# Navier Stokes: what about angular momentum?

I play with CFD for a while, and suddenly, a transcendantal question raises: :-) Navier Stokes is basically Newton applied on a continuum in Eulerian. For solids, we would consider linear, but also angular momentum. Why don't we have to do that for fluids ?

Conversely, you can take the curl of Navier Stokes and have an equation expressed in vorticity, which looks like our angular momentum world. Does it mean that the equation with velocity somehow embed the both kind of momentum, and they are totally correlated for fluids ?

But how it's not the same for solids ? i.e., where is the intrinsic difference that makes it different degrees of freedom in one case and equivalent in the second ?

## 2 Answers

The basic answer was given here: In a fluid, why are the shear stresses $\tau_{xy}$ and $\tau_{yx}$ equal?. Angular momentum conservation follows from linear momentum conservation (expressed by the Euler/Navier-Stokes equation) combined with the symmetry of the stress tensor.

Momentum conservation is the equation $$\frac{\partial}{\partial t}\pi_i + \nabla_j\tau_{ij} = 0$$ where $\pi_i=\rho v_i$ is the momentum density. Using $$\tau_{ij} = P\delta_{ij}+\rho v_iv_j$$ this equation is equivalent to the Euler equation, and including dissipative stresses gives the Navier-Stokes equation.

The density of angular momentum (about the origin) is $l_i=\epsilon_{ijk}x_j\pi_k$ and $l_i$ is conserved if $\epsilon_{ijk}\tau_{jk}=0$. We get $$\frac{\partial}{\partial t}l_i + \nabla_j m_{ij} = 0$$ where $m_{ij}=\epsilon_{ikl}x_k\tau_{lj}$ is the angular momentum flux.

Of course, the angular momentum of the fluid can change because of external torques, and the angular momentum of a fluid cell can change because of surface stresses. (That is, I can integrate the conservation law over a volume inside the fluid, and the angular momentum of the fluid volume changes because of surface torques. Of course, the total angular momentum of the fluid is conserved.)

• Do I interpret correctly the symmetry if I say that for a field made of solids particles (e.g. some asteroid or ice belt), any angular momentum induced on a particle due to the interaction with its neighborhood (e.g., chock) occurs with the symmetrical momentum on the other particle ? Feb 11, 2016 at 21:41

For the exact equation, that's almost true (one should look at a derivation of the NS equation from the more fundamental Boltzmann equation, where the conservation laws are build into the particle-particle interactions -- but the closure in the approximation of the continuity and NS equations might spoil this if not done correctly).

There is but also a practical question behind here for CFD, imho, where the NS equations are solved only approximately on discretized regions/equations - e.g. using Finite Differences, Finite Volume, or Finite Elements (or, more generally, some other sort of ansatz function Galerkin-like discretization or so). With none of those, you really can conserve of angular momentum (currently). This usually does not matter on large scare vortices. But for turbulence e.g. in DNS simulations, it matters, where a lot of small vertices are generated and transported.

So, even if the NS equations are made angular momentum conserving, the practical (discretized) numerical solution is most probably not exactly.

• I'm not sure this is directly relevant to the question - please make sure that if you're answering questions in the future that your answers pertain to the specific question, and not only the context around it. Apr 2, 2022 at 21:30