Biomedical Engineering Reference
In-Depth Information
Problem
5.2.1. Record a rigorous definition of the neighborhood of the particle X , N
( X ), using a reference particle X*, a small positive number e and a length
measure || X - X *||. Note that this neighborhood is, in general, a three-
dimensional neighborhood.
5.3 Localization
A constitutive equation valid at the particle X of a material object can depend upon
the behavior of the material in the neighborhood of the particle X , N ( X ), but is
unlikely to depend upon the behavior of the material in regions of the object far
removed from the particle X . The localization guideline for the development of
constitutive relations restricts the dependence of constitutive equations valid for a
particle X to events that occur in N ( X ). The application of the localization guideline
to the four constitutive equations described in the previous section is described in
the next paragraph.
The constitutive idea for Darcy's law is considered first. The pressure field
p ( X* , t ) at a particle X* in N ( X ) may be related to the pressure field p ( X , t )ata
particle X by a Taylor series expansion about the point by
X ;
X Þþ
p
ð
t
Þ¼
p
ð
X
;
t
Þþðr
p
ð
X
;
t
ÞÞ ð
X
higher order terms
;
(5.2)
where it is assumed that the pressure field is sufficiently smooth to permit this
differentiation. With the Taylor theorem as justification, the N ( X ) may always be
selected sufficiently small so the value of the pressure field p( X* , t ) at a particle X*
in N ( X ) may be represented by p ( X , t ) and
p ( X , t ). Thus, by localization, ( 5.1D )
may be rewritten as
q ¼ fr f v=r o ¼ qð
p
ðX;
t
Þ; r
p
ðX;
t
Þ; XÞ
(5.3D)
Exactly the same argument is applicable to the other three constitutive ideas;
thus we have that
T
¼ T
ð
u
ð
X
;
t
Þ;r
u
ð
X
;
t
Þ;
X
Þ;
(5.3H)
T
pU
þ T v ð
¼
v
ð
X
;
t
Þ;r
v
ð
X
;
t
Þ;
X
Þ;
(5.3N)
and
1
T
T
¼
ð
v
ð
X
;
t
s
Þ;r
v
ð
X
;
t
s
Þ;
X
;
s
Þ
d s
:
(5.3V)
s
¼
0
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