Geoscience Reference
In-Depth Information
Acceleration due to an imposed
force changing the otherwise
constant field
Acceleration due to moving in a
velocity field that changes with
position
u
(velocity
along the
X axis)
u
(velocity
along the
X axis)
Final
position in
velocity
field
Initial
velocity
field at
time
t
Initial
position in
velocity
field
d
u
t
δ
u
x
Final velocity
field at time
t
+
d
t
δ
x
X axis
X axis
x
x
Figure 16.1
Schematic
diagram illustrating the
contributions to velocity
changes in a moving
fluid field.
u
Force
at x
u
→
u
+
d
u
t
u
+
d
u
x
from changes in the velocity field in the moving air as it passes the point is given
by the product of the local velocity in the X direction,
u
, with the gradient of the
velocity
u
with respect to X in the moving air. Thus, in this simple case the
total
rate of change in
u
with time is given by the sum of two partial derivative terms,
i.e., by:
du
∂ ∂
=+
∂
u
u
(16.1)
u
dt
t
∂
x
However, more generally, the value of the instantaneous velocity component
u
within the moving air may be changing not only in the direction of the X axis but
also in the direction of the Y and Z axes, and the air may not just be moving only
in the direction of the X axis. The expression for the total rate of change in
u
with
time therefore recognizes these two additional possible causes of change and
the full expression for the total derivative is:
du
∂ ∂ ∂ ∂
=+ + +
∂
u
u
u
u
(16.2)
u
v w
dt
t
∂
x
∂
y
∂
z
Note that if the vector algebra representation were used, the last equation would be
written more concisely as:
⎡
⎛
⎞
⎤
du
∂
u
vu
∂
∂
∂
(16.3)
=+∇
( .
)
where
∇
is the vector operator
,
,
⎢
⎥
⎜
⎟
dt
∂
t
⎝
∂
x
∂
y
∂
z
⎠
⎣
⎦
