Geoscience Reference
In-Depth Information
3.10.1 Net force and the rate of change of velocity
close to an interface
2
The rate of change of velocity with distance may
decrease away from the boundary (Fig. 3.45). This possi-
bility is discussed next.
We can imagine that the further we go away from an
interface the less likely it will be that the flow “feels” the
influence of the surface; it will be increasingly retarded by
its own constant internal property of viscosity. This is our
introduction to the concept of a
boundary layer
, being that
part of a flowing substance close to the boundaries to the
flow where there is a spatial change in the flow velocity
(Section 4.3). Such boundary layer gradients were first
investigated systematically by Prandtl and von Karman in
the early years of the twentieth century. At this stage we
are not concerned with calculating or predicting the exact
nature of the change in the rate of flow in a boundary
layer, but are content to accept that the field and experi-
mental evidence for such change is in no doubt. We shall
look at the question in more detail in Sections 4.3-4.5.
We make use of thought experiments at this point: let
velocity stay constant, increase, or decrease away from a
flow boundary (Fig. 3.44). In the first case no viscous
stress or net force exists. In the second and third cases vis-
cous stresses exist. There are two further possibilities:
1
The velocity of flow may decrease linearly from any bound-
ary so that the rate of change of velocity is constant. Here there
can be no
net
force acting across the constant velocity gradient,
d
u
/d
y
. This is because there is no rate of change, d/d
y
, of the
gradient, that is, d
2
u
/d
y
2
3.10.2
Net viscous force in a boundary layer
Careful measurements of flow velocity at increments up
from the bed of a river or through the atmosphere demon-
strate how the shape of a boundary layer is defined and
that while the velocity slows down through the boundary
layer toward the boundary itself, the velocity gradient
actually increases (Section 4.3). If we now consider an
imaginary infinitesimal plane in the
xy
plane of this bound-
ary layer flow (Fig. 3.45) it is immediately apparent that
the viscous stress,
zx
acting on unit area will be greater on
one side than the other, because the velocity gradient is
itself changing in magnitude. We call this difference in
stress the
gradient of the stress per unit area
, or d
zx
/d
y
. We
have already come across the concept of stress gradients in
our development of the simple expression that determines
the force due to static pressure (Section 3.5). Since a stress
is, by definition, force per unit area, any change in force
across an area is the
net
force acting.
Since we already have Newton's relationship for viscous
stress,
d
u
/d
z
(Section 3.9), we can combine the
previous expressions and write the net force per unit area
as d/d
z
(
zx
0 and the applied Newtonian
viscous stresses acting on both sides of an imaginary infinitesi-
mal plane normal to the
y
-axis are equal and opposite.
d
u
/d
z
), more concisely written as the constant
molecular viscosity times the second differential of the
velocity,
d
2
u
/d
z
2
(Fig. 3.45). This is the second time we
(a)
(b)
(c)
u
3
u
3
Velocity
u
3
Viscosity,
m
u
2
u
2
Velocity
u
2
u
1
u
1
Velocity
u
1
Velocity,
u
Velocity,
u
Velocity,
u
Fig. 3.44
By Newton's relationship,
d
u
/d
y
, viscous frictional forces can only be present if there is a gradient of mean flow velocity in any
flowing fluid. The three graphs are sketches of simple hypothetical velocity distributions. (a) has no gradient and therefore no viscous stresses;
(b) has a positive linear velocity gradient, that is, velocity increasing at constant rate upward, and hence has viscous stresses of constant magni-
tude; (c) has a negative linear velocity gradient, that is, velocity decreasing at constant rate upward, and hence also has viscous stresses of con-
stant magnitude.
Search WWH ::
Custom Search