Biomedical Engineering Reference
In-Depth Information
Moreover, nondimensional numbers comparing different physical phenomena
occurring in the transport process are also introduced:
Pe
=(
u
0
×
a
)/
D
O
2
;Re=
u
0
×
a
)/
r
Da
=(
σ
cel
×
V
max
×
a
)/(
c
0
×
D
O
2
);
λ
M
=
K
M
/
c
0
(3.11)
As already mentioned, the Peclet number
Pe
represents the ratio of con-
vective and diffusive effects for problems in mass transport. It is the equivalent
to the Reynolds number Re for the momentum transfer in fluid dynamics phe-
nomena, with v standing for the kinematic viscosity of the fluid in place of
the solute diffusion coecient
D
O
2
. Furthermore, the Damkolher number
Da
represents the ratio of reactive (for instance, O
2
consumption of the cell layer
with surface density
σ
cel
) and diffusive effects. And finally the dimensionless
Michaelis constant
λ
M
is reduced using the reference concentration.
In this framework, the distribution of the oxygen concentration within the
channel is governed by the following stationary diffusion-convection equation,
written in its dimensionless form:
∇
.
C
= 0
U
×
∇
(
C
)+
Pe
−
×
(3.12)
Since cells are only attached on the pore wall, their consumption is rep-
resented by a flux (noted
N
O
2
) boundary condition following the Michaelis-
Menten kinetics:
N
O
2
·
f
(
C
)
−
n
=
−
Da
×
(3.13)
where
f
(
C
)=
C
/(
C
+
λ
M
) (approximately,
f
(
C
)=
C
for
C
≤
λ
M
and
f
(
C
)=1 for
C
≥
λ
M
) and
n
is the unit vector normal to the element of wall
surface.
Other boundary conditions associated with this transport equations are as
follows:
- At the inlet, the given reference oxygen concentration:
C
inlet
=
C
0
= 1
(3.14)
- A convective flux at the outlet of the pore:
n
outlet
.
∇
C
= 0
−
(3.15)
- Symmetric boundary conditions everywhere else on connecting pore surfaces:
n
sym
.
C
U
×
∇
C
+
Pe
−
×
(3.16)
The velocity field
U
appearing in the convective term of equation (3.12)
is obtained by solving the dimensionless Navier-Stokes equations, considering
the steady flow of an incompressible Newtonian fluid:
Re
U
.
∇
U
=
∇
(
P
)+∆
U
∇
. U
= 0
−
and
(3.17)
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