Biomedical Engineering Reference
In-Depth Information
é
2
2
2
ù
¶
c
¶
c
¶
c
¶
c
=
D
+
+
ê
ú
2
2
2
¶
t
¶
ξ
¶
η
¶
ζ
ë
û
1
1
1
(5.17)
Equation (5.17) has an isotropic value for the diffusion constant
D
to the price
of a rotation plus a homothetic transformation of the axes. In Section 5.3.8, for rea-
sons that we discuss, we proceed differently. We modify the computational domain
by a homothetic transformation to the price of a change of the isotropic diffusion
coefficient into a matrix of anisotropic diffusion coefficients.
5.3.3 Spreading from a Point Source — 1D Case
We analyze here the diffusion of a substance (tracers or nanoparticles) in a one-
dimensional geometry. Suppose that a very small spot of concentration of tracer
particles has been initially placed in a rectangular capillary of a very small cross
section (Figure 5.5).
In such a case, the diffusion may be considered one-dimensional and depends
one two variables: the time
t
and the axial coordinate
x
. The initial condition may
be approximated by:
c x t
( ,
)
=
c
δ
( )
x
(5.18)
0
0
where
d
(
x
)is the Dirac function. With such an initial condition, the solution to (5.5)
is
2
x
Dt
-
c
0
4
c x t
( , )
e
(5.19)
=
4
π
Dt
The solution (5.19) shows that the distribution profile of concentration in tracers is
Gaussian with
x
. In Figure 5.6, we have plotted the solution of (5.19) with the scal-
ing
c
/
c
0
, for
D
= 10
-10
m
2
/s at three different times (0.2, 1, and 10 seconds).
Remark that the characteristic nondimensional group
2
x
Dt
appears in the solu-
4
tion (5.19) to (5.5). This group represents in fact a characteristic diffusion length.
A characteristic diffusion length may be defined by
x
» 4
Dt
(5.20)
Figure 5.5
Schematic view of the diffusion of tracers in a one-dimensional geometry.
