Mechanics of Fluids

3. dp/dz =−ϱg. (Since the pressure varies only in the vertical (z) direction,
the partial derivative in eqn 2.2 may give way to the full derivative.)
To determine the pressure at any point in a fluid in equilibrium, eqn 2.2
must be integrated:
�
p = −ϱgdz
Evaluation of the integral is not possible, however, unless the variation of ϱ
with z known.
2.2.1 The equilibrium of a fluid of constant density
Since for all practical differences in height the variation of g is negligible,
integration of eqn 2.2 for a homogeneous fluid of constant density gives
p + ϱgz = constant (2.3)
This result is valid throughout a continuous expanse of the same fluid since,
in deriving eqn 2.2, no restriction at all was placed on the value of θ.
The value of the constant in eqn 2.3 is determined by the value of p at a
point where z is specified. If the fluid is a liquid with a horizontal free surface
at which the pressure is atmospheric (pa) this free surface may be taken
as the datum level z = 0. For equilibrium of the surface the pressure immediately
below it must equal that immediately above it, and so the pressure
in the liquid at the surface is pa. Then, for a point at a depth h below the
surface, h =−z (since h is measured downwards whereas z is measured
upwards) and, from eqn 2.3,
Variation of pressure with position in a fluid 45
p = pa + ϱgh (2.4)
The pressure therefore increases linearly with the depth, whatever the shape
of any solid boundaries may be.
Equation 2.4 shows that the pressure at a point in a liquid in equilibrium
is due partly to the weight of the liquid. Thus atmospheric pressure is usually
effective, even if indirectly, on all surfaces, and over the differences of height
normally encountered it is sensibly constant. Consequently it is often simpler
to regard atmospheric pressure as the zero of the pressure scale. A pressure
expressed relative to atmospheric pressure is known as a gauge pressure.
Equation 2.4 then reduces to p = ϱgh. As we shall see in Section 2.3, this
relation forms the basis of a number of methods of measuring pressure.
The direct proportionality between gauge pressure and h for a fluid of
constant density enables the pressure to be simply visualized in terms of
the vertical distance h = p/ϱg. The quotient p/ϱg is termed the pressure
head corresponding to p. So useful is the concept of pressure head that it
is employed whether or not an actual free surface exists above the point
in question. For a liquid without a free surface, as for example in a closed
pipe, p/ϱg corresponds to the height above the pipe to which a free surface
would rise if a small vertical tube of sufficient length and open to atmosphere
– known as a piezometer tube – were connected to the pipe (Fig. 2.2).
Fig. 2.2