Information Technology Reference
In-Depth Information
space dimensions. Or perhaps we should be frank and say that it is straightforward
in principle
, since there are no new physical, mathematical, or numerical
ideas
,but
the book-keeping and the amount of technical details are increased significantly, so
getting all details right in a computer program can be quite challenging. The more
you work with algorithms and implementations, the easier it will be to meet such
challenges.
7.3
Derivation of Diffusion Equations
Although the same diffusion equation can be applied to model phenomena of differ-
ent physical nature, the
derivation
of the equation from physical principles depends,
not surprisingly, on the physical context. We will therefore quickly go through how
the diffusion equations arise in three different physical contexts: the diffusive trans-
port of a substance, heat conduction, and viscous fluid flow. These three cases are
of increasing complexity.
7.3.1
Diffusion of a Substance
Conservation of Mass
We consider a one-dimensional medium where a substance undergoes diffusive
transport. An example can be ink spreading in a long, thin tube filled with water.
The motion of the substance will be determined by two physical laws: (a) the con-
servation of mass and (b) Fick's law relating the velocity (flux) to the concentration.
Conservation of mass is a principle that has been verified in physical experiments
for centuries, and is hence used without uncertainty. Fick's law, on the other hand, is
a simple empirical relation, based on physical reasoning and experiments, and with
potentially significant uncertainty.
Deriving a One-Dimensional PDE
In a part ˝
D
Œa; b of the medium, we shall express the principle of mass con-
servation. Now, the world is not one-dimensional, so we imagine a tube with the x
axis as the center line and that there are no variations in the y and
z
directions. Only
variations in the x direction, inside the interval ˝
D
Œa; b, are taken into account.
Let c.x; t/ be the concentration of the substance, and let q.x; t/ be the velocity of
the substance. In a small time interval t the net mass flow of the substance into
˝ must lead to an increase in the total mass in ˝. The mass flow, during the time
interval t ,into˝ can be expressed as %q.a/t ,where% is the mass density of
the pure substance. (In time t , the substance travels a distance qt, which upon
