Mechanics of Fluids

54 Fluid statics
Micro-manometers
for a small value of p ∗ 1 − p∗ 2
The interface between liquids of closely similar
densities, however is very sensitive to changes in surface tension and therefore
to traces of grease and other impurities. Air may be used as fluid B: it may
be pumped through the valve V at the top of the manometer until the liquid
menisci are at a suitable level. Then, of course, ϱB is negligible compared
with ϱA.
Certain practical considerations arise in the use of manometers. (1) Since
the densities of liquids depend on temperature the temperature of the liquids
should be known for accurate results. (2) Some liquids, otherwise suitable
for use in manometers, give ill-defined menisci. (3) Fluctuations of menisci
reduce accuracy; such movements may be reduced by restrictions in the
manometer connections (e.g. lengths of small-diameter pipe) which, under
equilibrium conditions, do not affect the pressure. (4) The density of the
fluid in the connecting tubes must be uniform; for example, water must not
contain air bubbles, nor must air contain ‘blobs’ of water. The layout of
the connecting tubes should be such as to minimize the possibility of trapping
air bubbles, and means should be provided for flushing the connecting
tubes through before the manometer is used. A valve by which the pressure
difference may be reduced to zero and the zero reading thus checked
is also desirable. (5) In tubes of less than about 15 mm diameter surface
tension effects may be appreciable and the meniscus is either raised above
or lowered below its ‘correct’ position. For example, for pure water in a
clean, vertical, glass tube, 6 mm diameter, the capillary rise is about 5 mm.
The corresponding depression for mercury is about 1.25 mm. Because of
the uncertain degree of cleanliness of tubes used in practice, however, it is
difficult to allow for surface tension effects. Fortunately these effects can be
nullified, for example in a U-tube manometer where the limbs are of equal
diameter and cleanliness, or where measurements are made of the movement
of a single meniscus in a uniform tube. Alcohol, being a solvent of grease, is
less sensitive to the cleanliness of the tube and so is frequently preferred to
water in manometers.
For measuring very small pressure differences, a wide variety of special
manometers has been developed. Several devices may be used to increase
the accuracy of a reading. For example, a meniscus may be observed
through a small telescope containing a horizontal cross-wire, and the
assembly may be raised or lowered by a slow-motion screw with a micrometer
scale. Or a scale floating on the surface of a liquid may be optically
magnified.
When an additional gauge liquid is used in a U-tube a large difference
of meniscus level may be produced by a small pressure difference.
One arrangement is illustrated in Fig. 2.11. The appropriate equilibrium
equation is
�
p1 + ϱAg(h + �z) + ϱBg z − �z + y
�
2
�
= p2 + ϱAg(h − �z) + ϱBg z + �z − y
�
+ ϱCgy (2.10)
2