Biomedical Engineering Reference
In-Depth Information
The entropy inequality for a mixture may now be formulated using the entropy
inequality for the single component continuum (
10.50
) as the guide. The term
D
s
D
t
in (
10.50
) is eliminated using (
10.63
). The heat supply density
r
in (
10.50
)is
replaced by that for the mixture given by the second of (
10.37
), thus entropy
inequality for a mixture takes the form
r
"
#
X
D
a
X
X
X
N
ðaÞ
D
t
N
N
N
h
ð
a
Þ
y
r
ðaÞ
ðaÞ
u
ðaÞ
1
y
1
r
ðaÞ
þr
þ
1
r½r
ðbÞ
u
ðbÞ
1
r
ðaÞ
r
ðaÞ
0
:
a
¼
a
¼
1
a
¼
a
¼
(10.64)
It is important to note that, while there were forms of each of the conservation
principles for each of the constituents (
10.18
), (
10.20
), (
10.21
) that were summed
over to obtain statements of those principles that applied to the mixture as whole,
(3.6), (3.29), (3.52), respectively, it was not assumed that there were constituent-
specific forms of the entropy inequality (
10.64
). The literature is somewhat divided
on the use of constituent-specific forms of the entropy inequality (Bowen,
1976
,
Sect. 1.7). The conservative position is to assume only the mixture level inequality.
Thus the entropy inequality employed here only makes a statement for the entire
mixture, not for any particular constituent.
The remainder of this section presents the development of an alternate form of
the entropy inequality (
10.64
). First, the product
r
(
a
)
r
(
a
)
is eliminated between
(
10.21
) and (
10.64
) and then, second, the result is multiplied by
y
, third, it is
assumed that the constituent-specific flux vectors,
h
(
a
)
and
q
(
a
)
, are related by
h
ðaÞ
¼
q
ðaÞ
þ r
ðaÞ
y
ðaÞ
u
ðaÞ
;
(10.65)
thus
"
#
X
X
N
D
a
D
a
N
ðaÞ
D
t
e
ðaÞ
D
t
q
ð
a
Þ
y
1
r
ðaÞ
y
þ yr
a
¼
a
¼
1
X
N
a¼
1
yr½r
ðbÞ
X
N
^
þ
u
ðbÞ
þ
T
ðaÞ
:
D
ðaÞ
r
q
Þ
þ e
ðaÞ
0
;
(10.66)
ð
a
a¼
1
then using (
10.38
), (
10.60
) and
X
N
q
¼
q
ðaÞ
;
(10.67)
a
¼
1
(The expression (
10.67
) for the heat flux is an approximation that neglects several
terms associated with diffusion velocities. This point is discussed on page 27 of
Bowen (
1976
)) it follows that
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