From 1D to cylindrical co-ordinates in PDEs

In summary, this book explains how to go from a one dimensional case to a cylindrical case in coordinates, but it doesn't explain how to solve the equation.
  • #1
gareth
189
0
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
 
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  • #2
gareth said:
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.

gareth said:
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
 
  • #3
Thanks for the reply Phioder, I'll let you know how I get on.
 

1. What is the significance of using cylindrical coordinates in PDEs?

Cylindrical coordinates are useful in solving certain types of partial differential equations (PDEs) that have cylindrical symmetry. This is often the case in problems involving rotation, such as fluid flow or heat transfer in cylindrical objects.

2. How do you convert from 1D to cylindrical coordinates in PDEs?

To convert from 1D to cylindrical coordinates, you can use the following equations:
x = rcosθ
y = rsinθ
z = z
where r is the distance from the origin, θ is the angle from the x-axis, and z remains the same. These equations can then be substituted into the original PDE to transform it into cylindrical coordinates.

3. Can you provide an example of solving a PDE using cylindrical coordinates?

One example is solving the heat equation for a cylindrical object with a constant temperature at its surface. In this case, the PDE can be transformed into cylindrical coordinates, and the solution can be found using separation of variables and Bessel functions. The resulting solution will be in terms of the radial distance and the angle around the cylinder.

4. Are there any limitations to using cylindrical coordinates in PDEs?

Cylindrical coordinates are most suitable for problems with cylindrical symmetry. If a problem does not have this symmetry, it may be more difficult to solve using cylindrical coordinates. Additionally, some PDEs may not have an analytical solution in cylindrical coordinates and may require numerical methods.

5. How do cylindrical coordinates compare to other coordinate systems in solving PDEs?

Cylindrical coordinates are one of several coordinate systems that can be used to solve PDEs. The choice of coordinate system depends on the problem at hand and the type of symmetry present. Other commonly used coordinate systems include Cartesian, spherical, and polar coordinates. Each has its own advantages and limitations in solving different types of PDEs.

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