Chemistry Reference
In-Depth Information
⎛
⎝
⎜
⎞
⎠
⎟
=
∂
2
u
x
∂
1
H
A
⋅
∂
u
x
∂
x
∂ε
⊥
∂
0
.
(7)
−
+
x
2
k
t
t
Using new variable
u
(
x
,
t
)
=
u
x
(
x
,
t
)
+ ε
⊥
(
t
)
⋅
x
, this equation may be transformed
into:
2
u
u
∂
∂
1
H
A
⋅
k
∂
−
=
0
(8)
x
2
∂
t
Note, that the apparent fluid velocity relative to the solid can be calculated using the
u
(
x
,
t
)
function, because
2
u
H
A
k
∂
x
2
,
v
=−
∂
as may be deduced from left part of the Equation (6).
Equation (8) is the ¿ rst equation that describes porous gel mechanics under load.
Interestingly this equation coincides with the equation derived for con¿ ned compres-
sion of articular cartilage [18].
7.3.2 The Governing Equation for Molecular Transport
Total solute concentration is the sum of
c
F
- free solute, dissolved in the fluid, and
c
B
- solute that is bound to the solid phase:
c
w
=
c
F
+
c
B
,
where
c
F
and
c
B
are defined as moles of substance contained in ɚ liter of fluid phase.
Note, that all concentrations should be calculated relative to coordinate system that
is tied with solid phase. The purpose of the model is to calculate
c
i
(
x
,
t
) -concentra-
tion dependence on time
t
and special variable
x
of a bound (
i
=
B
) or free (
i
F
) or total
=
(
i
w
) solute within the tissue. The volume ¿ lled by matrix is considered as “tissue”,
but not the À uid, that À ows out of the pores and becomes the bath solution. As the tis-
sue undergoes deformation, each particular point on matrix is displaced relative to the
¿ xed frame of reference used. That is why the concentration of the solute should be
considered relative to a ¿ xed REV. To put it another way, the solid phase in the current
model represents extracellular matrix with cells, and the aim of the study is to estimate
concentrations relative to cells.
The equations for mass balance for each type of solute in each phase can be intro-
duced. As concentrations in the bath do not depend on
z
, the continuity equation for
a free solute reads:
=
∂φ ⋅
c
F
∂
∂
(
)
=
q
(
v
C
F
v
s
)
+
φ ⋅
c
F
⋅
−
(9)
∂
t
x
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