Biomedical Engineering Reference
In-Depth Information
This result
t
should be interpreted as the change (per unit time) of the tem-
perature at a fixed point in space
δ
T
/δ
x
, in which at consecutive time values
t
different
material points will be found.
The temperature field at time
t
, used as an example above, can be written in
a Lagrangian description:
T
x
0
,
t
) and thus be mapped on the reference
configuration with the domain
V
0
. The field can also be written in an Eulerian
description:
T
=
T
(
x
,
t
) and be associated with the current configuration with
domain
V
(
t
). It should be noticed that a graphical representation of such a field in
both cases can be very different, especially in the case of large deformations and
large rotations (both quite common in biological applications).
In Section
9.4
we focus on the relation between the time derivatives discussed
above. In Section
9.6
the relation between gradient operators applied to both
descriptions will be discussed.
=
T
(
9.4
The relation between the material and spatial time derivative
For the derivation of the relation between the material and spatial time derivative
of, for example, the temperature (as an arbitrary physical state variable, associated
with the material) we start with the Eulerian description of the temperature field
T
=
T
(
x
,
t
), in components formulated as
T
=
T
(
∼
,
t
)
=
T
(
x
,
y
,
z
,
t
). For the total
differential
dT
it can be written:
∂
T
∂
x
∂
T
∂
y
dT
=
dx
+
dy
y
,
z
,
t
constant
x
,
z
,
t
constant
∂
T
∂
z
∂
T
∂
t
+
dz
+
dt
x
,
y
,
t
constant
x
,
y
,
z
constant
(9.11)
and in a more compact notation, using the gradient operator (see Chapter
7
):
+
δ
T
δ
t
dt
+
δ
T
δ
t
dt
.
· ∇
T
dT
=
d
x
T
and also
dT
=
d
∼
∼
T
(9.12)
This equation describes the change
dT
of
T
at an arbitrary, infinitesimally small
change
d
x
(with associated
d
∼
) of the location in space, combined with an
infinitesimally small change
dt
in time.
Now the change
d
x
, in the time increment
dt
, is chosen in such a way that the
material is followed:
d
x
=
vdt
. This implies a change in temperature according to
=
Tdt
. Substituting this special choice in Equation (9.12), directly leads to
Tdt
=
v
· ∇
Tdt
+
δ
T
δ
dT
t
dt
,
(9.13)
