Biomedical Engineering Reference
In-Depth Information
Let us further assume that, due to cylindrical symmetry, the velocity vector
v
=
(
v
r
,
v
θ
,
v
z
) is such that only
v
z
depends on
r
and, furthermore,
v
θ
= 0. Thus, the
governing equations for
v
z
=
v
reduce to
2
∂
∂
v
t
∂
∂
v
r
∂
∂
v
r
∂
p
(
+
−
1
ρ
=
frt
,
µ
+
r
−
(6.79)
2
∂
r
Let us assume a negligible radial pressure gradient and assume the following
boundary and initial conditions:
(
)
At
t
= 0,
r
i
≤
r
≤
r
o
,
v
= 0; at
r
=
r
i
,
∀
t
,
v
(
r
i
) = 0; and
rr t v r
o
=∀∂∂
,
,
/
=
0
r
o
=
Furthermore, the function
f
(
r
,
t
) is given by:
{
}
−
1
()
=
()
(
)
−
22
2
2
an
−
1
frt
,
nErt
ε
,
κ
r
/
2
β
Sinh
β
n r
/
r
o
t
β
(6.80)
where
2
/
DkT
).
An exact solution to the given set of equations can be shown:
κ
2
= (
n
ε
∞
t
∑
()
=
−
(
)
(
)
(
)
(
)
2
∫
2
µρβ
/
t
µρβ ξ
/
vrt
,
e
m
κβ
r
e
m
A
βξ ξ
,
d
(6.81)
0
m
m
0
m
=
1
where
β
m
,
s
are the positive roots of the following transcendental equation:
()
′
()
()
′
()
Jr
Jr
β
β
Yr
Yr
β
β
0
i
0
i
(6.82)
−
=
0
0
0
0
0
′
0
are the Bessel functions of zero order of first and second kind
and their derivatives evaluated at
r
o
, respectively, and
where
J
0
,
Y
0
,
J
′
0
,
Y
{}
′
(
)
(
.
2
β
ββ
Jr
Jr
Yr
Y
β
0
0
0
0
(
)
=
)
)
(
)
−
(
0
0
−
(
12
/
12
/
κβ
rN
−
=
NR r
m
β
(6.83a)
0
m
,
)
(
)
0
,
β
′
β
r
m
0
0
m
0
0
(
)
)
−
(
)
(
(
)
Nr
=
0
2
/
2
R
2
β
r r
2
/
2
R
0
2
β
r
′
(6.83b)
m
,
0
i
m
,
i
r
(
)
=
(
)
(
) (
)
∫
0
A
βξ
1
/
µρ
ςκ βς
f
ςξ ς
,
d
(6.83c)
m
,
0
m
,
r
i
