PDE: Heat equation in cylindrical coordinates

Hi,

Can you please help me with Q5?

Any thoughts ? I would imagine your lecture notes give some ideas.

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.

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?

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.

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.

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.

(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.

The vertical yellow axis is your z-axis. You can see that r (or rho) is clearly measured from it.

"a" is the radius of the cylinder

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.

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)

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