Biomedical Engineering Reference
In-Depth Information
At a certain time the edges AB, AD and AE are exactly oriented in the
directions of the
x
-,
y
- and
z
-axes, respectively (see figure). Determine for
that time instant the velocity vector
∼
B
of the point B.
7.3
Bone mineral density is often measured, because this quantity can be
related to the strength and stiffness of bone. The measurement can also
be used as a diagnostic tool for osteoporosis. Consider a rectangular piece
of bone in the
xy
-plane of a Cartesian
xyz
-coordinate system, with varying
density
(
x
,
y
). For the corner points ABCD of the specimen the den-
sity is given (expressed in [kg dm
−
3
), see figure. Prove, that it is impossible
for the gradient
ρ
=
ρ
∇
ρ
of the density field in the figure to be constant.
y
ρ
D
=
1.02
ρ
C
=
1.09
ρ
A
=
1.00
ρ
B
=
1.05
x
7.4
Consider a domain in the form of a cube in three-dimensional space, given
by
a constant and
x
,
y
and
z
the Cartesian coordinates. Within this domain a temperature field
exists with known gradient
−
≤
x
≤
3
,
−
2
≤
y
≤
2
and
−
2
≤
z
≤
2
with
∇
T
which is given by
∇
T
=
θ
e
y
with
θ
a constant.
In the domain a curve is given. The points on the curve satisfy
)
2
y
2
2
(
x
−
+
=
and
z
=
.
Calculate the coordinates of the point (or points) on the curve where the
derivative of the temperature in the direction of the curve is equal to zero.
7.5
In a two-dimensional
xy
-coordinate system (mutually perpendicular unit
vectors
e
y
along the axes) a two-dimensional stationary fluid flow
is considered. The fluid flow is caused by a fluid source in the origin. An
arbitrary point in the coordinate system is given by the vector
e
x
and
x
=
x
e
x
+
y
e
y
.
At some distance from the origin (with
|
x
|
>λ
) the velocity field
v
(
x
) can
be written as
x
|
v
(
x
)
=
α
2
.
x
|
The location of a point P is defined by the position vector
x
P
=
(
e
x
+
e
y
)
with
>λ
. Determine the velocity gradient tensor
L
in point P.
