Biomedical Engineering Reference
In-Depth Information
Accordingly, the Navier-Stokes equation (1.9) may be integrated to give
∂x
j
ν
f
∂
f
f
∂ u
i
∂
∂x
j
1
ρ
f
∂
p
∂
u
i
∂x
j
+
∂
u
j
∂x
i
f
f
=
+
u
j
u
i
−
+
∂t
∂x
i
ρ
+
ν
f
∂u
i
n
j
dA
p
1
V
f
∂x
j
+
∂u
j
∂
∂x
j
f
+
−
−
u
j
u
i
(1.14)
∂x
i
A
int
To close the foregoing macroscopic momentum equations (1.14), the terms
associated with the surface integral are modeled according to Vafai and Tien
(1981) as
+
ν
f
∂u
i
n
j
dA
1
V
f
p
ρ
f
∂x
j
+
∂u
j
∂
∂x
j
f
−
−
u
j
u
i
∂x
i
A
int
bε
2
f
1
/
2
ν
f
K
εu
i
f
f
f
=
−
−
u
k
u
k
u
i
(1.15)
such that
∂
f
f
f
f
∂
u
i
∂
∂x
j
1
ρ
∂
p
∂
∂x
j
ν
f
u
i
+
∂
u
j
f
f
=
+
u
j
u
i
−
+
∂t
∂x
i
∂x
j
∂x
i
b
ij
ε
2
f
1
/
2
ν
f
K
ij
ε
f
f
f
−
u
j
−
u
k
u
k
u
j
(1.16)
where
K
ij
and
b
ij
are the permeability and Forchheimer tensors, respectively.
These tensors, which depend on the anatomical structure, can be deter-
mined following the procedure established for anisotropic porous structure
(Nakayama et al. 2004), as sucient information on the anatomical structure
and properties is provided. For the vessels of suciently small diameter, the
foregoing equation reduces to Darcy's law:
f
1
ρ
∂
p
ν
f
K
ij
−
−
u
j
= 0
(1.17)
∂x
i
f
is the Darcian velocity (i.e., apparent velocity). We may
use the Darcy law for most tissue regions except for the regions where large
arteries or veins are located.
where
u
j
=
ε
u
j
1.4 Two-Energy Equation Model for Blood Flow
and Tissue
1.4.1 Related Work
Pennes (1948) carried out temperature measurements in the limb and found
that the maximum muscle temperature is located very close to the axis of
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