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1.3.2 Diffusion
Slide : [
diffusion 
Green function 
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At the microscopic scale, diffusion is in fact related to a
random motion leading to the prototype equation (exercise 1.04)

(1.3.2#eq.1) 
where
is the diffusion coefficient.
For a homogeneous medium, the combined advectiondiffusion equation

(1.3.2#eq.2) 
can be solved analytically in terms of the Green function
(exercise 1.03)
A numerical solution is generally required to solve the equation in an
inhomogeneous medium, where
,
Below is an example of a
numerical solution describing the diffusion of an initial box function
computed using the finite element method from chapter 3.
JBONE applet: press Start/Stop
to simulate the diffusion of a box function in a periodic domain.

A harmonic ansatz
can be used to analyze the
diffusion equation and leads to the dispersion relation

(1.3.2#eq.4) 
This shows that short wavelengths (large
) are more strongly damped
(large negative imaginary
) than long wavelengths, with a decay
rate proportional to
.
This explains the numerical experiments above.
Note that the heat equation, which describes the evolution of the
temperature
in a medium with a heat conductivity
in the
presence of heat sources and sinks

(1.3.2#eq.5) 
is one particular application of the diffusion equation. It can easily be
reduced to (1.3.2#eq.1) by substituting the differential operator
in cartesian coordinates for a 1D slab.
SYLLABUS Previous: 1.3.1 Advection
Up: 1.3 Prototype problems
Next: 1.3.3 Dispersion
