Biomedical Engineering Reference
In-Depth Information
Observing from (
10.33
) that the selected components of the stress
T
ðsÞ
may be
expressed in terms of the remaining components and the total stress.
X
Ns
T
ðsÞ
¼
T
T
ðbÞ
;
(10.73)
b¼
1
then the first two terms of the sum in (
10.70
) have the alternate representation
X
X
N
a¼
1
½
N
s
b¼
1
½
^
T
ðaÞ
:
D
ðaÞ
v
ðaÞ
ðaÞ
¼
T
:
D
ðsÞ
þ
T
ðbÞ
: ½r
u
ðbÞ
þ
u
ðbÞ
r
T
ðbÞ
(10.74)
where (
10.72
) has been employed. Substitution of (
10.74
) into (
10.70
) yields the
inequality
X
N
a¼
1
er½r
ðbÞ
D
s
D
s
D
t
r
y
D
t
1
y
r
q
ry þ
u
ðbÞ
þ
T
:
D
ðsÞ
X
Ns
þ
1
½
T
ðbÞ
: ½r
u
ðbÞ
þ
u
ðbÞ
r
T
ðbÞ
r½C
ðbÞ
r
ðbÞ
u
ðbÞ
0
:
(10.75)
b
¼
The inequality (
10.75
) for accelerationless processes in the absence of action-at-
a-distance forces will be applied, after further specializing it for a porous medium,
in the section after next.
10.8 Constitutive Equations for a Multi-Constituent
Porous Medium
In this section the conservation principles of mass, linear momentum and energy,
and the entropy inequality are first specialized to the development of a continuum
mixture model of a fluid-saturated porous medium containing a porous solid and
N-1 other species including a pore fluid and neutral solutes. Then the constitutive
equations for the multicomponent porous medium are developed following these
the four linear continuum theories considered, but here placing a greater emphasis
on the use of the entropy inequality to restrict the constitutive assumptions. At the
end of the section the reduced entropy inequality is employed to restrict the values
of the material coefficients in Fourier' and Darcy' laws.
The porous solid is taken to be the selected constituent in the mixture, hence
the “
s
” subscript or superscript notation now means solid as well as “selected”.
The porous solid constituent is special because it contains all the other constituents
and limits their behavior. From the viewpoint of the modeler it serves as a structure
Search WWH ::
Custom Search