# Analogy between fluid dynamics and electromagnetism

I've been reading a set of notes (Chapter 13 of Caltech Ph 136 notes from 2004, by Blandford and Thorne) that draw an analogy between fluid dynamics and electromagnetism, identifying the magnetic vector potential $\mathbf{A}$ with the velocity flow field $\mathbf{u}$, the magnetic field with vorticity $\mathbf{\omega}$, and the electric field with $\mathbf{\omega} \times \mathbf{u}$. This is different from the hydraulic analogy for electric circuits, which I think identifies the velocity field with current density. Several things go through this analogy in a reasonable way, like $\nabla \cdot \mathbf{B} = 0$, and incompressibility corresponding to the gauge choice of $\nabla \cdot \mathbf{A} = 0$.

But it looks to me like this analogy is not an "isomorphism". For example, does the condition $\mathbf{E} = \mathbf{B} \times \mathbf{A}$ (which follows, in the analogy, from the definition of vorticity) have any interpretation in electromagnetism? If our flow is irrotational, what does the velocity potential correspond to (e.g. does a scalar potential for the magnetic vector potential ever come up/have a name)?

I'd be interested in any reference that thoroughly explores this analogy. The best I've found is here, but as far as I can tell it does not address the above issues (and similar).

## 3 Answers

I looked briefly at the Blandford and Thorne notes. The analogy between $\omega$ and B appears to be mainly illustrative and not to be extended too far. (I don't see the reference to the analogy between E and $\omega \times \bf{u}$ there at first glance, but it does not seem to be apt at all.) It seems to be intended to draw upon any previous intuition you might have about the relationship between the vector potential and the magnetic field to help you understand how fluid velocity and vorticity are related, and I think that's all.

Actually, I think a more apt analogy is to relate: 1) $\bf{u}$ to $\bf{B}$, 2) $\omega$ to $\bf{J}$, the current density, since $\nabla \times \bf{v} = \bf{\omega}$ and (at least in magnetostatics) $\nabla \times \bf{B} = \mu_\circ \bf{J}$, and 3) the vector stream function $\bf{\Psi}$, where $\bf{v} = \nabla \times \bf{\Psi}$, to the magnetic vector potential $\bf{A}$, with $\bf{B} = \nabla \times \bf{A}$. This analogy is actually useful, because you can apply the tools of magnetostatics to the fluid flow, e.g. using the Biot-Savart law to write $\bf{v}$ in terms of $\bf{\omega}$, and writing Poisson's equation to relate $\bf{\Psi}$ to $\bf{\omega}$, etc.

Faber chapter 4 expands on the analogy in your first reference, but is an analogy with magnetostatics i.e. The divergence free part of the velocity field and with electrostatics I.e. The curl free part of the velocity field. The first will have a vector potential A (as the one of Magnetostatics) and the second a scalar potential phi (as the one of electrostatics which obey to the Laplace equation)

The curl of the potential A is parallel to the B field and to the v field, not as you mention above A parallel to v and B parallel to the curl of V. How you derive the parallel for the E field is also unclear to me. The curl free part of the velocity field, has a scalar potential which obey to the Laplace eq.

Once you have the potentials I am not sure if you can choose a gauge for these, and how you can interpret it. Namely stating that the vector potential is incompressible, may not be of much use. Also is questionable if a potential itself has a physical meaning...

If I were you I would stay off the paper of your second reference, because as from above you don't have a strong grip of div curl potentials, and of the continuity equation (I.e of standard EM stuff).

The second reason is that a deposit on airxiv is not peer reviewed, and it looks to be published on a strange journal. the third reason is that they use quaternions notation, which Does not look familiar at all, the last reason is that they refer to a set of preceding papers that you need to go through first.

In there they do a different analogy, that they call EMH, and is about dynamics this time, but it does not look so rigorous. (I.e Euler equation is a derivation of Newton law in any case so what? Em is all about fluids so what? Why they throw in a undocumented reference to de broglie and GR? ) and I would not be able to tell rapidly especially without the references.

The analogy looks more similar to what you have above, but is questionable and unclear to me if it holds.

• Going through a couple of things in the papers of the second reference, in paper one 2009 in para graph 2, he derives maxwells from maxwells, so he substantially derives nothing. The notation mess up things and he ends up with a non compensated field due to the reference frame. Then in paper 2, he derives from this field due to the reference frame, the analogy, which then depends on the reference frame, then there is a wild flipping of concepts about the de broglie, an unjustified analogy for the Lorentz gauge, a large copy editing error where he has the table ... I just deleted it. – flyredeagle Jun 13 '14 at 3:06

The analogy actually holds remarkably far, see section 5.2 of this Treatise on the Magnetic Vector Potential: https://skemman.is/bitstream/1946/29521/3/KOK_MagneticVectorPotential.pdf

Therein you can find a detailed table of comparison, see table 5.2.5.

The tricky part is to distinguish the particle velocity from the background field velocity and/or vorticity. Therefore, velocity can not be transformed to the magnetic vector potential in all cases. Thus the expression $$\mathbf{L}=\mathbf{v} \times \mathbf{\omega}$$ is anagolous to $$\mathbf{E}=\mathbf{v} \times \mathbf{B}$$ which is the well known Lorentz force, a convective term like the Lamb vector, $$\mathbf{L}$$.

As the above reference argues, the magnetic vector potential has a clear physical meaning as the momentum per charge. In simple terms $$\mathbf{A}=\frac{\mathbf{p}}{q}\ ,$$ analogously, velocity is momentum per mass unit $$\mathbf{v}=\frac{\mathbf{p}}{m} \ .$$