Heat diffusion equation with small angular variation in spherical coordinates

[hunt_mat]

Suppose I am considering the diffusion of heat in three dimensions:
$$\frac{\partial T}{\partial t}=\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\frac{\partial T}{\partial r}\right)+\frac{1}{r^{2}\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial T}{\partial\theta}\right)+\frac{1}{r^{2}\sin^{2}\theta}\frac{\partial^{2}T}{\partial\varphi^{2}}$$
and I am interesting in the case where the diffusion is ``almost spherical'', with azimuthal symmetry that is all derivatives of $\varphi$ vanish and the derivatives in $\theta$ are small. I'm pretty sure that I can't simply scale everything out but is there a functional form of $T$ which I can assume which will do the job?

 

[Chestermiller]

Don't you think that all this depends on the boundary conditions? What boundary conditions did you have in mind?

 

[hunt_mat]

Having thought about this, I think that this is a job for multiple scale analysis.

The boundary conditions are Neumann conditions and so can be expanded via perturbation to be on the sphere.

 

[Chestermiller]

Please write out the exact boundary comditiom you wish to use.

 

[hunt_mat]

This is only a test problem to see if my idea comes across as sensible. Take for example $\hat{\mathbf{n}}\cdot\nabla T=g(x)$ let's say.

 

[Chestermiller]

This is only a test problem to see if my idea comes across as sensible. Take for example $\hat{\mathbf{n}}\cdot\nabla T=g(x)$ let's say.

What's x?

 

[hunt_mat]

$x$ is a point on the boundary.

 

[Chestermiller]

$x$ is a point on the boundary.

So the heat flux at the surface of the sphere is a function of $\phi$?

 

[hunt_mat]

No, I am considering azimuthal symmetry.

As I said before, I think the method of multiple scales works fine for this problem.