Geoscience Reference
In-Depth Information
We are not quite finished, since the term given in (2.11) is not the only contri-
bution to the momentumflux tensor. Consider the situation illustrated in Fig. 2.2,
where the
x
component of the mean fluid velocity increases in the
z
direction. Par-
ticles randomly crossing the plane
z
z
0
from above that collide with particles
below will, on average, contribute more
x
momentum to the fluid below
z
=
z
0
than particles crossing in the other direction will contribute above. This momen-
tum transfer results in forces on the fluid such that the velocity gradient in this
direction is reduced.Unlike themomentumfluxdescribedby (2.11), this “viscous”
force depends on collisions and is a dissipative process. Energy in the mean flow
is converted into heat when viscosity is important. The exact form of the viscous
momentum flux density tensor is quite complicated, and we refer the reader to
a discussion by Landau and Lifshitz (1959) for details. In its simplest form, the
momentum flux density depends on the first derivatives of the velocity with two
associated proportionality constants
=
η
.When the divergence of themomen-
tum flux tensor is taken, as required by (2.10), the viscous force
F
v
is given by
η
and
2
U
+
η
∇
(
∇·
F
v
=
η
∇
U
)
(2.12)
For incompressible flow
(
∇·
U
)
=
0 and (2.12) reduces to the more familiar
form
2
U
F
v
=
η
∇
(2.13)
where
η
is called the dynamic viscosity coefficient. Equation (2.10) becomes
∂(ρ
U
)/∂
t
+
U
·∇
(ρ
U
)
+
ρ
U
(
∇·
U
)
=−∇
F
−∇·
π
w
+
p
+
viscous term
(2.14)
Using the continuity equation, the left-hand side is just
ρ
d
U
/
dt
. Finally,
using (2.13)
2
U
ρ
d
U
/
dt
=−∇
p
+
F
−∇·
π
w
+
η
∇
(2.15)
ˆ
x
U
5
u
≠
u
(
z
)
≠
z
.
0
Z
5
Z
0
ˆ
z
ˆ
x
Figure 2.2
If the wind increases with increasing
z
, the viscous force will tend to reduce
the velocity for
z
>
z
0
and to increase it for
z
<
z
0
.
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