Biomedical Engineering Reference
In-Depth Information
Note that the functional dependence of the terms in (
10.90
) is not uniform. In
particular, note that
m
ðbÞ
in (
10.90
) depends only upon
ð
E
; y; f
ðbÞ
Þ
, the independent
variables of the free energy, while
q
depends upon
by
assumption (
10.84
). In the case when the anisotropic form of the Fourier law
of heat conduction gives the heat flux,
q
ð
E
; y; ry; f
ðbÞ
;
v
ðb=sÞ
Þ
¼
K
∇
y
, the inequality reduces to
(
1
/
0 and it requires only that
K
be positive definite, since the
temperature is positive as a consequence of its definition. The entropy inequality
then reduces to the following:
y
)
∇
y
K
∇
y
X
Ns
b¼
1
½
v
ðb=sÞ
ðf
ðbÞ
rm
ðbÞ
Þ
0
:
(10.91)
Developments beyond this point could include the specification of constitutive
equations for the gradients of the chemical potentials,
f
(
b
)
∇
m
(
b
)
. The inequality
(
10.91
) may then be used to restrict
the coefficients in these constitutive
relationships.
As an example of the application of this result consider Darcy's law (5.36D)
which, in the case of an incompressible fluid,
r
f
¼ r
o
, is written
H
T
q
¼ f
v
¼
H
ð
p
Þr
p
ð
x
;
t
Þ;
H
ð
p
Þ¼
ð
p
Þ:
(10.92)
Considering the chemical potential (
10.87
) to be a function of the single constit-
uent pressure only, then (
10.91
) may be written as
f
v
r
p
0
:
(10.93)
Substituting
f
v
from (
10.92
) into (
10.93
) it follows that
r
p
H
r
p
0
;
(10.94)
which requires that
H
(
p
) be positive definite. This result is equivalent to (5.29D).
Problem
10.8.1 Derive the statement of mass balance for the entire mixture, (
10.76
), by
summing (
10.76
) over all constituents and employing (
10.77
).
10.9 Relevant Literature
Some of the literature relevant to mixture theory is discussed in Sect. 9.14. Bowen
(
1967
) summarized the formative years of mixture theory. A readable history of the
subject and its applications in the period 1957-1975 is given by Atkin and Craine
(
1976a
,
b
). de Boer (
1996
,
2000
) has presented more up-to-date histories. Of key
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