Page 88 - Aerodynamics for Engineering Students

Governing equations of fluid mechanics 71

4

U-

Fig. 2.12 Rectangular space of volume 6x x Sy x 1 at the point P (x, y) where the velocity components
are u and v and the density is p

2.4.2 The equation of continuity or conservation of mass
Consider a typical elemental control volume like the one illustrated in Fig. 2.8. This is
a small rectangular region of space of sides Sx, Sy and unity, centred at the point
P(x, y) in a fluid motion which is referred to the axes Ox, Oy. At P(x, y) the local
velocity components are u and v and the density p, where each of these three
quantities is a function of x, y and t (Fig. 2.12). Dealing with the flow into the box
in the Ox direction, the amount of mass flowing into the region of space per second
through the left-hand vertical face is:

mass flow per unit area x area
i.e.

(2.38)

The amount of mass leaving the box per second through the right-hand vertical face
is:

(2.39)

The accumulation of mass per second in the box due to the horizontal flow is the
difference of Eqns (2.38) and (2.39), Le.

(2.40)

Similarly, the accumulation per second in the Oy direction is

(2.41)

so that the total accumulation per second is

(2.42)