Information Technology Reference
In-Depth Information
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Fig. 7.1
Diffusion of ink in a long, thin tube simulated by a three-dimensional mathematical
model. The
top figure
shows the initial concentration (
dark
is ink,
white
is water). The
lower three
figures
show the concentration of ink at (
scaled
) times t
D
D
D
D
0:25, t
0:5, t
1,andt
3,
respectively. The evolution is clearly one-dimensional
One-Dimensional Variation Only
A closely related problem is our introductory example of bringing two metal pieces
at different temperatures into contact. If we assume that the surfaces of the metal
pieces are
insulated
, i.e., covered by some isolation material such that heat cannot
escape from the metal, except at the end surface that is in contact with the other
piece, see Fig.
7.4
, the heat transport vanishes at the surfaces of the combined piece.
If the temperature is constant throughout the pieces before they come into contact,
the heat will be transported in a normal direction to the contact surface, since this
is the only direction where we encounter temperature differences. Figure
7.5
depicts
the temperature evolution, which is clearly one-dimensional. The strong smoothing
effect of diffusion is evident from this figure, but it is perhaps even more evident
from the corresponding curves through the two materials in Fig.
7.6
.
On the other hand, if the surfaces are not insulated, there would be a difference
between the temperatures inside and outside the piece, resulting in a heat flow out
of the surface, and the problem would be three dimensional. A specific example
where the surfaces are in contact with air temperature at a fraction 0.8 of the ini-
tial temperature difference is shown in Fig.
7.7
. This plot, at time t
D
0:1,shows
