Biomedical Engineering Reference
In-Depth Information
υ
=
ω
×
(
x
−
x
S
)
z
ω
x
y
x
S
S
x
Figure 7.11
Rotation of material.
velocity field
v
=
v
(
x
) and on the velocity gradient tensor
L
that can be derived
from it. Figure
7.11
illustrates the considered problem. The axis of rotation is
defined by means of a point S, fixed in space, with position vector
x
S
and the
constant angular velocity vector
ω
. The associated columns with the components
with respect to the Cartesian basis can be written as
⎡
⎤
⎡
⎤
x
S
ω
x
⎣
⎦
⎣
⎦
∼
S
=
y
S
,
∼
=
ω
y
.
(7.25)
z
S
ω
z
The velocity vector
v
at a certain point with position vector
x
satisfies
v
=
v
(
x
)
=
ω
×
(
x
−
x
S
) .
(7.26)
It can be derived that the components of the velocity vector
v
and the spatial
coordinate vector
x
, stored in the columns
⎡
⎤
⎡
⎤
v
x
x
⎣
⎦
⎣
⎦
∼
=
v
y
and
∼
=
y
,
(7.27)
v
z
z
