Biomedical Engineering Reference
In-Depth Information
Boundary conditions associated with Navier-Stokes equations are as
follows:
- Blunt profile at the inlet:
U
=
U
0
X
with
U
0
= 1
(3.18)
- Arbitrary pressure at the outlet:
P
= 0
(3.19)
- No-slip boundary condition at the cell layer:
U
= 0
(3.20)
- Symmetric boundary conditions everywhere else on connecting pore surfaces:
P
I
+
T
n. U
= 0
t
and
·
−
∇
U
+
∇
U
·
n
= 0
(3.21)
where
t
is a unit vector tangent to the element of surface.
Moreover, a fair approach in mean oxygen concentration repartition along
the principal
X
perfusion direction of the pore channel can be obtained in
one-dimension formulation (Pierre and Oddou 2007) by solving the following
stationary diffusion-convection-reaction equation:
d
2
C
dX
2
dC
/
dX
−
f
(
C
) = 0
−
Pe
×
Da
×
(3.22)
In this equation, the main transport mechanisms are diffusion and
hydraulic convection (quantified by the Peclet number), associated with reac-
tion due to the cell oxygen consumption (quantified by the Damkohler num-
ber).
Similar equation can be obtained for ionic nutrients when considering
porous media presenting large pores. Nevertheless, for small hydraulic per-
meability, such as in the case of the mature cartilage for instance, the other
physicochemical phenomena described in Section 3.2.4 have to be taken into
account. Indeed, in addition to the Fickian diffusion involving the gradient of
the ionic force
C
i
, the electromigration of c
h
arged species in response to the
gradient of the reduced streaming potential
ψ
=
F ψ/RT
has to be considered.
Thus, the ionic flux becomes (Lemaire et al
.
2008):
−
J
±
=
ϕ
)
ψ
−
D
±
exp (
∓
∇
C
i
±
C
i
∇
(3.23)
Owing to the double-layer effects, the cationic or anionic diffusion coef-
ficient
D
±
is weig
h
ted by a Boltzmann-like term involving the double-layer
reduced potential
ϕ
. Moreover the velocity used in the definition of the Peclet
number is the sum of three contributions in response to hydraulic, osmotic,
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