Information Technology Reference
In-Depth Information
The real benefit of cylindrical coordinates occurs when we have
radial symme-
try
, i.e., no quantities in the problem depend on the angle . This is often the case
(cooling a can of beer is one example). One can then utilize @=@
D
0,which
means that the three-dimensional problem in Cartesian coordinates is reduced to a
two-dimensional problem in cylindrical coordinates. If, in addition, we have small
variations in the
z
direction, such that @=@
z
D
0 is a reasonable approximation, we
get a one-dimensional problem involving r as the only space coordinate.
The term
r
2
u
can be shown to be reduced to the simple form
r
@
u
@r
1
r
@
@r
(7.7)
when
u
depends on the cylindrical coordinate r and time t only.
Spherical Symmetry
Diffusion in spherical geometries can benefit from switching to spherical coordi-
nates, at least if it is reasonable to assume that the unknown depends only on the
spherical radial coordinate r in addition to time. The
r
2
u
term can in this case be
showntoreduceto
r
2
:
1
r
2
@
@r
@
u
@r
(7.8)
A nice feature of a spherically symmetric diffusion equation is that it can be reduced
to a standard one-dimensional equation by a simple transformation. We introduce a
new function
v
related to
u
by
u
D
v
=r. Inserting
u
D
v
=r into
r
2
C
f.r/
@
u
@t
D
k
1
r
2
@
@r
@
u
@r
(7.9)
yields
@t
D
k
@
2
v
@
v
C
rf .r / :
(7.10)
@r
2
We can therefore use a simulation program (or known analytical solutions) of the
standard one-dimensional diffusion equation to find solutions of spherically sym-
metric
three-dimensional
diffusion. This is a quite convenient observation, since
computing with a Cartesian coordinate is always easier than computing with the
radial spherical coordinate.
From One to Three Dimensions
As soon as one has understood the basics of diffusion in one space dimension, and
learned the technicalities of the solution procedures and their implementations, the
knowledge can be generalized in quite a straightforward manner to two or three
