The Student Room Group

PDE: Heat equation in cylindrical coordinates

Hi,

Can you please help me with Q5c? Can you check my solution for part a and b? I know for an infinite cylinder the U(phi,phi) term disappears but what about the U(zz) term? Does it also vanishes?

50C3BDC8-D8EF-43F8-8E2C-CAC239BF23DB.jpeg

1791E308-7D78-4064-8CD1-E38F3B436730.jpeg

2B2FA771-5751-492A-ACD5-E8DF9133B992.jpeg
(edited 11 months ago)

Scroll to see replies

Original post by Shaonx
Hi,

Can you please help me with Q5?
Attachment not found


Any thoughts ? I would imagine your lecture notes give some ideas.
Reply 2
I kind of know how to do a and b but not sure with the solution? Is my solution correct for a and b?


Original post by RDKGames
Any thoughts ? I would imagine your lecture notes give some ideas.
Original post by Shaonx
I kind of know how to do a and b but not sure with the solution? Is my solution correct for a and b?


Not quite.

(a) if the cylinder is infinitely long then the heat transfer U is axisymmetric meaning you lose phi and z derivatives. The function U only depends on rho and t.

Try part b again with that.
(edited 11 months ago)
Reply 4
Original post by RDKGames
Not quite.

(a) if the cylinder is infinitely long then the heat transfer U is axisymmetric meaning you lose phi and z derivatives. The function U only depends on rho and t.

Try part b again with that.


U is Axisymmetric along which axis? I have done part b correctly now, I also did part c but not sure of the solution. Can you please check my part c solution?

B823816E-4729-489B-8BDE-AA66098D1F93.jpeg
(edited 11 months ago)
Original post by Shaonx
U is Axisymmetric along which axis? I have done part b correctly now, I also did part c but not sure of the solution. Can you please check my part c solution?

B823816E-4729-489B-8BDE-AA66098D1F93.jpeg


Axisymmetric along z-axis. Spatially U only depends on the distance from z-axis, this is your rho.

While b looks mostly OK, it should say what lambdas are ?

Also, U should be bounded, right? What is the issue with the current solution at the moment? [hint: plot bessel functions J and Y to see]


EDIT nevermind you moved all this to part (c) which looks OK on first glance.
(edited 11 months ago)
Reply 6
Original post by RDKGames
Axisymmetric along z-axis. Spatially U only depends on the distance from z-axis, this is your rho.

While b looks mostly OK, it should say what lambdas are ?

Also, U should be bounded, right? What is the issue with the current solution at the moment? [hint: plot bessel functions J and Y to see]


EDIT nevermind you moved all this to part (c) which looks OK on first glance.


Thanks for your help. Appreciated
Reply 7
Original post by RDKGames
Axisymmetric along z-axis. Spatially U only depends on the distance from z-axis, this is your rho.

While b looks mostly OK, it should say what lambdas are ?

Also, U should be bounded, right? What is the issue with the current solution at the moment? [hint: plot bessel functions J and Y to see]


EDIT nevermind you moved all this to part (c) which looks OK on first glance.

I am just wondering why rho is the distance from z-axis? Isn’t rho supposed to be the radius of the cylinder and not the height z? In the sketch below r=rho.

311207A6-6DBE-4AE3-B1A6-65D74F143969.jpeg
Original post by Shaonx
I am just wondering why rho is the distance from z-axis? Isn’t rho supposed to be the radius of the cylinder and not the height z? In the sketch below r=rho.

311207A6-6DBE-4AE3-B1A6-65D74F143969.jpeg


The vertical yellow axis is your z-axis. You can see that r (or rho) is clearly measured from it.
Reply 9
Original post by Shaonx
I am just wondering why rho is the distance from z-axis? Isn’t rho supposed to be the radius of the cylinder and not the height z? In the sketch below r=rho.

311207A6-6DBE-4AE3-B1A6-65D74F143969.jpeg

"a" is the radius of the cylinder
Reply 10
Original post by RDKGames
(a) if the cylinder is infinitely long then the heat transfer U is axisymmetric meaning you lose phi and z derivatives. The function U only depends on rho and t.


Don't we need further assumptions to know this? We haven't been told anything about the boundary conditions.
Reply 11
Original post by RDKGames
The vertical yellow axis is your z-axis. You can see that r (or rho) is clearly measured from it.


Thanks
Reply 12
Original post by RichE
"a" is the radius of the cylinder

Yes, 0<=rho<=a
Reply 13
Original post by RichE
Don't we need further assumptions to know this? We haven't been told anything about the boundary conditions.

Agreed. As it stands I don't think you can rule out dependence on either z or theta.
Reply 14
Original post by DFranklin
Agreed. As it stands I don't think you can rule out dependence on either z or theta.

Having said that, I have no idea how to answer the question as given. So maybe RDK knows better what's expected by the setter.
Reply 15
Original post by RichE
Having said that, I have no idea how to answer the question as given. So maybe RDK knows better what's expected by the setter.

I know for certain that U(phi,phi) goes away but wasn’t sure about U(z,z) but it makes sense that U(z,z) also disappears since part b says to find U in terms of T(t)R(rho)
Reply 16
Original post by Shaonx
I know for certain that U(phi,phi) goes away but wasn’t sure about U(z,z) but it makes sense that U(z,z) also disappears since part b says to find U in terms of T(t)R(rho)

Do you mean U_zz rather than U(z,z) which is something different and likely meaningless?

What DFranklin and I are saying is that the question is not specified enough to be answered? But there may be a few things that are being assumed which aren't explicit in the question.
Reply 17
Original post by RichE
Do you mean U_zz rather than U(z,z) which is something different and likely meaningless?

What DFranklin and I are saying is that the question is not specified enough to be answered? But there may be a few things that are being assumed which aren't explicit in the question.

Yes, I meant U_zz
Reply 18
There is another PDE question I posted today, it’s about the heat equation. The initial condition doesn’t get me anywhere. I would be grateful if anyone can help me with that
Reply 19
Original post by Shaonx
Yes, I meant U_zz

Properly understood the question doesn't give you enough information to conclude U_zz=0, but maybe you're expected to assume something that isn't explicit in the question.

Quick Reply