Thread: Diffusion equation with cylindrical coordinates - Bessel function

Hi,

I'm trying to describe diffusion through a solid cylinder by following Crank's "The Mathematics of Diffusion". I more or less follow the method of adapting the diffusion equation for a cylinder, and using separation of variables to get the general equation. But when it comes time to introduce the boundary conditions we see Bessel's functions appear and this is where I get lost. I have never seen these before (I am not a mathematician!) and I was wondering if someone could give me some tips for how to use an equation like the one attached (sorry, I don't know how to write it in the post). Specifically, what are the alpha terms? Crank's book says they are the roots of the bessel function, but I feel like this means you go around in circles? Any help would be greatly appreciated.

Hey hannah89.

Can you post the derivation included in your book?

Hi chiro,

Thanks for your reply.

I think the easiest way might be to look at this online version:

The Mathematics of Diffusion

and head to page 79, eqn 5.22.

Thanks

Unfortunately I don't think I have the experience to give advice on this subject: I'm not familiar enough with it.

If I had to guess it would have to do with the roots of the function satisfying a zero that define the actual function (i.e. re-arranging all the terms to get a zero on the RHS).

You can expand the Bessel functions using a series expansion or you can relate the Bessel function J1 to J0 (or vice versa).

I'm sorry I can't help you out much more than this.