fluid_mechanics

2.3 Pressure Variation in a Fluid at Rest 41
The resultant surface
force acting on
a small fluid element
depends only
on the pressure
gradient if there are
no shearing
stresses present.
or
where î, ĵ, and kˆ are the unit vectors along the coordinate axes shown in Fig. 2.2. The group
of terms in parentheses in Eq. 2.1 represents in vector form the pressure gradient and can be
written as
where
dF s a 0p
0x î 0p
0y ĵ 0p
0z kˆb dx dy dz
0p
0x î 0p
0y ĵ 0p
0z kˆ §p
§ 1 2 01 2
0x î 01 2
0y ĵ 01 2
0z kˆ
and the symbol § is the gradient or “del” vector operator. Thus, the resultant surface force per
unit volume can be expressed as
(2.1)
Since the z axis is vertical, the weight of the element is
where the negative sign indicates that the force due to the weight is downward 1in the negative z
direction2. Newton’s second law, applied to the fluid element, can be expressed as
where dF represents the resultant force acting on the element, a is the acceleration of the element,
and dm is the element mass, which can be written as r dx dy dz. It follows that
or
and, therefore,
dF s
dx dy dz §p
dwkˆ g dx dy dz kˆ
a dF dm a
a dF dF s dwkˆ dm a
§p dx dy dz g dx dy dz kˆ r dx dy dz a
§p gkˆ ra
Equation 2.2 is the general equation of motion for a fluid in which there are no shearing stresses.
We will use this equation in Section 2.12 when we consider the pressure distribution in a moving
fluid. For the present, however, we will restrict our attention to the special case of a fluid
at rest.
(2.2)
2.3 Pressure Variation in a Fluid at Rest
For a fluid at rest a 0 and Eq. 2.2 reduces to
or in component form
0p
0x 0
§p gkˆ 0
0p
0y 0
0p
0z g
These equations show that the pressure does not depend on x or y. Thus, as we move from
point to point in a horizontal plane 1any plane parallel to the x–y plane2, the pressure does not
(2.3)