- #1
happyparticle
- 456
- 21
- Homework Statement
- What is the temperature distribution inside a sphere and fluid
- Relevant Equations
- $$\kappa \nabla^2 T = 0$$
$$T(r,\theta) = \sum_{m=0}^{\infty} = (A_m r^m + \frac{B_m}{r^{m+1}}) P_m(cos \theta)$$
Consider a sphere of thermal diffusivity ##\kappa_2## is submerged in an incompressible and stationary fluid of thermal diffusivity ##\kappa_1##.
The fluid is held between 2 large plates ( at T_0 for the top plate and T_1 for the bottom plate).
What is the stationary temperature distribution inside the sphere and in the fluid ?
Since ##\frac{\partial}{\partial t} = 0## and we have symmetry in ##\phi## coordinates, the heat conduction equation becomes:
##\kappa \nabla^2 T = 0##
With the following solution in spherical coordinates :
$$T(r,\theta) = \sum_{m=0}^{\infty} = (A_m r^m + \frac{B_m}{r^{m+1}}) P_m(cos \theta)$$.
Also, since ##T \rightarrow \infty## at ##r \rightarrow 0##
B must be 0.
This is as far as I can go. I'm not sure what are the boundary conditions to find the value of A and m.
The fluid is held between 2 large plates ( at T_0 for the top plate and T_1 for the bottom plate).
What is the stationary temperature distribution inside the sphere and in the fluid ?
Since ##\frac{\partial}{\partial t} = 0## and we have symmetry in ##\phi## coordinates, the heat conduction equation becomes:
##\kappa \nabla^2 T = 0##
With the following solution in spherical coordinates :
$$T(r,\theta) = \sum_{m=0}^{\infty} = (A_m r^m + \frac{B_m}{r^{m+1}}) P_m(cos \theta)$$.
Also, since ##T \rightarrow \infty## at ##r \rightarrow 0##
B must be 0.
This is as far as I can go. I'm not sure what are the boundary conditions to find the value of A and m.