Biomedical Engineering Reference
In-Depth Information
where terms of order of the diffusion velocity
v
ða=sÞ
squared have been neglected or
when
v
v
ðsÞ
.
With the results (
10.31
) and (
10.34
) in hand it is now possible to return to the
development of the sums of the constituent-specific balance equations. Recall that it
is required that the summation of the forms of the balance of mass (
10.20
), the
balance of momentum (
10.26
) and the balance of energy (
10.27
) for each constitu-
ent over all the constituents is required to produce again the single constituent
continuum forms of the balance of mass (3.6), the balance of momentum (3.29) and
the balance of energy (3.52), respectively, to within the supply terms for each
constituent and the supply term for the mixture. The summation of the component-
specific form of the conservation of linear momentum (
10.26
), employing the
representation (
10.35
) for the sum of the density-weighted, component-specific
time derivatives of the component-specific velocities, one obtains a result that is
similar to the single component form (3.29),
¼
D
s
v
Dt
¼r
^
r
T
þ r
d
þ
ð
t
Þ;
(10.36)
if the squares of the diffusion velocities are neglected. The total stress
T
is defined by
X
N
T
¼
T
ðaÞ
;
(10.37)
a¼
1
and the sum of the action-at-a-distance forces by
X
N
1
r
d
¼
¼
1
ðr
ðaÞ
d
ðaÞ
Þ;
(10.38)
a
and the sum of the constituent momentum supplies
^
is denoted by
^
,
ðaÞ
X
N
a¼
1
f
^
v
ðaÞ
^
v
^
^
ðaÞ
þ
ðaÞ
g
ð
t
Þ¼
:
(10.39)
If the velocity of the selected component is equal to the mean velocity of the
mixture,
v
ðsÞ
¼
v
, the results (
10.35
) through (
10.39
) will coincide with results that
appear in Bowen (
1967
,
1976
,
1980
,
1982
).
The summation of the constituent-specific form of the balance of energy (
10.27
)
over all the constituents, and subsequently employing the formula (
10.29
) with
ˆ
ðaÞ
e
ðaÞ
, yields
replaced by
X
N
D
s
e
Dt
¼ r
^
r
r
þ e
ð
t
Þþ
T
ðaÞ
a¼
1
n
o
;
X
N
:
D
ðaÞ
r
q
ðaÞ
þðe
ðaÞ
eÞr
ðaÞ
v
ða=sÞ
(10.40)
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