Page 89 - Aerodynamics for Engineering Students

72 Aerodynamics for Engineering Students

As mass cannot be destroyed or created, Eqn (2.42) must represent the rate of
change of mass of the fluid in the box and can also be written as
a(p x volume)
at
but with the elementary box having constant volume (Sx by x 1) this becomes
aP (2.43)
-6xSy x 1
at
Equating (2.42) and (2.43) gives the general equation of continuity, thus:
-+- a(Pu) a(P.1 - 0
+--
aP
at
ay
ax
This can be expanded to:
ap
aP ax ay (E i;)
ap
-+-
-+u-+v-+p =o (2.45)
at
and if the fluid is incompressible and the flow steady the first three terms are all zero
since the density cannot change and the equation reduces for incompressible flow to
au av
-+-=o (2.46)
ax ay
This equation is fundamental and important and it should be noted that it expresses
a physical reality. For example, in the case given by Eqn (2.46)

This reflects the fact that if the flow velocity increases in the x direction it must
decrease in they direction.
For three-dimensional flows Eqns (2.45) and (2.46) are written in the forms:
ap ap ap ap au av aw
-+-+-
-+u-+v-+w-+p az (ax ay az 1 =o (2.47a)
at
ay
ax
au av
-+-+-=o aw (2.47b)
ax ay az
2.4.3 The equation of continuity in polar coordinates
A corresponding equation can be found in the polar coordinates r and 0 where the
velocity components are qn and qt radially and tangentially. By carrying out a similar
development for the accumulation of fluid in a segmental elemental box of space, the
equation of continuity corresponding to Eqn (2.44) above can be found as follows.
Taking the element to be at P(r, 0) where the mass flow is pq per unit length
(Fig. 2.13), the accumulation per second radially is:

(2.48)