Biomedical Engineering Reference
In-Depth Information
Quintard and Whitaker (1993), and many others, we decompose a variable
into its intrinsic average and the spatial deviation from it:
f
+
φ
φ
=
φ
(1.4)
We shall exploit the following spatial average relationships:
f
=
φ
1
f
f
+
φ
1
φ
2
f
φ
1
φ
2
φ
2
(1.5)
∂φ
∂x
i
=
∂
or
∂φ
∂x
i
f
f
∂x
i
φ
1
V
=
1
ε
∂ε
φ
1
V
f
+
φn
i
dA
+
φn
i
dA
∂x
i
A
int
A
int
(1.6a,b)
and
∂φ
∂t
=
∂
φ
(1.7)
∂t
where
A
int
is the local interface between the blood and solid matrix, while
n
i
is the unit vector pointing outward from the fluid side to the solid side. The
similarity between the volume averaging and the Reynolds averaging used in
the study of turbulence is quite obvious. However, it should be noted that
the present volume averaging procedure is somewhat more complex than the
Reynolds averaging procedure, since it involves surface integrals, as clearly
seen from equation (1.6). It should also be noted that biological tissues in
reality are highly compliant. To include the compliance of the tissues, the fore-
going spatial averaging relationships must be modified accordingly to account
for the deformation of the elementary control volume. In the present study,
we neglect such effects for simplicity.
We subdivide the anatomic structure into the blood phase (fluid phase) and
the tissue and other solid tissue phase (solid matrix phase), in which metabolic
reactions may take place. We shall consider the microscopic governing equa-
tions, namely, the continuity equation, the Navier-Stokes equation, and the
energy equation for the blood phase and the heat conduction equation for the
solid matrix phase.
For the blood phase:
∂u
j
∂x
j
= 0
(1.8)
ν
f
∂u
i
∂x
j
∂u
i
∂t
∂
∂x
j
1
ρ
∂p
∂x
i
∂
∂x
j
+
∂u
j
∂x
i
+
u
j
u
i
=
−
+
(1.9)
ρ
f
c
p
f
∂T
∂x
j
u
j
T
=
k
f
∂T
∂x
j
∂
∂
∂x
j
+
(1.10)
∂t
For the solid matrix phase:
ρ
s
c
s
∂T
∂t
k
s
∂T
∂x
j
+
S
m
∂
∂x
j
=
(1.11)
Search WWH ::
Custom Search