Biomedical Engineering Reference
In-Depth Information
Similarly, we define
2
J
1
(δj
3
/
2
)
(J
0
(δj
3
/
2
)δj
3
/
2
)
M
1
e
jε
1
=
1
−
(4.21)
2
J
1
(δj
3
/
2
y)
(J
0
(δj
3
/
2
)δj
3
/
2
)
M
2
(y)e
jε
2
(y)
=
1
−
denoting the modulus and phase of the Bessel functions of first kind
J
i
and
i
th
order [
1
].
For an
elastic pipeline
,the
no-slip
condition is still valid (
w
=
0for
y
=±
1),
such that the radial velocity is
y
−
A
P
e
jω(t
−
2
J
1
(δj
3
/
2
y)
J
0
(δj
3
/
2
y)δj
3
/
2
jωR
2
ρ
z
c
)
u(y)
=
c
˜
y
M
P
e
j(ωt
−
Φ
P
)
2
J
1
(δj
3
/
2
y)
J
0
(δj
3
/
2
)δj
3
/
2
Ry
2
ρ
=
−
(4.22)
c
˜
and using (
4.21
), the equivalent form of (
4.22
) becomes
R
2
ρ
c
M
P
M
2
(y)e
j(ωt
−
Φ
P
+
ε
2
(y))
u(y)
=
(4.23)
˜
The flow is given by
πR
2
M
P
ωρ
πR
4
M
P
μδ
2
M
1
e
j(ωt
−
Φ
P
−
π/
2
+
ε
1
)
M
1
e
j(ωt
−
Φ
P
−
π/
2
+
ε
1
)
Q
=
=
(4.24)
The effective pressure wave has the general form of
z
c
)
A
P
e
j(ω(t
−
−
φ
P
)
,
p(z,t)
=
(4.25)
where
φ
P
can be a phase shift for
z
=
0at
t
=
0. It follows that
dp
dz
=
A
P
ω
˜
z
˜
M
P
e
j(ωt
−
Φ
P
)
e
j(ω(t
−
c
)
−
φ
P
+
π/
2
)
−
=
(4.26)
c
For
z
=
0, it follows that
A
P
ω
c
0
√
M
1
M
P
e
j(ωt
−
Φ
P
)
e
j(ωt
−
φ
P
+
π/
2
−
ε
1
/
2
)
=
(4.27)
from which we have
A
P
ω
M
P
=
c
0
√
M
1
(4.28)
´
and
Φ
P
=
φ
P
−
π/
2
+
ε
1
/
2
(4.29)
