Continuity equation

If we consider the machine to be an open-ended duct, we find that the mass flow per second will depend on the density of the fluid and the volume flowing per sec:

Now volume flow = Area of duct x distance travelled (L)/ Time (sec)
But the distance travelled per second = Velocity.
Therefore, Mass flow = density x area x velocity.
This is known as the ‘continuity equation' and it is true for any steady flow system regardless of changes in the cross-sectional area of the duct.

INCOMPRESSIBLE FLUID FLOW.

Now consider an incompressible fluid as it flows through the duct system shown in the fig. 1.7. We know that the mass flow is of a constant value and, naturally, as the fluid enters the larger cross sectional area it will take up the new shape and the initial volume will now occupy less length in the duct. Therefore, in a given time, less distance is travelled and the velocity is reduced.
Thus we conclude that if the mass flow is to remain constant, as it must, an increase in duct area must be accompanied by a reduction in flow velocity, and a decrease in duct area must bring about an increase in velocity; we can express this action as - velocity varies inversely with changes