A Almost spherical flow

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1. Jun 1, 2017

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?

2. Jun 1, 2017

Staff: Mentor

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

3. Jun 1, 2017

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.

4. Jun 1, 2017

Staff: Mentor

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

5. Jun 1, 2017

hunt_mat

This is only a test problem to see if my idea comes across as sensible. Take for example $\hat{\mathbf{n}}\cdot\nabla T=g(x)$ lets say.

6. Jun 1, 2017

What's x?

7. Jun 2, 2017

hunt_mat

$x$ is a point on the boundary.

8. Jun 2, 2017

Staff: Mentor

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

9. Jun 2, 2017

hunt_mat

No, I am considering azimuthal symmetry.

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