Introduction - An Introduction to Dynamic Meteorology - page 9

expressed in the form F u 0 δz/l is the velocity shear

across the layer δz. The viscous force per unit area, or shearing stress , can then be
defined as

= µAδu/δz where δu =

µ δu µ ∂u

∂z

τ zx = lim

δz δz = → 0

where subscripts indicate that τ zx is the component of the shearing stress in the x
direction due to vertical shear of the x velocity component.
From the molecular viewpoint, this shearing stress results from a net downward
transport of momentum by the random motion of the molecules. Because the
mean x momentum increases with height, molecules passing downward through a
horizontal plane at any instant carry more momentum than those passing upward
through the plane. Thus, there is a net downward transport of x momentum. This
downward momentum transport per unit time per unit area is simply the shearing
stress.
In a similar fashion, random molecular motions will transport heat down a mean
temperature gradient and trace constituents down mean mixing ratio gradients. In
these cases the transport is referred to as molecular diffusion . Molecular diffusion
always acts to reduce irregularities in the field being diffused.
In the simple two-dimensional steady-state motion example given above there
is no net viscous force acting on the elements of fluid, as the shearing stress acting
across the top boundary of each fluid element is just equal and opposite to that acting
across the lower boundary. For the more general case of nonsteady two-dimensional
shear flow in an incompressible fluid, we may calculate the viscous force by again
considering a differential volume element of fluid centered at (x,y,z) with sides
δxδyδz as shown in Fig. 1.4. If the shearing stress in the x direction acting through
the center of the element is designated τ zx , then the stress acting across the upper
boundary on the fluid below may be written approximately as

δz
2
while the stress acting across the lower boundary on the fluid above is

∂τ zx
∂z

τ zx +

τ zx −

∂τ zx
∂z

δz
2

−

(This is just equal and opposite to the stress acting across the lower boundary
on the fluid below .) The net viscous force on the volume element acting in the x
direction is then given by the sum of the stresses acting across the upper boundary
on the fluid below and across the lower boundary on the fluid above:
τ zx + δx δy τ zx − δx δy

∂τ zx
∂z

δz
2

∂τ zx
∂z

δz
2

−