Biomedical Engineering Reference
In-Depth Information
w is a scalar length parameter in the n direction. The volume flow rate q
¼ fr f v /
r o
projected in the n direction,
fr f v /
r o , is then given by
n
fr f v
n
=r o ¼ð@
p
=@
w
Þ
n
H
(5.28D)
r o in the n direction to be
pointed in the direction of decreasing pressure, it is necessary to require that
In order for the volume flux per unit area q
¼ fr f v /
n H n> 0 for all unit vectors n:
(5.29D)
If the fluid flowed the other way, all the mass of the fluid would concentrate itself
at the highest-pressure location and we know that that does not happen. The
condition ( 5.29D ) is the condition that the symmetric tensor H be positive definite.
This condition is satisfied if all the eigenvalues of H are positive.
The tensor of material coefficients for the Newtonian law of viscosity is positive
definite also. To see this, the local stress power tr( T
T : D is calculated using
the constitutive equation ( 5.11N ) and the decomposition (A18) of the rate of
deformation tensor,
D )
¼
D
¼
ðÞ
1
=
3
ðÞ
tr D
1
þ
dev D
;
dev D
¼
D
ðÞ
1
=
3
ðÞ
tr D
1
:
(5.30N)
The stress due to viscous stresses may be recast in the form
m Þ
T
þ
p 1
¼ðð
3
2
3
Þ
ðÞ
tr D
1
þ
2
dev D
;
(5.31N)
l
m
using ( 5.11N ) and ( 5.30N ). Calculation of the viscous stress power tr{( T
þ
p 1 ) D }
using the two equations above then yields
2
2
m
ð
T
þ
p 1
Þ :
D
¼ðð
3
2
3tr D
ðÞ
þ
2
trdev D
Þ
:
(5.32N)
l
m
Note that the terms in ( 5.32N ) involving D , and multiplying the expressions
3
2
m
and 2
m
are squared; thus if the viscous stress power tr{( T
þ
p 1 )
D }
¼
l
( T þ p 1 ): D is to be positive it is necessary that
> m
>:
3
2
(5.33N)
l
m
p1 ): D must be positive for an inert material as
the world external to the material is working on the inert material, not the reverse.
The inequalities restricting the viscosities ( 5.33N ) also follow for the condition that
the 6 by 6 matrix ( 5.10N ) be positive definite. Finally, to see that the tensor of
elastic coefficients is positive definite, the local form of the work done expressed in
terms of stress and strain, T : E ¼ T E is employed. Since T ¼ C E it follows that
T : E
The viscous stress power ( T
þ
E thus from the requirement that the local work done on an
inert material be positive, T : E
¼ T
E
¼ E
C
¼ T
E
>
0, it follows that
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