Biomedical Engineering Reference
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the exit pressure of
p
o
.At
t
¼
0 the pore fluid pressure in the reservoir at
x
1
¼
L
is
raised from
p
o
to
r
gh
þ
p
o
and held at
r
gh
þ
p
o
for all subsequent times,
1 >
t
>
0. The pressure at the entrance to the second reservoir at
x
1
¼
0, or the exit from
the layer, is held at
p
o
for all times,
. (a) Show that the steady-state
solution (that is to say after the startup effects of the pressure changes at
t
1 >
t
> 1
¼
0 have
r
gh
þ
p
o
at
x
1
¼
L
to
vanished) is a linear variation in the pore fluid pressure from
p
o
at
x
1
¼
0. (b) How does the pressure
p
(
x
1
,
t
) in the layer evolve in time to the
linear long-term steady-state solution?
Solution
: The problem is one-dimensional in the direction of
x
1
. The one-
dimensional form of the differential equation (
6.8
)is
2
p
@
p
@
K
11
t
@
t
¼
x
1
:
ð
u
Þ
unsteady
@
In the special case of steady states it reduces to
2
p
@
x
1
¼
0
:
ð
s
Þ
steady
@
The solution to the steady-state equation (s) subject to the condition that
p
¼
p
o
at
x
1
¼
0 and
p
¼ r
gh
þ
p
o
at
x
1
¼
L
for all
t
for all times,
1 >
t
>>
0is
p
(
x
1
)
¼ r
gh
(
x
1
/
L
)
þ
p
o
. This result represents a linear variation in the pore fluid
pressure from
0. In the case of unsteady flow a
solution to the differential equation (u) for the unsteady situation is sought, subject
to the conditions that
p
r
gh
þ
p
o
at
x
1
¼
L
to
p
o
at
x
1
¼
¼
p
o
everywhere in the medium and on its boundaries for
t
<
0, that
p
¼
p
o
at
x
¼
0 for all times
1 >
t
>
0 and
p
¼ r
gh
þ
p
o
at
x
1
¼
L
for all
0. Before solving the differential equation (u), it is first rendered
dimensionless by introducing the dimensionless pressure ratio
P
, the dimensionless
coordinate
X
, and the dimensionless time parameter
T
, thus
1 >
t
>
p
p
o
x
1
L
;
K
11
L
2
t
t
;
P
¼
;
X
¼
and
T
¼
r
gh
respectively. These equations are solved for
p
,
x
1
, and
t
,
L
2
t
K
11
T
p
¼ r
ghP
þ
p
o
;
x
1
¼
LX
;
and
t
¼
;
and substituted into the differential equation (u) for the unsteady situation which
then converts to the dimensionless version of this differential equation
2
P
@
P
T
¼
@
X
2
:
@
@
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