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Fig. 7.2 The same diffusion problem and three-dimensional mathematical model as in Fig. 7.1 ,
but the initial concentration of ink at the left end does not fill the tube cross-section entirely, thus
inducing some small initial three-dimensional effects. How appropriate a one-dimensional model
is for the present case becomes evident by comparing the plots with those in Fig. 7.1
that, in comparison with the corresponding plot in Fig. 7.5 (second from the top),
there are some visible three-dimensional (two-dimensional in a flat plot) effects.
The final temperature will in this case stabilize at the surrounding air temperature
(cf. Fig. 7.8 ). The evolution as a whole looks one-dimensional, except at the early
stages (e.g., t D 0:025).
Cylindrical and Spherical Symmetry
Replacing the familiar Cartesian coordinates x, y,and z by another coordinate sys-
tem can lead to one-dimensional models, although the mathematical problem in
Cartesian coordinates are two or three dimensional. Consider computing the tem-
perature inside a very long cylinder, where none of the input data varies along
the surface of the cylinder. Switching to cylindrical coordinates, we can omit the
dependence on all coordinates, except the radial one. This leaves a one-dimensional
problem on an interval from zero to the radius of the cylinder in cylindrical coor-
dinates. In Cartesian coordinates, the problem is two dimensional in a circular
domain. Similarly, heat conduction in a hollow sphere (e.g., a gas container) is a
 
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