Page 76 - Aerodynamics for Engineering Students

Governing equations of fluid mechanics 59

But
dv dvds dv
-
-
dt - ds dt = 'ds
and therefore
dv dp
pv-+-+pg cosa! = 0
ds ds
or
dv 1 dp
v- + - - + gcos a! = 0
ds Pds
Integrating along the stream tube; this becomes

f + vdv + g / cos ads = constant
but since
I cos ads = increase in vertical coordinate z

and

then

/ f + v2 + gz = constant

This result is known as Bernoulli's equation and is discussed below.

The conservation of energy
Conservation of energy implies that changes in energy, heat transferred and work
done by a system in steady operation are in balance. In seeking an equation
to represent the conservation of energy in the steady flow of a fluid it is useful
to consider a length of stream tube, e.g. between sections 1 and 2 (Fig. 2.6), as

Fig. 2.6 Control volume for the energy equation