Biomedical Engineering Reference
In-Depth Information
The relative change in
r
is
ρ ρ
-
ρ ρ
-
L
P
L
=
φ
ρ
ρ
L
L
For the typical value
f
= 0.02,
r
P
= 2,000 and
r
L
=1,000 kg/m
3
, the relative
change is 2%.
6.2.5.4 Transport System of Equations
In the case where the concentration effect on the fluid properties is not negligible,
the advection-diffusion equation is not decoupled anymore. It has been seen in Sec-
tion 6.1.5.1 that concentration is proportional to volume fraction. Taking advan-
tage of the linearity of the transport equation, we obtain the following system for a
creeping flow (Stokes hypothesis)
�
Ñ =
.
U
0
�
(6.26)
Ñ = D
P
η
U
�
¶
φ
+
U
.
Ñ =
φ
D
D +
φ
S
¶
t
In the case where the concentration of species is sufficient to affect the proper-
ties of the liquid, we have to solve (6.26) using the constitutive relations
5
é
ù
ηφ η
( )
=
1
+
φ
0
ê
ú
2
ë
û
(6.27)
k T
1
D
B
0
D
( )
φ
=
=
5
6
π
R
é
5
ù
H
1
+
φ
η
1
+
φ
0
ê
ú
2
2
ë
û
The system is now strongly coupled and no more linear due to terms of the form
φ
φ
1
. The numerical solution requires a coupled multiphysics approach where the
unknowns are the vectors (
u
,
v
,
w
,
P
,
f
) at each node of the computational domain
and the use of a nonlinear solver.
6.2.6 Dimensional Analysis and Peclet Number
Let us start from the usual form of the diffusion-advection equation without source
or sink.
é
2
2
2
ù
¶
c
¶
c
¶
c
¶
c
¶
c
¶
c
¶
c
+
u
+
v
+
w
=
D
+
+
(6.28)
ê
ú
2
2
2
¶
t
¶
x
¶
y
¶
z
¶
x
¶
y
¶
z
ë
û
In this problem, there are four parameters: the velocity
U
¥
, the length scale
L
,
the incoming concentration
c
0
, and the diffusion coefficient
D
. These four param-
eters contain three different units: m, s, kg (or mole). In such a case, Buckingham's
