Chapter 5 Steady and unsteady diffusion

16 CHAPTER 5. STEADY AND UNSTEADY DIFFUSION
condition is called the Neumann Greens function G N ,
G N (x =
1
4πK|x − x s | + 1
4πK|x − x I |
(5.56)
A similar procedure could be used for more complicated geometries. The
Greens function for a source in a corner with zero flux conditions, as shown in
figure 5.7(a), could be obtained by using four sources of equal strength placed
symmetrically in an infinite domain. Similarly, the Greens function for a
source in a corner with zero temperature conditions, as shown in figure 5.7(b),
could be obtained by placing two sources and two sinks symmetrically in an
infinite domain. The Greens function for a source in a finite channel, as
shown in figure 5.8, would require an infinite number of sources.
5.2.5 Greens function for a sphere
The Dirichlet Greens function G D for a sphere is the solution for the temperature
field due to a source of unit strength which satisfies the zero temperature
condition at the surface of the sphere. This Greens function can be derived
using spherical coordinates (r, θ, φ). Consider a source of strength 1, which
is located, without loss of generality, at x S = (0, 0, r) in a sphere of radius
1, as shown in figure 5.9. The image, by symmetry has to be located along
the line joining the source point and the origin, at x I = (0, 0, r ′ ), but the
strength Q I can, in general, be different from that of the source. The temperature
at a point on the surface, x = (r, θ, φ) in spherical coordinates,
or x = (sin (θ)cos (φ), sin (θ) sin (φ), cos (θ)), due to the source and sink, is
given by
T =
=
=
1
4πK|x − x I | + Q
4πK|x − x I |
(
1
1
4πK (sin (θ) 2 cos (φ) 2 + sin (θ) 2 sin (φ) 2 + (cos (θ) − r) 2 1/2
)
Q
+
(sin (θ) 2 cos (φ) 2 + sin (θ) 2 sin (φ) 2 + (cos (θ) − r ′ ) 2 ) 1/2
1
4πK
(
1
(1 + r 2 − 2r cos (θ)) 1/2 + Q
(1 + r ′2 + 2r ′ cos (θ)) 1/2 )
(5.57)
The values of Q and r ′ are determined from the boundary condition on the
surface of the sphere, which requires that T = 0 for all values of θ and φ.