Biomedical Engineering Reference
In-Depth Information
Thus the mesh size must be quite small. For D = 10 -10 m 2 /s and u =1 mm/s, one
finds D x = 4 m m. This is approximately the same size as that of the meshes in the
boundary layer. The problem could be rapidly intractable, especially if it is a three-
dimensional problem. In order to avoid this difficulty and relax the size of the mesh
outside the boundary layer, an artificial diffusion is added to the diffusion constant
in the meshes outside the boundary layers. If this additional diffusion is correctly
chosen, it reduces the value of the mesh Peclet number, allows for larger meshes, and
does not affect the solution very much since it applies in a domain where the gradient
of concentration is small. It is usual to choose the value of the added diffusion by
β D
u x
D
=
(6.42)
add x
,
2
where b is a coefficient depending on the problem. Such a formulation respects the
condition imposed by the boundary layer because u and D x are small in the bound-
ary layer. Note that the equivalent numerical diffusion coefficient is anisotropic
é
u x
D
ù
D
+
β
0
0
ê
ú
2
ê
ú
v y
D
ê
ú
D
=
0
D
+
β
0
(6.43)
num
ê
ú
2
ê
ú
w z
D
ê
ú
0
0
D
+
β
ê
ú
2
ë
û
6.2.8.3 Time Step
Time step and mesh size are not independent. If an explicit formulation is chosen,
the Courant condition yields
u t
D < D
x
(6.44)
and introducing the limitations on the mesh size
4
D
( )
x
æ
δ
ö
num x
,
u t
D < D <
x
min
,
(6.45)
ç
÷
è
ø
u
2
Thus,
æ
4
D
ö
δ
( )
x
num x
,
D <
t
min
, 2
(6.46)
ç
÷
2
è
u
ø
u
This condition is usually very restrictive. Typically it yields values of the time
step of the order of 10 -2 seconds. Having in mind that the duration of a biological
reaction is of the order of 10 minutes to 10 hours, the number of time step is quite
large. This is why semi-implicit or implicit algorithms are preferred for solving
transient concentration problems in microsystems.
To conclude, it is important for the numerical modeling of mass transfer in
microsystems to have very small meshes in the boundary layers, to use an added
diffusion coefficient to avoid numerical instabilities, and to choose a semi-implicit
or implicit solution scheme.
 
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