Biomedical Engineering Reference
In-Depth Information
Thus, applying the constitutive relation (
3
)toEq.(
1
), the Navier-Stokes
equations for incompressible Newtonian fluids are obtained:
⎧
⎨
ρ
∂
u
∂
t
+
ρ
(
u
·
∇
)
u
+
∇
p
−
div
(
2
μ
D
(
u
)) =
0
,
(4)
in
Ω
,∀
t
>
0
,
⎩
div
u
=
0
.
3.2
Generalized Newtonian Fluids
The most general form of Eq. (
2
), for isotropic symmetric tensor functions, under
frame invariance requirements [
21
], is given by
+
φ
2
D
2
σ
=
φ
0
I
+
φ
1
D
,
(5)
with
φ
0
,
φ
1
,
and
φ
2
dependent on the density
ρ
and on the three principal invariants
2
(
)
,and
III
D
=
1
2
D
2
of
D
,
I
D
=
tr
(
D
)
,
II
D
=
tr
(
D
))
−
tr
(
det
(
D
)
,where
tr
(
D
)
and
det
(
D
)
denote the trace and the determinant of tensor
D
, respectively. By setting
φ
2
=
φ
1
constant, we obtain the relation for a Newtonian fluid, governed by
the Navier-Stokes equations (
4
). Considering
0, and
0 does not correspond to any
existent fluid under simple shear, so that the constitutive relation (
5
) is often used in
the reduced general form, with
φ
2
=
Moreover, respecting
the frame invariance requirements and the behavior of real fluids,
φ
2
=
0[
21
]:
σ
=
φ
0
I
+
φ
1
D
.
φ
1
becomes the
viscosity function [
21
], and the following general constitutive relation is obtained:
σ
=
−
p
I
+
2
μ
(
II
D
,
III
D
)
D
,
(6)
where the viscosity function
μ
might depend on the second and third invariants of
D
.
0 in simple shear, as well as in other viscometric flows, it is
reasonable to neglect the dependence of
Since
III
D
=
μ
on
III
D
.Furthermore,
II
D
is negative
for isochoric motions, where
tr
(
D
)=
0, so the positive metrics of the rate of
deformation
2
tr
γ
≡
˙
−
4
II
D
=
(
D
2
)
,
also known as the shear rate, may be defined. Using the definition of the shear rate
as a function of the second invariant of
D
, relation (
6
) can be rewritten as follows:
σ
=
−
p
I
+
2
μ
(
γ
)
˙
D
.
(7)
This equation defines the constitutive equation for the generalized Newtonian fluids,
such that the equations of motion for these fluids are of the form
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