Heat diffusion in a spherical shell

1. Sep 10, 2014

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.

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. Sep 12, 2014

Staff: Mentor

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

3. Sep 12, 2014

Orodruin

Staff Emeritus
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.