Heat conduction equation in cylindrical coordinates

[shreddinglicks]

TL;DR Summary

Using Bessel functions to solve the heat equation for hollow cylinders.

I've been studying a few books on PDE's, specifically the heat equation. I have one book that covers this topic in cylindrical coordinates. All the examples are applied to a solid cylinder and result in a general Fourier Bessel series for 3 common cases that can be found easily with an online source.

Using separation of variables I have an equation,

J(0,alpha*r) + Y(0,alpha*r)

Then it says the 2nd term is eliminated due being bounded at r = 0. The boundary condition at the outer wall of the cylinder is then evaluated for alpha.

Afterwards I'm presented with the three Fourier Bessel solutions for the boundary conditions,
J'=0
hJ + alpha*b*J' = 0
and J = 0

I want to know how would I solve this problem if I had a hollow cylinder. I've attempted it on my own with poor results.

I assume in the hollow cylinder case Y(0,alpha*r) does not go to 0. How would I obtain my solution in this case?

 

[pasmith]

Define "hollow". Do you mean a circular annulus lying between $0 < a \leq r \leq b$? In that case yes, both $J$ and $Y$ are admissible solutions. For orthogonality relations, see section 11.4 of Abramowitz & Stegun, available at https://www.cs.bham.ac.uk/~aps/research/projects/as/resources/AandS-a4-v1-2.pdf.

 

[shreddinglicks]

Define "hollow". Do you mean a circular annulus lying between $0 < a \leq r \leq b$? In that case yes, both $J$ and $Y$ are admissible solutions. For orthogonality relations, see section 11.4 of Abramowitz & Stegun, available at https://www.cs.bham.ac.uk/~aps/research/projects/as/resources/AandS-a4-v1-2.pdf.

Interesting. I tried to apply what is shown in 11.4.2. How would I solve for A and B that is shown in 11.4.1?