Biomedical Engineering Reference
In-Depth Information
There are a number of alternative ways to write this equation, some of which may
be more advantageous for interpreting and applying this axiom in mixture analyses.
These alternative representations employ the deformation gradient for constituent
α
, given by
F
α
∂
χ
α
/∂
X
α
. For example, when using the material time derivative
in the spatial frame following constituent
α
, the axiom of mass balance reduces to
D
α
=
Dt
J
α
ρ
α
=
0
,
(17.2)
where
J
α
v
α
. This relation clearly
shows that
J
α
ρ
α
remains invariant in the absence of mass supply to constituent
α
.
For example, in the case of the solid matrix, this relation reproduces the classical
statement
J
s
ρ
s
det
F
α
and
D
α
(
=
·
)/Dt
≡
∂(
·
)/∂t
+
grad
(
·
)
·
ρ
r
, where
ρ
r
is the solid density in the reference configuration
=
(when
χ
α
X
α
and
J
s
=
=
1). This means that the mass balance for the solid may
also be expressed as
ϕ
s
ϕ
r
/J
s
, where
ϕ
r
is the solid volume fraction in the refer-
=
ence state.
When using motion relative to the solid constituent, the mass balance equation
for any constituent may also be written as
D
s
Dt
J
s
ρ
α
+
J
s
div
m
α
=
0
.
(17.3)
This relation is useful in finite element modeling of mixtures, when the mesh is
defined on the solid matrix.
When the mixture constituents are intrinsically incompressible, the relation
ρ
α
=
ϕ
α
ρ
T
may be substituted into Eq. (
17.1
) and the invariant
ρ
T
may be canceled out
from the resulting expression. Taking the sum over all constituents and using the
mixture saturation condition,
α
ϕ
α
=
1, produces a mass balance relation for the
mixture,
div
v
s
w
α
+
=
0
.
(17.4)
α
When the solute volume fraction is negligible, the volume flux of solutes is neg-
ligible in comparison to that of the solvent, thus the net volume flux of interstitial
fluid (solvent
≡
α
w
α
, reduces to the flux of the
+
solutes) relative to the solid,
w
w
w
.
In a finite element modeling framework for intrinsically incompressible con-
stituents, the mass balance equations that need to be enforced are Eq. (
17.3
)for
each of the solutes, Eq. (
17.4
) for the mixture, and Eq. (
17.2
) applied to the solid.
The intrinsic incompressibility constraint eliminates the need to explicitly enforce
the balance of mass for the solvent.
solvent,
w
≈
17.2.2 Electroneutrality
When mixture constituents carry an electric charge, a charge number
z
α
may be
associated with each constituent, which represents the number of charges per mole
