Biomedical Engineering Reference
In-Depth Information
For example, consider a mixture consisting of any number of constituents
α
.The
true density
of a specific constituent,
ρ
T
, represents the mass d
m
α
of
α
, per volume
d
V
α
of
α
, in an elemental region d
V
; thus,
ρ
T
=
d
m
α
/
d
V
α
. The true density is an
intrinsic property of constituent
α
. In a saturated mixture, the volume of the ele-
mental region is the sum of the constituent volumes, d
V
=
α
d
V
α
. Therefore, it is
also possible to define the
apparent density
of constituent
α
as
ρ
α
d
m
α
/
d
V
.The
apparent density represents a measure of the relative content of
α
in the mixture.
An alternative measure of relative content is the
volume fraction ϕ
α
=
d
V
α
/
d
V
.
This measure is most useful when constituents are intrinsically incompressible, in
which case
ρ
α
=
ϕ
α
ρ
T
;
ρ
α
and
ϕ
α
may be used interchangeably since
ρ
T
is invari-
=
ant in that case. In a saturated mixture, the volume fractions satisfy
α
ϕ
α
=
1; if
the volume fraction of solutes is negligible, this relation reduces to
ϕ
s
ϕ
w
+
≈
1,
where
α
w
represents the sol-
vent. Another common measure of relative content, normally used for solutes, is
the
molar concentration c
α
, defined as the number of moles d
n
α
in the elemen-
tal region, per volume of the solution; thus,
c
α
=
s
represents the porous solid matrix and
α
=
d
n
α
/(
d
V
d
V
s
)
. Since the mo-
=
−
lar mass of
α
is defined as
M
α
d
m
α
/
d
n
α
(an invariant property), it follows that
=
ρ
α
ϕ
w
M
α
c
α
. Many classical relations from chemistry employ
c
α
as measure of solute content; some applications in bone mechanics employ
ρ
s
to
describe trabecular bone density; and applications in cartilage mechanics use
ϕ
s
in
the formulation of various constitutive relations. Other measures of relative content
are also commonly used (e.g., molality, molar fraction, or mass fraction for solute
content); thus, users of mixture theory must be comfortable switching among these
various measures, depending on the needs of a particular analysis.
The motion of each constituent
α
of a mixture is given by
χ
α
(
X
α
,t)
, where
χ
α
represents the position at time
t
of a material point initially located at
X
α
.The
velocity of
α
is given by
v
α
ϕ
s
)M
α
c
α
=
(
1
−
≈
∂
χ
α
/∂t
. In biological tissues, the mixture includes
a solid matrix and the boundaries of the tissue are normally defined on that solid.
Therefore, the solid may be viewed as a special constituent that may serve as a
reference for the motion of the others. The mass flux of constituent
α
relative to
the solid is given by
m
α
=
v
s
)
; note here that the mass flux is defined as
the rate of mass of
α
crossing a unit area of the mixture normal to the direction
of
v
α
ρ
α
(
v
α
=
−
v
s
. The volumetric flux of constituent
α
relative to the solid is given by
−
w
α
ϕ
α
(
v
α
v
s
)
; and the molar flux is
j
α
ϕ
s
)c
α
(
v
α
v
s
)
ϕ
w
c
α
(
v
α
v
s
)
.
=
−
=
(
1
−
−
≈
−
These fluxes are related by
m
α
ρ
T
w
α
M
α
j
α
.
=
=
17.2.1 Mass Balance
In the absence of chemical reactions, the differential statement of the axiom of mass
balance for constituent
α
maybegivenby
∂ρ
α
∂t
+
div
ρ
α
v
α
=
0
.
(17.1)
