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upward, while denser and heavier fluid sinks. That is, gravity and density differ-
ences set up a flow, leading to transport of the dissolved substance by convection.
This convection is usually small in liquids, but can often be substantial in gases,
which flow much more easily. In air, for example, there is always some minor flow,
either due to temperature and thereby density differences, or due to some external
forcing. When you take your shoes off, you can sometimes quickly smell your feet.
If the smell molecules were brought to your nose by diffusion only, it would take
a very long time (about a year!) to notice the smell. Minor air flow will introduce
convection of the scent molecules and contribute to a transport that takes just a sec-
ond. Similarly, heating up a room by diffusion would take a very long time, but the
temperature differences introduced by a heating device set up density differences
and a corresponding flow that efficiently distributes the warmed air throughout the
room - by convection. The diffusion process can in such cases be neglected.
Contrary cases also exist: Convective transport can be approximated by diffusion.
The ground, consisting of media such as soil and rock, is porous. That is, there is a
network of very small pores in which fluid can flow. The transport of pollution in
the pores is dominated by convection. However, this network, when viewed from a
macroscopic
level (containing a large number of pores), makes the transport look
like a diffusion process. Hence, mathematical models for the diffusive transport of a
substance are highly relevant for a wide range of transport problems in the ground.
Sediment transport and the formation of geological layers involve the convective
transport of sediments in water. Nevertheless, on large spatial and temporal time
scales, the cumulative effect of the transport can well be described as a diffusion
process. Diffusion equations are therefore used to simulate the geological evolution
of sedimentary basins that may contain oil and gas.
The above discussion reveals that building mathematical models and applying
them correctly to solve problems in nature and technology can be complicated.
Many consider this topic as beyond the scope of scientific computing, meaning
that scientific computing only deals with the numerical treatment of a given model.
However, the formulation of a sound numerical method is tightly connected to the
formulation of the model, which is connected to the derivation of the model and the
physics of the problem. The boundaries of scientific computing as a discipline are
therefore (also) diffuse.
7.1.4
The Relevance of One-Dimensional Models
Mathematicians, physicists, and computer scientists are often accused of having a
one-dimensional life. What is definitely true, is that they prefer to work in a one-
dimensional world whenever possible. We will follow this line of reasoning and
concentrate on one-dimensional diffusion problems in this topic. Since the world is
definitely three-dimensional, this restriction deserves some justification.
It turns out that many three-dimensional physical problems are well described
by one-dimensional models, and since one space dimension is much easier to deal
