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 further that
v
θ
= 0. Thus, the governing
equations for
v
z
=
v
reduce to:
−
2
∂
∂
v
t
=
∂
∂
v
1
∂
∂
v
r
∂
∂
p
ρ
(,)
+
µ
+
frt
(4.23)
2
r
r
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 at
r
=
r
o
,
∀
t
,
∂∂
vr
r
o
/
=
0
.
=
Furthermore, the function
f
(
r, t
) is given by:
{
}
−
1
22
2
2
an
−
1
frt
(,)
=
nErt
ε
(,)
kr
/
2
β
Sinh
β
nr
o
( / ) t
−
β
(4.24)
where
k
2
= (
n
2
/
DkT
).
An exact solution to the given set of equations can be shown to be:
ε
t
∞
∑
0
−
(
)
2
(
)
2
(
)
µρβ
/
mt
(/)
µρβ
ξ
vrt
(,)
=
e
k
β
r
e
A
β
ξξ
d
(4.25)
m
0
m
m
m
=
1
where
β
m
,
s
are the positive roots of the following transcendental equations:
Jr
J
()
(
β
β
Yr
Yr
()
(
β
β
0
i
0
i
−
=
0
(4.26)
′
′
r
)
)
0
o
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
,
.
/
1
β
ββ
Jr
Jr
()
(
Yr
Y
()
β
−
(/ )
12
0
0
−
(/ )
12
k
(, )
β
r
=
N
−
=
NR r
m
(, )
β
(4.27)
0
m
0
′
)
β
′
(
β
r
)
m
0
0
m
0
0
where
2
2
2
Nr R
=
(
2
)
(
β
,
r r
)
−
(
/)
2
R
( ,)
β
r
,
(4.28)
′
0
0
m
0
i
0
m
i
r
0
∫
A
(,
βξ
)(
=
1
µρ ς βς ςξς
)
k
(, )(,
f
)
d
.
(4.29)
m
0
m
r
i
