16.6 : Continuity Equation

The continuity equation asserts that the mass flow rate must remain constant for a steady flow of an incompressible fluid within a confined system. This principle applies to systems where fluid passes through varying cross-sectional areas, such as nozzles, syringes, and pipes.

The mass flow rate is expressed as:

For incompressible flow, where the density remains constant, the continuity equation simplifies to:

This equation illustrates that the velocity increases when the cross-sectional area decreases and vice versa, ensuring mass conservation. An example is the flow through a syringe. When the plunger compresses the fluid through a nozzle, the continuity equation shows that if the outlet area is half the inlet area, the outlet velocity must be twice the inlet velocity.

Another example is flow in a gravity-driven tank. Applying both Bernoulli's equation and the continuity equation to points along the flow helps to determine outlet velocity and volumetric flow rate:

where h is the fluid's height, D is the diameter of the tank, and d is the outlet diameter. This relationship shows that the flow rate converges to a stable value as the outlet diameter becomes small relative to the tank diameter. These derivations exemplify the utility of the continuity equation in predicting fluid behavior under various constraints, ensuring mass conservation in dynamic systems.