Biomedical Engineering Reference
In-Depth Information
are objective tensors. In Sections
12.6
and
12.7
two types of constitutive behaviour
for fluids will be discussed by means of a specification of
d
(
D
).
σ
12.6
Newtonian fluids
For a Newtonian fluid the relation between the deviatoric stress tensor and the
deformation rate tensor is linear, yielding:
σ
=−
p
I
+
2
η
D
and also
σ
=−
p
I
+
2
η
D
,
(12.54)
with
η
the
viscosity
(a material parameter that is assumed to be constant) of the
fluid. The typical behaviour of a Newtonian fluid can be demonstrated by applying
Eq. (
12.54
) to two elementary examples of fluid flow: pure shear and uniaxial
extensional flow.
A pure shear flow 'in the
xy
-plane' can be created with the following velocity
field (in column notation):
⎡
⎣
⎤
⎦
=
γ
⎡
⎣
⎤
⎦
v
x
v
y
v
z
y
0
0
∼
=
,
(12.55)
with
(the shear velocity) constant. For the associated deformation rate matrix
D
it can easily be derived that
γ
⎡
⎣
⎤
⎦
0
γ
0
1
2
D
=
γ
00
000
(12.56)
and verified that the constraint tr(
D
)
0 is satisfied. Applying the
constitutive equation it follows for the relevant shear stress
=
tr(
D
)
=
σ
xy
=
σ
yx
in the fluid:
σ
xy
=
η γ
,
(12.57)
which appears to be constant. Thus, the viscosity can be interpreted as the
'resistance' of the fluid against 'shear' (shear rate actually).
To a uniaxial extensional flow (incompressible) in the
x
-direction the following
deformation rate matrix is applicable:
⎡
⎣
⎤
⎦
˙
0
0
D
=
0
20
00
−˙
/
,
(12.58)
−˙
/
2
with
(the rate of extension) constant. For the stress matrix it is immediately
found that
˙
