Geoscience Reference
In-Depth Information
Moving plate
U
h
Z
Figure 16.3
Velocity and
shearing stress generated by
laminar flow in a fluid
between when two plates, one
stationary and one moving in
the X direction at a velocity
u
t
U
h
.
X
Fixed plate
The resistance to continued motion between the plates per unit area, which is
called the
shearing stress
,
t
, is proportional to the (in this case uniform) gradient
of the velocity in the fluid. More generally, Newton proposed that even if the
gradient were not uniform, the
local
shearing stress between layers of fluid at
z
is proportional to the
local
gradient in velocity along the X direction at that
point, i.e., that:
∂
u
z
(16.15)
tmr
=
a
∂
where
m
is a property of the fluid called the
dynamic viscosity
. In the application
for which Equation (16.15) is required here, it is preferable to re-write the equation
to provide a description of the diffusion in air of the
kinematic flux of momentum
,
t
k
(=
t /r
a
), by defining the kinematic viscosity,
u
(=
m /r
a
). The resulting equation
has the form:
∂
u
z
tu
=
(16.16)
k
∂
Note that the dimensions of
t
k
in Equation (16.16)
are
(m s
−1
)(m s
−1
) as they must
be because the kinematic flux of momentum
k
uw
.
The above description is of a steady state in which the rate at which momentum
in the X direction is diffusing vertically does not change with distance along the Z
axis. In this example the desired term to be included in the sum of 'forces' on the
right hand side of Equation (16.4) that correspond to unbalanced diffusion of
horizontal momentum would be zero. However, it is when there is imbalance in
diffusion of horizontal momentum that is of general interest. This will occur when
the gradient of
u
in the Z direction is not uniform. Then the flow of momentum in
the X direction entering a small parcel of air at the point of interest from below will
not necessarily equal that leaving from above, and the parcel will change its veloc-
ity. Figure 16.4 illustrates this in the X-Z plane for the one-dimensional case of
non-uniform flow in the X direction.
τ= ′′
