Heat Diffusion In Spherical Shell

[lavaca_86]

Hello, I have been going round in circles with this problem and any help would be greatly appreciated!

 

I have du/dt = K*laplacian(u) inside of a spherical shell a<r<b. Boundary conditions: u=const=C at r=b, du/dr=0 at r=a Initial conditions: r=0 for a<r<b

 

Are my boundary and inital conditions set up correctly? My shell should have a totally insualting inner the boundary at r=a and u held constant by a resevoir at the outer boundary r=b, with u = 0 between a and b at time t=0. I would like to calculate the heat profile as a function of time and radius.

 

I have worked out that I have to use separation of variables and then use Bessel functions.

 

u=T(t)R®

 

T is just an exponential

T=Aexp(-k^2*K*t)

 

R I end up with as:

R = 1/(r*sqr(k)) * (c1*sin(kr)+c2*cos(kr)) (this is different to the link I posted below by a factor of 1/sqrt(k))

 

I am confused now as to how to fit the boundary conditions to find c1,c2 and k. I found a similar thread at http://scienceforums.com/topic/12240-heat-diffusion-equation-in-a-3d-sphere/ which mentions an eigenfunction expansion using all allowed values of k. How do I go about this and how then can I match coefficients of sine and cosine terms to the boundary conditions?

 

Thank you!