Page 75 - Aerodynamics for Engineering Students

58 Aerodynamics for Engineering Students

t
W
Fig. 2.4 The stream tube and element for the momentum equation

w

Fig. 2.5 The forces on the element

the curved surface the pressure may be taken to be the mean value p + $ (dp/ds)Ss.
In addition the weight W of the fluid in the element acts vertically as shown.
Shear forces on the surface due to viscosity would add another force, which is
ignored here.
As a result of these pressures and the weight, there is a resultant force F acting
along the axis of the cylinder where F is given by

SA-WCOS~ (2.5)

where Q is the angle between the axis of the stream tube and the vertical.
From Eqn (2.5) it is seen that on neglecting quantities of small order such as
(dp/ds)SsSA and cancelling,

since the gravitational force on the fluid in the element is pgA Ss, i.e. volume x
density x g.
Now, Newton's second law of motion (force = mass x acceleration) applied to the
element of Fig. 2.5, gives
dP dv
-pgASs COS^ --ASS = PASS-
ds dt
where t represents time. Dividing by A 6s this becomes
dp dv
-pgcos a - - = p-
ds dt