction, but it would also have a shear stress on the z-y plane in the positive z direction - without any physical mechanism to account for this.

Since this is a truly continuous fluid, the inwards diffusion of angular momentum leads to an infinite angular speed at r = 0 for t > 0. Roughly, the viscous torque on a cylindrical volume element initially scales as r-squared and is equal to the rotational inertia of the axially centered fluid element (which scales as the density times r to the 4th power) times the angular acceleration which must then blow up as 1/r-squared. If such a fluid could be created in the physical world, angular speed would in fact blow up at the origin and this model would then be correct.

In the real world however, fluids are composed of particles that have non-zero radii and non-zero rotational inertias that prevent viscous torques from creating infinite angular speeds (there is an additional length scale that is relevant as well – the average inter-particle distance…). Consider replacing the above fluid with radon atoms. The same initial velocity gradient would cause the radon spheres to acquire an average spin. As these atoms collide, these spins would then average back to zero and indirectly mediate a transition of angular momentum of the fluid inward. If the relaxation rate (from intrinsic to extrinsic angular momentum) is sufficiently high, one can get away with symmetrizing the extrinsic stress tensor, avoiding the troublesome infinities, and allowing the intrinsic stress tensor to indirectly approximate the evolution of the fluid flow.

The continual relaxation of the spins in the case of the pure shear flow also indirectly accounts for the additional perpendicular shear stress - which resolves the paradox mentioned above.

The total angular momentum of a continuum is the vector sum of net angular momenta of the particles it is comprised of. The net angular momentum of a particle accounts for its spin as well as its translational motion about the point about which angular momentum is to be calculated (moment of its momentum).

Vorticity is a kinematic quantity measuring (half of) the angular velocity of a point relative to a different point in the same continuum. As a result, vorticity equation is like a statement of conservation of moment of momentum.

However, the angular momentum corresponding to the spin of each particle is assumed to be 0 in most of the continuum treatment. This assumption follows the random distribution of molecular angular momenta (rather all vector quantities) inside the continuum. Very closely related to the assumption of kinetic theory of gases.

Angular momentum is about a fixed point (you decide it) that is the same for every lump of fluid in the domain. Vorticity is (1/2) the angular momentum of the fluid about EACH fluid lump's OWN center of gravity.

This is why a potential vortex has zero vorticity (except at the origin). But lots of angular momentum at each point in the flow (especially about the obvious axis of the center of the vortex, but also about any other axis).

They are related, but definitely not the same (or just off by a factor of 1/2). Vorticity is more useful. In physics - utility is always the final answer. Engineers use angular momentum control volume equations (when they are useful, like for turbine problems). An undergrad engineering fluids book (any) will have a section in the control volume chapter on the angular momentum equations.