Biomedical Engineering Reference
In-Depth Information
dT
d
ξ
d
x
d
ξ
· ∇
=
T
.
(7.10)
Along a contour line (isotherm) in a two-dimensional cross section
dT
/
d
ξ
=
0,
∇
T
(with components
so the gradient vector
∼
T
) must be perpendicular to the
contour line in each arbitrary point. In a three-dimensional configuration it can be
said that the gradient vector of a certain scalar property in a point of V will be
perpendicular to the contour plane through that point. The directional derivative
dT
de
of the temperature (the increase of the temperature per unit length along the
curve) in the direction of the unit vector
/
e
, with components
∼
can be written as
dT
de
=
∼
dT
de
=
e
· ∇
T
.
T
∼
T
and also
(7.11)
7.5
Vector fields
In the previous section, as an example, a scalar temperature field was considered.
Departing from a given temperature field, the associated (temperature) gradient
field can be determined. This gradient field can be regarded as a vector field (in
every point of the volume
V
the components of the associated vector can be deter-
mined). Considering the special meaning of such a gradient field, we will not use
that field as a typical example in the current section. Instead, the (momentary)
velocity field
x
) of the material in the volume
V
will be used as an illustra-
tion. With respect to the Cartesian
xyz
-coordinate system the velocity vector
v
=
v
(
v
and
the associated column
∼
can be written as
⎡
⎤
v
x
v
y
v
z
⎣
⎦
v
=
v
x
e
x
+
v
y
e
y
+
v
z
e
z
,
∼
=
.
(7.12)
The possibilities for a clear graphical illustration of a vector field (velocity) in a
three-dimensional configuration are limited. For a problem with a flat fluid flow
pattern, for example:
v
x
=
v
x
(
x
,
y
)
v
y
=
v
y
(
x
,
y
)
v
z
=
0,
(7.13)
a representation like the one given in Fig.
7.8
can be used. The arrows represent
the velocity vector (magnitude and orientation) of the fluid at the tail of the arrow.
At a given velocity
v
in a certain point of
V
, the velocity component
v
in
the direction of an arbitrary unit vector
e
(components in the column
∼
) can be
calculated with
