# Why does the Biot-Savart law give the induced velocity of a infinitesimal segment of a vortex filament at point P?

My question is about the proof of how the Biot-Savart law can be used for vortex filaments. This is what I have in my textbook:

However, I'm not certain how it was derived - I understand that the Biot-Savart law originally came from electromagnetic theory, but could anyone present the proof for it's use for vortex filaments?

Intuitively though, something seems off - a vortex filament is just made out of lots of individual vortexes put together in a filament, so shouldn't the induced velocity at a point P of an infinitesimal section of the filament be zero unless $$d\vec{l}$$ and $$\vec{r}$$ are perpendicular... but clearly it's not the case, $$d\vec{V} = \frac{ d\vec{l} \times \vec{r} }{ \lvert \vec{r} \lvert ^3}$$ , which is nonzero even when $$d\vec{l}$$ and $$\vec{r}$$ aren't perpendicular

• I think this might be a better fit for physics.stackexchange. The vortices may be related to aviation, but proving a theorem is better suited for physics. – Bianfable Oct 25 '19 at 16:22
• The whoda whatzit law for vortex what-the-hecks? Over my head... – John K Oct 25 '19 at 16:37

Why would

the induced velocity at a point P of an infinitesimal section of the filament be zero unless $$d\vec{l}$$ and $$\vec{r}$$ are perpendicular

?

The cross product in $$\frac{ d\vec{l} \times \vec{r} }{ \lvert \vec{r} \lvert ^3}$$

does not change signs when going over the $$\frac{\pi}{2}$$ angle in between vectors, so the infinitesimal filament sections $$d\vec{l^-}$$ (immediately before $$d\vec{l}$$) and $$d\vec{l^+}$$ (immediately after $$d\vec{l}$$) have contributions to $$d\vec{V}$$ that do not cancel each other out. Therefore, $$\vec{V}$$ is not just a function of a single filament section, but of the entire filament, hence the integral.

If you check the next section in the source of your image, titled Straight-vortex case you will see this in action.

Nevermind guys, I finally found a proof for this, after a long time of googling:

https://www.whoi.edu/fileserver.do?id=218347&pt=10&p=116694

Basically, you use the differential definition of vorticity (w=curl of velocity) and the incompressibility condition (divergence of velocity = 0), then combine them and solve.

• Can you please summarize the results into your answer? Otherwise, it defies the purpose of SE. – JZYL Oct 28 '19 at 20:03