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Heat diffusion in a spherical shell

  1. Sep 10, 2014 #1
    Hey guys, I have a problem that is giving me trouble.

    1. The problem statement, all variables and given/known data

    I have to solve time dependent diffusion equation ##D\nabla^2 T(r,t)=\frac{\partial T}{\partial t}## (##D## is diffusion constant and ##T(r,t)## is temperature function) for a spherical shell of radii ##r_1## and ##r_2## in a where inner and outer temperatures are constant at ##T_0## as shown in the picture, with initial condition ##T(r_1<r<r_2,t=0)=T_1##. Since problem is spherically symmetrical function ##T## is not dependent on asimutal and polar angle.
    COnUOfd.png

    2. Relevant equations

    By separating the variables (##T(r,t)=R(r)\tau (t)##) we obtain two equations:
    $$\nabla^2 R(r) + k^2 R(r) = 0$$
    $$\frac{\partial \tau(t)}{\partial t}\frac{1}{\tau(t)} = - k^2 D$$
    For time dependent part we get ##\tau(t)=C e^{-k^2DT}## and for the radial part a linear combination of spherical bessel functions of first (##j_0##) and second type (##n_0##) (only 0-th order because of symmetry) $$R(r) = \sum\limits_n [A_n j_0(k_n r) + B_n n_0(k_n r)]$$

    Complete solution can then be written as:
    $$T(r,t) = \sum\limits_n [A_n j_0(k_n r) + B_n n_0(k_n r)]e^{-k^2_n DT} + T_0,$$
    with initial condition $$T(r_1<r<r_2,t=0)=T_1,$$
    and boundary conditions $$ T(r=r_1,t)=T(r=r_2,t)=T_0$$

    3. The attempt at a solution

    I don't know how to get coefficients ##A_n##, ##B_n## and ##k_n##. I tried getting ##k_n## from zeroes of spherical bessel function ##j_0## but since the center of sphere is hollow I must not set ##B_n## to zero as I would in the case of a full sphere.

    I can't seem to get any further than this and would appreciate any suggestion. Thanks for the help.
     
  2. jcsd
  3. Sep 12, 2014 #2
    At each value of n, the term in brackets must be zero at the two boundaries. This leads to two equations in two unknowns, for kn and Bn/An.

    Chet
     
  4. Sep 12, 2014 #3

    Orodruin

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    Since you have a problem without angular dependence, I also suggest using the fact that ##j_0(x) \propto \sin x / x## and ##n_0(x) \propto \cos x / x##. This will let you work with trigonometric functions that you know rather than the spherical bessel functions.
     
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