Continuity Equation (Aerodynamics)

The law of Aerodynamics are formulated by applying to a flowing gas several basic principles from physics.

Physical Principle :- Mass can neither be created nor destroyed.

Fig. 1 Steam tube with mass conservation.

To apply this principle of flowing gas,

Consider an imaginary circle drawn perpendicular to the flow direction as shown in Fig. 1.
Now look at all the steam lines that go through the circumference of the circle. These streamlines form a tube called Stream tube.
As we move along with the gas confined inside the stream tube, we see that the cross sectional area of the tube may change, say in moving from point 1 to point 2 in Fig. 1.
However, as long as the flow is steady, the mass that flows through the cross section at point 1 must be the same as the mass that flows through the cross section at point 2, become the definition of stream line, there can be no flow across streamlines.
The mass flowing through the stream tube is confined by the streamlines of the boundary, much as the flow of water through a flexible garden hose is confined by the wall of the hose.

* This is a case of '' What goes in one end must come out the other end''.

Let,

After a lapse of time dt, these same fluid elements all move a distance.

In doing so elements have swept out a volume, downstream of point 1. The mass of gas dm in the volume is equal to the density times the volume that is,

This is the mass of gas that has swept through the cross section at point 1 during time interval dt.

*The mass flow Ṁ through area A is the mass crossing A per unit time.

Also, the mass flow through the other cross section A2, bounded by the same streamlines that go through the circumference of the A1, is obtained in the same fashion, as