Biomedical Engineering Reference
In-Depth Information
where
t ¼ m
f
r
o
@r
f
(6.9)
@
p
is a material constant of dimension time. The time constant
depends upon the
porosity of the medium and the viscosity and barotropic character of the pore fluid.
The constant intrinsic permeability tensor
K
is of dimension length squared; it
depends only upon the arrangement and size of the pores in the medium and, in
particular, is independent of the pore fluid properties. The differential equation (
6.8
)
is a typical diffusion equation for an anisotropic medium; in the case of an isotropic
medium the differential equation (
6.8
) becomes
t
@
p
k
t
r
2
p
t
¼
;
(6.10)
@
where
K
k1
. Mathematically equivalent differential equations for anisotropic
and isotropic heat conductors are obtained from the Fourier law of heat conduction
and the conservation of energy. The pore fluid density
¼
r
f
satisfies the same
diffusion equation (
6.8
) as the pore fluid pressure
p
,
@r
f
@
1
t
t
¼
K : ðr rÞr
f
;
(6.11)
a result that follows from the assumed barotropic character,
r
f
¼ r
f
(
p
), of the pore
fluid, and the assumption that
@p
@r
f
is constant.
The boundary conditions on the pore pressure field customarily employed in the
solution of the differential equation (
6.8
) are (1) that the external pore pressure
p
is
specified at the boundary (a lower pressure on one side of the boundary permits flow
across the boundary), (2) that the pressure gradient
p
at the boundary is specified
(a zero pressure gradient permits no flow across the boundary), (3) that some linear
combination of (1) and (2) is specified. The complete theory for the flow of a fluid
through a rigid porous medium consists of the differential equation (
6.8
) specified
for an object
O
and boundary and initial conditions. The boundary conditions
include the prescription of some combination of the pressure and the mass flux
normal to the boundary
r
O
as a function of time,
n
q
¼ fr
f
n
v(
x*
,t
)/
r
o
¼
∂
(1/
m
)
n
K
r
p
(
x*
,
t
),
x
*
∂
O
, thus
x
;
x
;
x
;
x
@
ð
c
1
=mÞ
n
K
r
p
ð
t
Þþ
c
2
p
ð
t
Þ¼
f
ð
t
Þ;
O
(6.12)
where
c
1
and
c
2
are constants and
f(
O
. In the case
when
c
1
is zero, this condition reduces to a restriction of the boundary pressure and,
in the case when
c
2
is zero, it is a restriction on the component of the mass flux
vector normal to the boundary.
x*
,t)
is a function specified on
∂
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