Chapter 5 Steady and unsteady diffusion

6 CHAPTER 5. STEADY AND UNSTEADY DIFFUSION
K p
∂T p
∂r
= K m
∂T m
∂r
(5.22)
Though the temperature field is disturbed by the presence of the particle near
the particle surface, it is expected that the temperature field is unaltered at
a large distance from the particle r → ∞, where it is given by
T = T ′ x 3 = T ′ r cos (θ) (5.23)
The temperature field far from the particle can be expressed in terms of
the Legendre polynomial solutions for the Laplace equation as
T = T ′ rP 0 1 (cos (θ)) (5.24)
Since the temperature field is driven by the constant temperature gradient
imposed at a large distance from the particle, it is expected that the temperature
field near the particle will also be proportional to P1 0 (cos (θ)), since
any solution which contains other Legendre polynomials is orthogonal to
P (0)
1 (cos (θ)). Therefore, the solutions for T p and T m have to be of the form
T p
T m
= A p1 rP1 0 (cos (θ) + A p2
r P 0 2 1 (cos (θ))
= A m1 rP1 0 (cos (θ) + A m2
r P 0 2 1 (cos (θ)) (5.25)
The constant A p2 is identically zero because T p is finite at r = 0, while
A m1 = T ′ in order to satisfy the boundary condition in the limit r → ∞.
The constants A p1 and A m2 are determined from the boundary conditions at
the surface of the sphers,
3T ′
A p1 =
2 + K R
A m2 = (1 − K R)R 3
(5.26)
2 + K R
The temperature fileds in the particle and the matrix are
T p =
3T ′ r
P1 0 (cos (θ)
2 + K R
T m = T ′ rP1 0 (cos (θ) + (1 − K R)R 3 T ′
P 0
(2 + K R )r 2 1 (cos (θ)) (5.27)