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