Can the Magnetic Reynolds Number be Proven with Simple Math?

[kd001]

Is there a simple mathematical way of proving the left hand side equals the right hand side (see attached image)?

Any help will be much appreciated as I'm desperately trying to work out exactly where the magnetic Reynolds number comes from. I can derive the left hand side (its simply the ratio of advection term to the diffusion term which themselves can be derived using Maxwell's equations) but I am stuck at this final step as I'm not very good at vector calculus.

Thanks in advance for any help.

 

[kd001]

More than 400 views and no one has an idea?

In case my initial post wasn't very clear, I'm just looking for a way of proving that the ratio of the vector quantities on the left hand side of the equation is equal to the term on the right hand side. The expression is definitely correct. I just want to understand why.

Thanks in advance.

 

[nasu]

I don't know what is this about, specifically. However there is no "r" in the left side and no B in the right hand side. You should have some additional relationship between them. Otherwise how can you get an "r" from an expression that does not contain it?

 

[kd001]

I don't know what is this about, specifically. However there is no "r" in the left side and no B in the right hand side. You should have some additional relationship between them. Otherwise how can you get an "r" from an expression that does not contain it?

That's exactly the problem that I'm having. As I said the relationship is definitely true:

http://scienceworld.wolfram.com/physics/MagneticReynoldsNumber.html

However, I haven't come across any books or other sources that explain why this is the case and what the 'additional relationship' might be. All the books that I've read simply quote the relationship without actually explaining it. But surely there must be a reason and I do not believe it is simply empirical.

Here's another link with some information but it doesn't explain it either:

http://books.google.co.uk/books?id=...netic reynolds number vector notation&f=false

 

[nasu]

The book link does not work for me but a little Google research did clarify the things.
The magnetic Reynolds number is defined as the ratio of the two terms (on the left side). Their sum is equal to the derivative of B in respect to time.
The value of the Reynolds number shows the relative contribution of the terms.
So here is nothing to prove here.
Inspection of the units shows that the ratio is of the order of (v r)/lambda where "r" is some characteristic radius (or length) of the system. As in the classic Reynolds number.
If you replace the "=" with "of the order of" you have a better representation.
Again, nothing to prove in general.
For a specific system, you can pick some specific dimension as r. The variation of the field or maybe v will depend on this r but the way they do is system dependent.