Geoscience Reference
In-Depth Information
⎛⎞
F
∂
v
1
∂
P
⎛⎞
(16.11)
y
=
= −
⎜⎟
⎜⎟
⎝⎠
r
∂
t
r
∂
y
⎝⎠
a
pressure
a
pressure
⎛⎞
F
⎛⎞
∂
w
1
∂
P
(16.12)
z
=
= −
⎜⎟
⎜⎟
r
⎝⎠
∂
t
r
∂
z
⎝⎠
a
pressure
a
pressure
Were we to use vector algebra formulation for Equations (16.10), (16.11), and
(16.12) these three equations would be written more concisely as a single vector
equation, thus:
⎛⎞
F
∂
⎛⎞
v
1
=
= −
∇
P
(16.13)
⎜⎟
⎜⎟
⎝⎠
⎝⎠
r
∂
t
r
a
pressure
a
pressure
Viscous flow in fluids
A second way that the velocity component of a small parcel of air might change at
a particular point is if, as a result of the molecular movements in the air, there is a
net transfer to that point of momentum in the direction of interest. In other words
if, for example, there is more momentum in the X direction diffusing into the par-
cel by molecular diffusion than is diffusing out of the parcel, the local velocity of
the parcel in the X direction will increase. We therefore expect that one of the
'force' terms needed on the right hand side of Equations (16.4), (16.5), and (16.6)
will quantify the effect of any imbalance in the amount of momentum in each
direction entering and leaving at the point where the equations are applied. The
next step is, therefore, to consider the equations which describe the molecular dif-
fusion of momentum in air and, from these, to define the term that calculates the
local imbalance for each axis.
The equation describing molecular transfer of momentum in fluids was written
by Newton many centuries ago. It is framed in terms of the
viscosity
of the fluid,
which is a basic molecular property of the fluid (in the present case, air) that is a
measure of the fluid's internal resistance to deformation. In other words, it defines
the ease with which (hypothetically) parallel layers of fluid can slip past each other.
Figure 16.3 illustrates a two-dimensional example in which a fluid is undergoing
smooth, streamlined, laminar (i.e., not turbulent) flow in the X direction between
two very large parallel plates separated by a distance
h
.
Consider the variation in velocity in the Z direction which is perpendicular to
the two plates. In this case, the velocity
u
in the X direction varies uniformly from
zero at the lower plate to
U
h
at the upper plate and:
U
∂
u
zh
(16.14)
=
h
∂
