Biomedical Engineering Reference
In-Depth Information
It will be clear that, in the context of the discussion above, the following formal
relations hold for the velocity field and the acceleration field:
v
=
v
(
x
0
,
t
)
and
a
=
a
(
x
0
,
t
) .
(9.6)
The configuration change of the material in a certain time interval can be asso-
ciated with deformation. We deal with deformation when the mutual distances
between material points change. The mathematical description of deformation and
deformation velocity is the major theme of Chapter
10
.
9.3
Lagrangian and Eulerian description
In the previous section the velocity and acceleration of the material are formally
written as functions of the material identification
x
0
with material coordinates
∼
0
in
V
0
and the time
t
. Obviously, this can also be done with other physical
properties associated with the material, for example the temperature
T
.Fora
(time-dependent) temperature field it can be written
T
=
T
(
x
0
,
t
) .
(9.7)
The temperature field in the current configuration
V
(
t
) is mapped on the reference
configuration. Such a description is referred to as
Lagrangian
. Partial differen-
tiation to time
t
at constant
x
0
results in the
material time derivative
of the
T
:
temperature,
∂
T
∂
t
T
=
.
(9.8)
x
0
constant
This variable
T
has to be interpreted as the change (per unit time) of the
temperature at a material point (moving through space) identified by
x
0
.
Another approach concentrates on a fixed point in three-dimensional space. At
every point in time a different material particle may be arrived at this location. For
the (time-dependent) temperature field it can be written:
T
=
T
(
x
,
t
) ,
(9.9)
indicating the temperature of the material being present at time
t
in the spatial
point
x
in
V
(
t
). This alternative field specification is called
Eulerian
. When the
partial derivative with respect to time
t
of the temperature field in the Eulerian
description is determined,
the spatial time derivative
δ
T
/δ
t
is obtained:
∂
T
∂
δ
T
δ
t
=
.
(9.10)
t
x
constant
