Mechanics of Fluids

558 Unsteady flow
✷
whence
du
dt
=−(8 m)g
l
� 0
− 2f
d u2
du
du
∴ t =−
=−d
u1 (8 m)(g/l) + (2f /d)u2 2f u1 gd/(fl)(4 m) + u2 = d
�
� � � �
� �
fl
fl
arctan u1
2f gd(4 m)
gd(4 m)
�
Substituting u1 = 2.829 m · s −1 and the values of d, f , l and g we get
t = 2.618 s.
12.3 PRESSURE TRANSIENTS
We now consider the third category of unsteady flow phenomena: those
in which the changes of velocity occur so rapidly that elastic forces are
important. As a result of the elasticity of the fluid – and also the lack of
perfect rigidity of solid boundaries – changes of pressure do not take place
instantaneously throughout the fluid, but are propagated by pressure waves.
A change of velocity at a particular point in a fluid always gives rise to a
change of pressure, and an important instance of such pressure changes is
the phenomenon commonly known as water hammer in pipe-lines. The name
is perhaps a little unfortunate because not only water but any fluid – liquid
or gas – may be involved.
It is common experience that when a domestic water tap is turned off very
quickly a heavy knocking sound is heard and the entire pipe vibrates. These
effects follow from the rise in pressure brought about by the rapid deceleration
of the water in the pipe when the tap is turned off. A similar phenomenon
may occur in a pumping station owing to the slamming shut of non-return
valves when a pumping set is shut down. Not infrequently the increases of
pressure caused by water hammer are sufficient to fracture the pipes, and for
this reason alone the study of the phenomenon is of considerable practical
importance.
Consider the simple case of a fluid, originally flowing with a certain
velocity in a pipe, being brought to rest by the closing of a valve at the
downstream end of the pipe. If the fluid were entirely incompressible and
the walls of the pipe perfectly rigid, then all the particles in the entire column
of fluid would have to decelerate together. From Newton’s Second Law, the
more rapid the deceleration the greater would be the corresponding force,
and with an instantaneous closure of the valve all the fluid would be stopped
instantaneously and the force would be infinite. In fact, however, even a
liquid is to some extent compressible, so its constituent particles do not
decelerate uniformly. An instantaneous closure of the valve would not bring
the entire column of fluid to a halt instantaneously: only those particles of
fluid in contact with the valve would be stopped at once, and the others
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