# Analogy between fluid dynamics and electromagnetism

1. May 4, 2014

### komdu

I've recently heard of an analogy between fluid dynamics and electromagnetism in which the velocity flow field is identified with the magnetic vector potential, (and therefore the vorticity is identified with the magnetic field), and the vector $\omega \times v$ is identified with the electric field.

I'm curious as to how far this analogy goes but haven't had much luck finding a completely systematic exploration of this. It looks to me like this isn't a perfect correspondence, because, for example, there seems to be a relationship between magnetic field, electric field, and the vector potential that doesn't exist in electromagnetism.

5 days ago I asked a very similar question on Stack Exchange but it hasn't attracted any attention.

2. May 4, 2014

### WannabeNewton

The analogy between the magnetic field $\vec{B} = \vec{\nabla}\times \vec{A}$ and the vorticity $\vec{\omega} = \vec{\nabla}\times \vec{v}$ is a very strong one; in fact in general relativity, the vorticity of the velocity field generating time-translations in space-time is identified with the gravitomagnetic field. But beyond that the analogy, as far as I know, is quite weak and as such you probably won't find any systematic exploration of it.

3. May 4, 2014

### komdu

Many thanks. Do you know of any reasonable interpretation of $E = B \times A$ in electromagnetism, which seems to arise from the definitions in the analogy?

4. May 4, 2014

5. May 4, 2014

### komdu

Ah, those links are excellent. Following some references in the links you provided I've found this paper which seems to say (top of section III) that in the inviscid case some fluid equations precisely coincide with the Maxwell equations. I will need to read this carefully.

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