Information Technology Reference
In-Depth Information
u(x,t=0)
u(x,t=0.001)
u(x,t=0.025)
u(x,t=0.25)
1
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
Fig. 7.6
Curve plots of the temperatures along the lines y
D
0:5 for the same physical problem
as in Fig.
7.5
0.5
0.4
0.3
0.2
0.1
0
0
1
Fig. 7.7
0:025 for the same problem as in Fig.
7.5
, but the bound-
aries are now in contact with the surrounding air instead of being insulated. This produces a slight
three-dimensional effect
Plot of the temperature at t
D
radial direction, which would utilize the radial symmetry of the problem and hence
actually model two-dimensional horizontal heat transport, disregarding the trans-
port in the vertical direction only. The actual temperature values we compute from
such one-dimensional models may not be very accurate, compared with real-world
observations, but the
qualitative features
of the temperature distribution may be
well represented. That is, we can use the crude one-dimensional model to achieve a
rough description of how the temperature inside the can develops in time. This gives
us a better understanding of the features of a diffusion process. Since many differ-
ent physical phenomena can be described by the same model, our understanding
from studying a particular example, even in a very simplified one-dimensional form,
reaches far beyond the example itself. In a nutshell, this is the power of mathematical
modeling.
