Mechanics of Fluids

98 The principles governing fluids in motion
Fig. 3.5
velocity u1 enters at a section where the (average) elevation is z1 and leaves
with pressure p2 and velocity u2 where the (average) elevation is z2.
As the fluid moves from inlet to outlet, its properties, in general, change
from one point to another. However, we assume they do not change
with time.
We fix attention on the body of fluid bounded originally by the device
itself and the planes A (at entry) and C (at exit). After a short time interval
δt this fluid has moved forward, and is then bounded by the device
itself, entry plane B and exit plane D. During this time interval an elemental
mass δm (originally between planes A and B) enters the devices and,
by the principle of continuity, a mass δm also leaves to occupy the space
between planes C and D. The elements are assumed small enough for their
properties to be uniform. The element at entry has internal energy δme1
(where e represents internal energy/mass), kinetic energy 1
2 δmu2 1
and gravita-
tional energy δmgz1. Changes of electrical, chemical, nuclear or free surface
energy are disregarded here. If the energy of the fluid in the device itself
(i.e. between B and C) totals E then the energy of the fluid between A and
CisE + δm(e1 + 1
2 u2 1
+ gz1).
After this fluid has moved to the position between B and D its energy is
that of the fluid between B and C plus that of the element between C and D.
In other words the total is E+δm(e2 + 1
2u2 2 +gz2). Consequently the increase
in energy which this particular body of fluid receives is
� �
E + δm e2 + 1
2 u2 �� � �
2 + gz2 − E + δm e1 + 1
�
= δm (e2 − e1) + 1
2
�
u 2 2 − u2 �
1 + g(z2 − z1)
�
2 u2 1
+ gz1
��