Biomedical Engineering Reference
In-Depth Information
and Darcy's law (2.47) becomes
∂σ
∂
∂
∂
∂σ
∂
∂
∂
=
C
(
Uu
−
),
=
C
(
Uu
−
)
(2.73)
r
r
z
z
r
t
z
t
Substituting Equation (2.71) into Equations (2.72) and (2.73), we have
(2
NAQu QR U
++ℵ++ℵ =
)
(
)
0,
r
r
ℵ+ℵ+
∂
Qu RU
C
∂
−=
(
uU
)
0
(2.74)
r
r
r
r
t
and
2
2
++
∂
∂
++
∂
u
z
U
z
z
z
(2
NAQ
)
(
QR
)
=
0,
2
2
∂
2
2
∂
∂
u
z
∂
∂
U
z
∂
∂
−=
z
z
Q
+
R
+
C
(
uU
)
0
(2.75)
z
z
2
2
t
where
2
ℵ=
∂
∂
1
∂
∂
1
+
−
(2.76)
2
2
r
r
rr
Substituting the assumed solution (2.70) into Equation (2.75), an identity
and a relationship are obtained as
∂
∂
Cz
(()
Dt Dt
−
( ))
=
0
(2.77)
1
2
t
which is equivalent to
D
1
(
t
) =
D
2
(
t
) + Θ
(2.78)
Making use of Equations (2.70), (2.71), and (2.78), the boundary conditions
(2.65)-(2.67) become
++
∂
u
r
AQ
u
r
r
r
(2
NAQ
)
∂
++ −
(
)
Dt
( )
1
(2.79)
−
pt
() at
ra
=
++
∂
U
r
U
r
=
r
r
(
QR
)
∂
+− −Θ
Dt
()
1
0at
rb
=
