Biomedical Engineering Reference
In-Depth Information
The formulae for the variation of
a
(
t
) and
b
(
t
) with time
t
can be obtained
by substituting Equation (4.45) into Equation (4.17) as
BB
t
′ +
(
=+ε− ε+
− ′
ε+ η
BB
21
)
1
2
at
aa
a
Bt
e
,
2
0
0
∞
0
∞
∞
2
∞
2
(4.46)
)
2
(
BB
t
′ +
=+η− η−
′ −
BB
B
ε−
′ −
BB
21
2
1
2
1
bt
()
bb
b
η
te
2
0
0
∞
0
∞
∞
∞
4
2
2
The final radii of the cylinder are then
()
()
(4.47)
a
=
lim
a t
=
a
(1),
+ε
b
=
lim
b t
=
b
(1
+η
)
∞
0
∞
∞
0
∞
t
→∞
t
→∞
Case C:
When
and
B
2
= 0, the solutions of the equations are
BB
0
1
= ′ <
2
(
)
1
(4.48)
ε=−ε
e
Bt
,
η=−
B
′
ε+η
t
e
Bt
1
∞
1
∞
∞
This can also be written as
1
(4.49)
(
)
()
ε=ε−ε
()
t
e
Bt
,
η =η −
t
B
′
ε+η
t
e
Bt
1
∞∞
∞
1
∞
∞
The formulae for the variation of
a
(
t
) and
b
(
t
) with time
t
can be obtained by
substituting Equation (4.49) into Equation (4.17), as follows:
(
)
1
(4.50)
(
)
()
Bt
Bt
at
()
=+ε− =+η− ′ε+η
∞
aa
1
e
,
b t
bb
b
Bt
e
1
0
0
0
0
∞
01
∞
∞
The final radii of the cylinder are then
()
()
a
=
lim
a t
=
a
(1
+ε
),
b
=
lim
b t
=
b
(1
+η
)
(4. 51)
∞
0
∞
∞
0
∞
t
→∞
t
→∞
Case D:
When
BB
0
1
= ′ <
and
B
′ =
0
, the solutions of the equations are
2
1
(
)
Bt
Bt
ε=−η+ε
Bt
e
,
η=−η
e
(4.52)
1
1
2
∞
∞
∞
This can also be written as
(
)
()
()
1
(4.53)
Bt
Bt
ε=ε− η+ε
t
B
t
e
,
η =η −η
t
e
1
∞
2
∞
∞
∞
∞
