The Fluid Mechanics Continuity Equation
The basic principle of conservation of mass as applied to a fluid system is typically called the continuity equation. In its most general form, the unsteady state, the compressible flow continuity equation states that the total rate of mass flow into a system minus the total rate of mass flow out of that system at any time must equal the rate of change of mass in the system. Written as an equation this is:
Σmin -Σmout - dM/dt, where M is the mass of fluid in the system.
Many fluid flow systems operate under steady state flow conditions. That is, the total mass flow into the system and total mass flow out of the system remain constant with time. In that case, the amount of mass in the system doesn't change, so dM/dt = zero, and the steady state form of the compressible flow continuity equation becomes:
Σmin = Σmout or ΣρQin = ΣρQout or ΣρVAin = ΣρVAout
Another simplification of the continuity equation is its application to incompressible flow. In fluid mechanics the term, 'incompressible flow,' doesn't really refer to a fluid that cannot be compressed. Rather it refers to a situation in which the density of a fluid is not changing for a given set of fluid flow conditions. In general, the flow of liquids can be treated as incompressible flow and many cases of flowing gases must be treated as compressible flow. For incompressible flow, the density, ρ, remains constant, so the incompressible flow continuity equation becomes:
ΣQin = ΣQout or ΣVAin = ΣVAout.