Geoscience Reference
In-Depth Information
∂
p
d
z
p
...
+ +
∂
d
x
d
y
z
2
∂
p
d
x
p
y
...
+
+
d
y
d
z
∂
x
2
C
d
z
D
d
x
B
∂
p
d
x
p
...
−
+
d
y
d
z
∂
x
2
q
A
r
g
d
x
d
y
d
z
∂
p
d
z
∂
...
p
d
x
d
y
−
+
z
∂
z
2
x
Fig. 1.4 Definition sketch for the conservation of momentum of a fluid element occupying the volume
δ∀
=
z
). The element is subject to pressure forces and the
acceleration of gravity. The
y
-coordinate, which is not shown, points into the plane of the
drawing.
(
δ
x
δ
y
δ
z
), with its center at (
x
,
y
,
Making use of Equation (1.4), one obtains immediately
D
v
Dt
=−
1
ρ
∇
p
+
g
(1.11)
which is a form of Euler's equation. Inclusion of the effect of viscosity into Euler's equation
produces the Navier-Stokes equation; expanding the substantial derivative according to its
definition (Equation (1.3)), one can write it as follows
∂
v
∂
t
+
v
·∇
v
=−
1
ρ
∇
p
+
g
+
f
(1.12)
where
f
denotes the frictional force (per unit mass); for an incompressible Newtonian
fluid it can be shown that this is given by
f
is the kinematic viscosity.
To repeat briefly, the first term on the left represents the change in momentum (per unit
mass) of the fluid due to local acceleration, i.e. changes in velocity at the point (
x
=
ν
∇
2
v
, where
ν
z
)
under consideration. The second term represents the momentum changes resulting from
acceleration (or deceleration) experienced by the fluid as it moves between points with
different velocities. The first term on the right represents the force resulting from the pressure
gradient, and the second the force resulting from the gravity field of the Earth. If the
z
-axis
represents the vertical and points upward (or
θ
=
0 in Figure 1.4), Equation (1.12) can be
written as
,
y
,
∂
v
∂
t
+
v
·∇
v
=−
1
ρ
∇
p
−
g
k
+
f
(1.13)
in which, it should be recalled,
k
is the unit vector in the
z
-direction.
