Page 74 - Aerodynamics for Engineering Students

Governing equations of fluid mechanics 57

Fig. 2.3 The stream tube for conservation of mass

The conservation of mass
This law satisfies the belief that in normally perceived engineering situations matter
cannot be created or destroyed. For steady flow in the stream tube shown in Fig. 2.3
let the flow properties at the stations 1 and 2 be a distance s apart, as shown. If the
values for the flow velocity v and the density p at section 1 are the same across the
tube, which is a reasonable assumption if the tube is thin, then the quantity flowing
into the volume comprising the element of stream tube is:
velocity x area = VIA]
The mass flowing in through section 1 is
PlVlAl
Similarly the mass outflow at section 2, on making the same assumptions, is

PzvzA2 (2.2)
These two quantities (2.1) and (2.2) must be the same if the tube does not leak or gain
fluid and if matter is to be conserved. Thus

PlVlAl = P2V2-42 (2.3)
or in a general form:

pvA = constant (2.4)

The conservation of momentum
Conservation of momentum requires that the time rate of change of momentum in
a given direction is equal to the sum of the forces acting in that direction. This is
known as Newton’s second law of motion and in the model used here the forces
concerned are gravitational (body) forces and the surface forces.
Consider a fluid in steady flow, and take any small stream tube as in Fig. 2.4. s is
the distance measured along the axis of the stream tube from some arbitrary origin.
A is the cross-sectional area of the stream tube at distance s from the arbitrary origin.
p, p, and v represent pressure, density and flow speed respectively.
A, p, p, and v vary with s, i.e. with position along the stream tube, but not with time
since the motion is steady.
Now consider the small element of fluid shown in Fig. 2.5, which is immersed in
fluid of varying pressure. The element is the right frustrum of a cone of length Ss, area
A at the upstream section, area A + SA on the downstream section. The pressure
acting on one face of the element is p, and on the other face is p + (dp/ds)Ss. Around