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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