Biomedical Engineering Reference
In-Depth Information
σ
∗
σ
∗
and so for the stress tensor
and the associated matrix
in the imaginary
configuration it is found that
σ
∗
=
P
T
σ
∗
=
P
T
P
·
σ
·
and also
P
σ
(9.43)
with
P
the matrix representation of the tensor
P
.
9.8
Fluid flow
For fluids it is not common practice (and in general not very useful) to define a
reference state. This implies, that the Lagrangian description (expressing prop-
erties as a function of
x
0
and
t
) is not commonly used for fluids. Related to
this, derivatives with respect to
x
0
(the gradient operator
∇
0
) and derivatives with
respect to time under constant
x
0
will not appear in fluid mechanics. The defor-
mation tensor
F
is not relevant for fluids. However, the material time derivative
(for example to calculate the acceleration) is important nevertheless. For fluids an
Eulerian description is used, meaning that all physical properties are considered
in the current configuration, so as functions of
x
in the volume
V
(
t
) and
t
.
The kinematic variables that generally play a role in fluid mechanics problems
are the velocity
x
,
t
), both in an Eulerian
description. Their relation is given by (see the end of Section
9.4
)
v
=
v
(
x
,
t
) and the acceleration
a
=
a
(
∇
v
T
δ
v
t
=
L
·
v
+
δ
v
a
=
·
v
+
t
.
(9.44)
δ
δ
Figure 9.4
Streamlines in a model of a carotid artery bifurcation.
