Biomedical Engineering Reference
In-Depth Information
Fig. 3.1
A potato-shaped object and a second potato-shaped object that is fully contained within
the first potato-shaped object. All the conservation principles may be applied to both objects
separately. Furthermore one may select or define these objects as one chooses
ð
O
ð rð
x
;
t
Þ
d
v
þ rð
x
;
t
Þ
d
v
_
Þ¼
0
:
(3.3)
Then, using the relationship relating the time rate of change in the volume to the
present size of the volume from (2.36), d
v
_
¼ðr
v
Þ
d
v
, it follows that
ð
O
f rð
x
;
t
Þþrð
x
;
t
Þðr
v
Þg
d
v
¼
0
:
(3.4)
The next step in the development of this continuum representation of the
conservation of mass is to employ the argument that the integral equation (
3.4
)
over the object
O
may be replaced by the condition that the integrand in the integral
equation (
3.4
) be identically zero, thus
r þ rðr
v
Þ¼
0
:
(3.5)
The argument that is used to go from (
3.4
) and (
3.5
) is an argument that will be
employed three more times in this chapter. The argument requires that the inte-
grand in the integral (
3.4
) be continuous. The argument is that any part or
subvolume of an object
O
may also be considered as an object and the result
(
3.4
) also holds for that sub-object. In Fig.
3.1
an object and a portion of an object
that may be considered as an object itself are illustrated. The argument for the
transition (
3.4
)
(
3.5
) is as follows: suppose it is not true that the integrand of
(
3.4
) is not zero everywhere (i.e., suppose the transition (
3.4
)
!
(
3.5
) is not true).
If that is the case then there must exist domains of the object in which the integrand
is positive and other domains in which the integrand is negative, so that when the
integration is accomplished over the entire object the sum is zero. If that is the case
!
Search WWH ::
Custom Search