Biomedical Engineering Reference
In-Depth Information
After an infinitesimally small increase in time
dt
the material line element
d
x
d
will change into
d
x
+
xdt
. By means of the figure it can be proven that:
d
d
∼
=
x
=
L
·
d
x
and also
L
d
∼
,
(7.20)
and so the tensor
L
(with matrix representation
L
) is a measure for the current
change (per unit of time) of material line elements. This holds for the length as
well as for the orientation.
The divergence (an operator that is often used for vector fields) of the velocity,
div(
v
), defined as
∇·
v
can be written as
e
x
∂
∂
x
+
e
y
∂
∂
y
+
e
z
∂
v
)
=
·
(
v
x
e
x
+
v
y
e
y
+
v
z
e
z
)
div(
∂
z
+
∂
v
y
∂
=
∂
v
x
∂
+
∂
v
z
∂
x
y
z
=
tr(
L
) ,
(7.21)
and also
div(
v
)
=
∂
v
x
∂
x
+
∂
v
y
∂
y
+
∂
v
z
T
)
=
∼
T
∼
=
tr(
L
) .
z
=
tr(
∼ ∼
(7.22)
∂
∇
For the sake of completeness the following rather trivial result for the gradient
applied to the position vector
x
is given:
=
∂
x
∂
x
e
x
+
∂
y
∂
x
e
y
+
∂
z
∂
x
∇
x
e
x
e
x
e
x
e
z
+
∂
x
e
x
+
∂
y
e
y
+
∂
z
∂
y
e
y
∂
y
e
y
∂
y
e
y
e
z
+
∂
x
e
x
+
∂
y
e
y
+
∂
z
∂
z
e
z
∂
z
e
z
∂
z
e
z
e
z
=
e
x
e
x
+
e
y
e
y
+
e
z
e
z
=
I
,
(7.23)
and the according matrix formulation yields
⎡
⎤
∂
x
∂
x
∂
y
∂
x
∂
z
∂
x
⎣
⎦
∂
x
∂
y
∂
z
T
∼
∼
=
=
I
.
(7.24)
∂
y
∂
y
∂
y
∂
x
∂
z
∂
y
∂
z
∂
z
∂
z
7.6
Rigid body rotation
In the present section it is assumed that the mass in the rigid volume
V
rotates
around a fixed axis in three-dimensional space. We focus our attention on the
