Geography Reference
In-Depth Information
Fig. 1.1
The x component of the pressure gradient force acting on a fluid element.
Neglecting the higher order terms in this expansion, the pressure force acting on
the volume element at wall
A
is
p
0
+
δy δz
∂p
∂x
δx
2
F
Ax
=−
where δyδz is the area of wall A. Similarly, the pressure force acting on the volume
element at wall B is just
p
0
−
δy δz
∂p
∂x
δx
2
F
Bx
=+
Therefore, the net x component of this force acting on the volume is
∂p
∂x
F
x
=
F
Ax
+
F
Bx
=−
δx δy δz
Because the net force is proportional to the derivative of pressure in the direction
of the force, it is referred to as the
pressure gradient force.
The mass m of the dif-
ferential volume element is simply the density ρ times the volume: m
=
ρδxδyδz.
Thus, the x component of the pressure gradient force per unit mass is
F
x
m
=−
1
ρ
∂p
∂x
Similarly, it can easily be shown that the y and z components of the pressure
gradient force per unit mass are
F
y
m
=−
1
ρ
∂p
∂y
F
z
m
=−
1
ρ
∂p
∂z
and
