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A Almost spherical flow

  1. Jun 1, 2017 #1

    hunt_mat

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    Suppose I am considering the diffusion of heat in three dimensions:
    [tex]\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}}[/tex]
    and I am interesting in the case where the diffusion is ``almost spherical'', with azimuthal symmetry that is all derivatives of [itex]\varphi[/itex] vanish and the derivatives in [itex]\theta[/itex] are small. I'm pretty sure that I can't simply scale everything out but is there a functional form of [itex]T[/itex] which I can assume which will do the job?
     
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  3. Jun 1, 2017 #2
    Don't you think that all this depends on the boundary conditions? What boundary conditions did you have in mind?
     
  4. Jun 1, 2017 #3

    hunt_mat

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    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.
     
  5. Jun 1, 2017 #4
    Please write out the exact boundary comditiom you wish to use.
     
  6. Jun 1, 2017 #5

    hunt_mat

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    This is only a test problem to see if my idea comes across as sensible. Take for example [itex]\hat{\mathbf{n}}\cdot\nabla T=g(x)[/itex] lets say.
     
  7. Jun 1, 2017 #6
    What's x?
     
  8. Jun 2, 2017 #7

    hunt_mat

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    [itex]x[/itex] is a point on the boundary.
     
  9. Jun 2, 2017 #8
    So the heat flux at the surface of the sphere is a function of ##\phi##?
     
  10. Jun 2, 2017 #9

    hunt_mat

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    No, I am considering azimuthal symmetry.

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