Biomedical Engineering Reference
In-Depth Information
Based on the velocity field in
V
(
t
) often streamlines are drawn. Streamlines are
representative for the current (at time
t
) direction of the velocity: the direction of
the velocity at a certain point
x
corresponds to the direction of the tangent to the
streamline in point
x
. Fig.
9.4
gives an example of streamlines in a flow through a
constriction.
For a stationary flow,
0, the streamline pattern is
the same at each time point. In that case the material particles follow exactly the
streamlines, i.e. the particle tracks coincide with the streamlines.
v
=
v
(
x
) and thus
δ
v
/δ
t
=
Exercises
9.1
The material points of a deforming continuum are identified with the
position vectors
∼
0
of these points in the reference configuration at time
t
=
0. The deformation (Lagrangian approach) is described with the current
position vectors
∼
as a function of
∼
0
and time
t
, according to
⎡
⎣
⎤
⎦
x
0
+
(
a
+
by
0
)
t
∼
(
∼
0
,
t
)
=
y
0
+
at
with
a
and
b
constant.
z
0
Determine the velocity field as a function of time in an Eulerian description,
in other words, give an expression for
∼
=
∼
(
∼
,
t
).
9.2
Consider a fluid that flows through three-dimensional space (with an
xyz
-
coordinate system). In a number of fixed points in space the fluid velocity
∼
is measured as a function of the time
t
. Based on these measurements
it appears that in a certain time interval the velocity can be approximated
(interpolated) in the following way:
⎡
⎣
⎤
⎦
ay
+
bz
1
+
α
t
∼
=
0
cx
.
1
+
α
t
Determine, based on this approximation of the velocity field as a function
of time, the associated acceleration field as a function of time, thus:
∼
(
∼
,
t
).
9.3
Consider a (two-dimensional) velocity field for a stationary flowing con-
tinuum:
x
y
2
v
x
=
1
y
with
x
and
y
spatial coordinates (expressed in [m]), while
v
x
and
v
y
are the
velocity components in the
x
- and
y
-direction (expressed in [ms
−
1
]). The
velocity field holds for the shaded domain in the figure given below.
v
y
=
