Continuity Equation

A system always entails the same quantity of matter. Therefore, the mass M would be constant. To go from system approach to control approach, we use Reynold transport equation in which N is for our case M, the mass of a fluid system and it is M = JJf pdV. Therefore, n = 1. Thus for a constant mass in a system at any time t, we can write the Reynold transport equation as following:

The velocity v and the time derivative are measured relative to the control volume. Above equation can be written in the following form:

cs cv

Equation 2.6 which is the final form of the continuity equation expresses that the net efflux rate of mass through the control surface equals the rate of decrease of mass inside the control volume. Two special cases of continuity can be considered. The case of steady flow in which all fluid properties at any fixed position in the reference must remain invariant, the continuity equation can be written as:

The other case is the case of incompressible fluid having constant p.The continuity equation can be written as:


A more detailed definition of compressibility is brought in the proceeding sections.

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