Biomedical Engineering Reference
In-Depth Information
!
X
N
1 r ðaÞ v ða=sÞ
@r
@
^
t þr r
v ðsÞ þ
¼
ð
t
Þ:
(10.23)
When the selected point ( s ) for velocity reference is the point where the velocity
is equal to the mean velocity, the statement of the conservation of mass above
reduces to the traditional formula below involving the mean velocity,
@r
@
^
t þrðr
v
Þ¼
ð
t
Þ:
(10.24)
The constituent form of mass balance ( 10.20 ) summed over all the constituents
produces the continuum statement (3.6) if the definitions ( 10.16 ) for the density of
the continuum mixture and the mean mixture velocity v ,
X
N
1 r ðaÞ v ðaÞ
1
r
v ¼
(10.25)
are employed. Recall from the introductory paragraph of this section that this
presentation of mixture-based poroelasticity will not employ this mean velocity
concept. An exception to this non-use is note the fact that some results simplify
when the velocity of a selected constituent v ðsÞ is set equal to the mean velocity
( 10.25 ). The conservation of momentum for a single constituent continuum,
r v ¼rT þ rd;
ð
3
:
29 repeated
Þ
may be written as
D ðaÞ v ð a Þ
D t ¼r
^
r ðaÞ ¼
T ðaÞ þ r ðaÞ
d ðaÞ þ
Þ ;
(10.26)
ð
a
where T ðaÞ is the partial stress, d ðaÞ is the action-at-a-distance force density and ^
is
ðaÞ
the momentum supply associated with constituent a . The momentum supply ^
is
the only term that is not directly associated with a term in (3.29); it represents the
transfer of momentum from the other constituents to constituent a . In this presenta-
tion it is assumed that all the partial stress tensors T ðaÞ are symmetric. The
assumption is consistent with the mixture theory applications that are to be consid-
ered here, but it is an assumption that may be avoided if necessary (Bowen 1976 ,
1980 ). The conservation of energy for constituent a is a similar generalization of the
single constituent continuum result (3.52),
ðaÞ
re ¼
T
:
D
r
q
þ r
r
;
ð
3
:
52 repeated
Þ
thus
D a
e ðaÞ
q ðaÞ þ r ðaÞ r ðaÞ þ e
^
r ðaÞ
D t ¼
T ðaÞ :
D ðaÞ r
Þ ;
(10.27)
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