Biomedical Engineering Reference
In-Depth Information
e
E
(
x
,
y
,
z
,
t
) = [
eE
(
x
,
y
,
z
,
t
)
x
,
eE
(
x
,
y
,
z
,
t
)
y
,
eE
(
x
,
y
,
z
,
t
)
z
]
T
,
(6.120)
is the force vector on an individual cation due to electro-osmotic motion of an ion
in an electric field,
k
is the Boltzmann's constant,
ν
(
x
,
y
,
z
,
t
) = [
ν
x
(
x
,
y
,
z
,
t
),
ν
y
(
x
,
y
,
z
,
t
),
ν
z
(
x
,
y
,
z
,
t
)]
T
(6.121)
is the velocity vector of the hydrated cations,
η
ν
(
x
,
y
,
z
,
t
) = [
ην
x
(
x
,
y
,
z
,
t
),
ην
y
(
x
,
y
,
z
,
t
),
ην
z
(
x
,
y
,
z
,
t
)]
T
(6.122)
is the force vector of the viscous resistance to the motion of individual hydrated
cations in the presence of a viscous fluid medium with a viscosity of
η
, and
kT
∇
[ln(
ρ
( ,
x y z t
, , )
+
n
ρ
( ,
xyzt
, , )]
(6.123)
M
+
w
is the force vector due to diffusion of individual cations and accompanying molecules
of hydrated water in the polymer network with the following
x
,
y
, and
z
components,
respectively:
∂
ln[
ρ
( , , , )
xyzt
+
n
ρ
( , , , )]
xyzt
M
+
w
kT
,
(6.124)
∂
x
∂
ln[
ρ
( , , , )
xyzt
+
n
ρ
( , , , )]
xyzt
M
+
w
kT
,
(6.125)
∂
y
∂
ln[
ρ
( , , , )
xyzt
+
n
ρ
( , , , )]
xyzt
M
+
w
,
(6.126)
kT
∂
z
The force vector due to inertial effects on an individual hydrated cation is
MnM
d
dt
ν
(
+
+
)(
)
(6.127)
M
w
such that, in a compact vector form, the force balance equation reads:
d
dt
ν
Ne
ρ
Ε
=
N
(
ρ
M
+
n
ρ
M
)(
)
+
N
ρην
+
M
+
M
+
M
+
w
w
M
+
(6.128)
N
ρ
kT
∇
n
(
ρ
+
n
ρ
)
+ ∇
P
−∇
.
σ
*
M
+
M
+
w
f
where the stress tensor
can be expressed in terms of the deformation gradients in
a nonlinear manner such as in neo-Hookean or Mooney-Rivlin types of constitutive
σ
*
