1 D axisymmetric heat conduction with neumann boundaries

variational formulation
1

Hi there,

I want to make a simple model of heat dissipating in a hetrogenous medium in a disk. The only source is a probe in the centre of the disk and there is a no flow (neumann boundary at the edge of the boundary. I understand the problem as follows:

d(KdT/dr)/dr + q = CdT/dt (inside the domain, K is conductivity and C is heat capacity)

boundary conditions : r = 0, r= 1, I have dT/dr = 0

q is a point source at r = 0

Can anyone help me create the variational form? When I evaluate the integrals I only end up with everything as zero.

I saw the pure neumann tutorial however I don’t know how to convert it for the time dependancy and for the axial condition.

Thanks for your time.

2

DISCLAIMER: This reply only handles integration by parts to show how to obtain the Neuman condition for this problem. To obain the axi-symmetric variational formulation, a change of integration coordinate system has to be applied as well, see the old Q&A-forum

So, you have the equation:
C\partial T/\partial t - \partial (K \partial T)/\partial r = q .
Using finite difference temporal discretization and Integration by parts yields
(C (T^{i+1}- T^i)/\delta t, v)_\Omega + K(\partial T^{i +1}/\partial r, \partial v/\partial r)_\Omega - (K n\partial T^{i +1}/\partial r, v )_{\partial\Omega}=(q,v)_\Omega, where n is ±1 depending on if you are at x=0 or x=1.
Since you have a zero neumann condition, the boundary terms falls away and you are left with volume terms.
To get the point source q, you can follow Help a starter! on how to make a point source (in that thread it has the name f).

3

Thank you for your prompt reply Dokken. One question on the formulation below:

Are the variables U with subscripts n meant to be T. Or is this a new variable?

I think I miss quoted the 1d axisymmetric problem in the first place. So if I understand what you are saying about the Neumann boundaries perhaps I can rewrite it as follows:

C(Tn+1−Tn)/δt,v)Ω+K(∂T/∂r,∂v/∂r)Ω−(K∂T/∂r,v)∂Ω + K/r(∂T/∂r,v/)Ω=(q,v)Ω

?

In writing the conduction equation for axisymmetric flow I think I was missing the first derivative with respect to r:

-K((∂2T/∂r2) + (1/r) (∂T/∂r)).

Am I right?

Many thanks for your input.

4

I am sorry for the typo, It should have been T. I’ve also now distinguished between the time step (i and i+1) and the normal vector.
However, you have to be quite careful when transforming the system to cylindrical coordinates, as the new term adds in a singularity.
Please see this old thread on how to derive the variational formulation for axi-symmetric problems, where both the derivatives and integration measure is transformed into cylindrical coordinates.