Biomedical Engineering Reference
In-Depth Information
function
ˆ
χ
(
)
in the region
P
and vanishing elsewhere. It is useful
to recall that in spherical coordinates and for a radial flow the tensor
D
C
has the
diagonal structure Diag
r
, equal to
χ
u
,
(
/
,
/
)
.
Neglecting body forces, and denoting by
d
t
the material derivative, we write
down the momentum balance equations for the two components (supposing they
have the same mass density
u
r
u
r
δ
):
d
u
d
t
=
∇
·
δν
T
C
+
m
C
,
(26)
d
v
d
t
=
∇
·
δ
(
1
−
ν
)
T
E
+
m
E
,
(27)
in which we define the interaction forces
m
C
,
m
E
to be
m
C
=
λ
C
(
v
−
u
)
,
(28)
m
E
=
λ
E
(
u
−
v
)
.
(29)
The coefficients
λ
C
,
λ
E
can be found by imposing two conditions:
1.
The global balance of momentum exchange rate
ˆ
=
λ
E
−
λ
C
1
χδν
1
=
m
C
+
ˆ
+
−
ˆ
+
.
0
χδν
u
m
E
χδν
v
u
u
(30)
−
ν
−
ν
2.
The Darcy's law for the flow of the extracellular liquid relative to cells
v
−
u
=
−
K
∇
p
E
,
(31)
where
K
(
1
−
ν
)
plays the role of hydraulic conductivity.
The final result deduced from Eqs. (
28
)to(
31
) is that the interaction forces have
the expressions:
1
K
+
u
ν
=
−
ˆ
,
m
C
χδ
(32)
1
−
ν
u
K
.
m
E
=
(33)
In practice Eq. (
32
) reduces to
m
C
=
−
/
K
with very good approximation.
Coming back to Eqs. (
26
)and(
27
), we note that the inertia terms can be neglected.
It is not difficult to show that those equations provide the governing differential
system for the two pressures
p
C
(
u
)
(
)
r
,
p
E
r
, namely
u
K
4
3
η
C
χδ
(
p
C
=
−
ν
+
r
−
ρ
P
)
,
(34)
u
p
E
=
−
ν
)
,
(35)
K
(
1
where
δ
(
·
)
denotes the Dirac function.
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