Biomedical Engineering Reference
In-Depth Information
constant even if the a th constituent is incompressible since the volume fraction
f ( a )
may change. The mixture is only incompressible when all the
, N ,are
incompressible. If less than N constituents are incompressible, ( 10.19 )isa
constraining relationship between the densities. Note that the mixture density
g ( a ) , a
¼
1, 2,
...
r
given by ( 10.19 ) may be variable even when all the constituents are incompressible,
that is to say all the
g ( a ) 's are constant, because the volume fraction ( 10.17 )ofthe
constituents present at a point x is variable.
10.5 The Conservation Laws for Mixtures
In this section equations are postulated equations for the balance of mass, momen-
tum, and energy for each constituent and then the necessary and sufficient
conditions are obtained so that the usual global balance of mass, momentum, and
energy for the entire mixture is satisfied. In order to postulate equations for the
balance of mass, momentum, and energy for each constituent X ( a ) , a
¼
1, 2,
...
, N ,
each constituent of the mixture is assigned a density
r ( a ) , an action-at-a-distance
force density d ( a ) , a partial stress T ðaÞ , a partial internal energy density e ( a ) , a partial
heat flux, and a partial heat supply density, r ðaÞ .
The local statement of mass conservation for a single constituent continuum,
@r
@
t þrðr
v
Þ¼
0
;
ð
3
:
6 repeated
Þ
may be written for each constituent a
¼
1, 2,
...
, N as
@r ð a Þ
@
^
t þr r ðaÞ v ðaÞ
¼
Þ :
(10.20)
ð
a
where ^
represents the mass supply to a constituent from other constituents and
from external sources. The local statement of mass conservation for a single
constituent continuum rewritten in terms of the selected velocity becomes
ðaÞ
@r ð a Þ
@
¼
^
t þrr ðaÞ
v ðsÞ þ
v ða=sÞ
Þ :
(10.21)
ð
a
The sum of all mass supplies to a constituent from other constituents is denoted
by ^ , thus
X
N
^
^
ðaÞ ð
t
Þ¼
ð
t
Þ:
(10.22)
1
The summation of ( 10.21 ) over all constituents and the use of ( 10.22 ) yields
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