Biomedical Engineering Reference
In-Depth Information
y
18
20
22
3
m
24
26
28
30
6
m
x
Figure 7.7
Isotherms (
◦
C) in a cross section with constant
z
.
by
∇
when vector notation is used and with
∼
when components are used. In the
example of the temperature field this yields
⎡
⎣
⎤
⎦
⎡
⎣
⎤
⎦
∂
T
∂
∂
∂
x
x
∂
T
∂
∂
∼
T
=
with
∼
=
,
(7.7)
∂
y
y
∂
T
∂
∂
∂
z
z
and also
e
x
∂
T
∂
x
+
e
y
∂
T
∂
y
+
e
z
∂
T
∂
z
e
x
∂
e
y
∂
e
z
∂
∇
∇=
T
=
with
∂
x
+
∂
y
+
∂
z
.
(7.8)
The gradient of a certain property is often a measure for the intensity of a physi-
cal transport phenomenon; the gradient of the temperature for example is directly
related to the heat flux. Having the gradient of a certain property (
T
), the deriva-
tive of that property along a spatial curve (given in a parameter description, see
Fig.
7.3
),
x
=
x
(
ξ
), can be determined (by using the chain rule for differentiation):
dT
d
=
∂
T
dx
d
+
∂
T
dy
d
+
∂
T
dz
d
ξ
∂
x
ξ
∂
y
ξ
∂
z
ξ
=
(
∼
T
)
T
d
∼
T
T
)
d
∼
d
=
(
∼
d
ξ
ξ
d
∼
d
ξ
T
=
∼
T
.
(7.9)
On the right-hand side of the equation the inner product of the (unnormalized)
tangent vector to the curve and the gradient vector can be recognized. Eq. (
7.9
)
can also be written as
