From 1D to cylindrical co-ordinates in PDEs

[gareth]

Hi all,

I have another post on here relating to Fick's law of diffusion, but before I asked that I really should have started with this question:

How do you go from a one dimensional version of the diffusion equation to a cylindrical co-ordinate system of the same equation?

I have found both equations in many books but they just seem to skip from the one or three dimensional case in Cartesian co-ordinates to the cylindrical version without any intermediate steps.

I have attached the diffusion equation in pdf entitled "ficks2" and also the other part which is an excerpt from a paper "diffusion equation". The paper is solving the PDE for the case of two dimensions in cylindrical co-ordinates but I fail to see how they get equation (1) in the paper from the Cartesian equivalent.

The next part of my question would then be how do they solve this version of the diffusion equation. (i.e. how do they get from (1) to (3))

I have very limited experience in this field so the more simple the replies are the better!

Thanks in advance.

Gareth

 

[phioder]

How do you go from a one dimensional version of the diffusion equation to a cylindrical co-ordinate system of the same equation?

I have found both equations in many books but they just seem to skip from the one or three dimensional case in Cartesian co-ordinates to the cylindrical version without any intermediate steps.

Hi,

The following book
https://www.amazon.com/dp/0534373887/?tag=pfamazon01-20

has a very good explanations on basic cylindrical problems, it shows some of the algebraical steps on how to go from Cartesian to cylindrical coordinates in 3D. The book also may give you some tips and could help you solving the diffusion equation.

The next part of my question would then be how do they solve this version of the diffusion equation.

My experience in this field is also very limited if you don't come further let me know I may have some notes on the diffusion equation

Hope it helps
Thank you and best regards
phioder

 

[gareth]

Thanks for the reply Phioder, I'll let you know how I get on.