Biomedical Engineering Reference
In-Depth Information
porous medium,
r
o
a reference value of density,
v
the velocity of the fluid passing
through the pore,
f
the porosity of the medium, and
q
the volume flux of fluid per
unit area,
q
¼ fr
f
v
/
r
o
, through the pores. This constitutive idea is that the fluid
volume flux
q
¼ fr
f
v
/
r
o
at a particle
X
is a function of the pressure variation in the
neighborhood of
X
,
N
ð
X
Þ
. The constitutive relation for the rigid porous continuum
is Darcy's law. Darcy's law relates the fluid volume flow rate,
q
¼ fr
f
v
/
r
o
, to the
r
p
) of the pore pressure
p
,
gradient (
H
T
q
¼ fr
f
v
=r
o
¼
H
ð
p
Þr
p
ð
x
;
t
Þ;
H
ð
p
Þ¼
ð
p
Þ; ð
5
:
36D
Þ
repeated
material symmetries greater than monoclinic. The conservation law that is com-
bined with Darcy's law is the conservation of mass (3.6) in a slightly rearranged
form. In (3.6) the density
r
is replaced by the product of the porosity and the fluid
density,
fr
f
, in order to account for the fact that the fluid is only in the pores of the
medium, and the resulting mass balance equation is divided throughout by
r
o
, thus
r
o
@fr
f
1
t
þrðfr
f
v
=r
o
Þ¼
0
:
(6.4)
@
In the case of compressible fluids it is reasonable to assume that fluid is barotropic,
that is to say that the fluid density
r
f
is a function of pressure,
r
f
¼ r
f
(
p
), in
which case (
6.4
) may be written as
r
o
@r
f
f
@
p
t
þr
q
¼
;
0
(6.5)
@
p
@
where
fr
f
v
/
r
o
has been replaced by
q
and where it has been assumed that the
porosity
is not a function of time. Substituting (5.36D) into (
6.5
), and multiplying
through by the inverse of the factor multiplying the partial derivative of the pressure
p
with respect to time, a differential equation for the pore pressure is obtained,
f
rð
@
p
r
o
f
p
@r
f
@
t
¼
H
r
p
Þ:
(6.6)
@
If it is assumed that
H
and
@
p
@r
f
are constants, and if the viscosity
m
of the pore
fluid is introduced by the substitution
1
m
H
K
;
(6.7)
then
@
p
1
t
t
¼
K
: ðr rÞ
p
;
(6.8)
@
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