Page 98 - Aerodynamics for Engineering Students

Governing equations of fluid mechanics 81

So the x component of Eqn (2.61) becomes
du sx du sx du sx du sx
p ( u+-- dx2) Syxl ( u+-- dx2) -p ( u--- dx2) Syxl ( u--- ax2)

( ;;:)
( :;;)
dv sy ( :;3
+p v+-- Sxxl u+-- -p v--- Sxxl u---
( dy2)
Cancelling like terms and neglecting higher-order terms simplifies this expression to
p(2ug+v$+ug)sxsyx 1

This can be rearranged as
( :: du {E 3)
-+-
p u-+v-+u SxSy x 1 (2.62a)
ay
- -
= 0 Eqn (2.46)
In an exactly similar way the y component of Eqn (2.61) can be shown to be
( ;: :;) sxsy x 1 (2.62b)
v-
p u-+
Term (iii) the body force, acting on the control volume, is simply given by the
weight of the fluid, i.e. the mass of the fluid multiplied by the acceleration (vector)
due to gravity. Thus

Normally, of course, gravity acts vertically downwards, so gx = 0 and g,, = -g.
The evaluation of Term (iv), the net pressure force acting on the control volume is
illustrated in Fig. 2.22. In the x direction the net pressure force is given by
dP
(P-ax~)syxl-(p+g~)s~xl=--SxS~x~
ap
sx
(2.64a)
dX

Fig. 2.22 Pressure forces acting on the infinitesimal control volume