Information Technology Reference
In-Depth Information
with than three, reduction to one dimension is a common task. Even if the one-
dimensional model reveals itself to be a crude quantitative approximation of the
real-world problem, we can gain a lot of qualitative insight and understanding
about the physics and mathematical description of the problem by studying the
one-dimensional model.
One-Dimensional Models are Convenient
There are two major practical advantages of one-dimensional models: Solutions can
be computed in a short time, allowing many experiments to be done, and visual-
ization of the solution is easy, since a simple curve plot will be sufficient. Another
advantage from a teaching point of view is that one-dimensional models contains
fewer numerical details than higher-dimensional models, making it easier to focus
on the principal ideas and to get a computer code to work.
Long and Thin Geometries
Some examples of reducing the three-dimensional world to a one-dimensional
approximate model will now be discussed. Our first example concerns “long and
thin geometries”, i.e., the geometry has a dominant dimension, which makes the
problem one-dimensional. Think of a long, thin tube filled with water and imagine
that we inject some ink at one end. If the injected ink expands the cross-section
of the tube, the transport of ink by diffusion will be directed along the tube. What
we observe is a diffusive front propagating away from the end where the ink was
injected. Figure
7.1
depicts this process. The underlying microscopic physics of this
problem, i.e., molecular vibrations, is highly three dimensional. The macroscopic
physics, i.e., diffusive transport, is also three dimensional, but the transport in the
directions perpendicular to the tube walls is very small if the tube is thin, since the
whole cross-section is then supposed to have approximately the same concentration.
In mathematical terms, this means that only the derivative in the direction of the tube
axis needs to be taken into account. This property makes the mathematical model
one-dimensional.
What happens if the injected ink does not fill the whole cross section? A spher-
ical droplet at the inlet makes the problem three-dimensional, at least in principle.
Figure
7.2
presents simulations of this case. Comparing Figs.
7.1
and
7.2
shows
that after a short time, the problem in Fig.
7.2
also seems to be well described by
a one-dimensional model, due to the smoothing effect of diffusion and the “thin”
geometry the tube. A more accurate assessment of the accuracy of a purely one-
dimensional model can be obtained by comparing c plotted along the tube axis and
a concentration function computed by a one-dimensional model. Figure
7.3
displays
the curves obtained by the two models: Some discrepancies are indeed observed, but
the one-dimensional model is clearly a reasonable approximation.
