The continuity equation is a consequence of the conservation of mass. It states that that the rate at which mass enters a system is equal to the rate at which mass leaves the system.
Imagine that a tube has two different sections, as shown in the figure below. It has fluid in a steady flow (that is, the density and the speed of the fluid at each point are constant). Since the flow is steady, the shape of the streamlines (in blue in the figure) remains constant.
If the fluid has no viscosity (it has no internal friction) every fluid element in the cross-sectional area A1 of the tube will move at the same speed (in red). Those fluid elements will reach surface A2 simultaneously.
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The flux or mass flow rate Φ is defined as the rate of flow of mass per unit area:
Since mass is conserved, the net flux has to be zero (the mass that enters is equal to the mass that leaves the tube).
The amount of fluid that crosses A1 in time dt is given by:
As for A2, in the same time dt we have:
And if we assume there is no mass loss along the tube:
If the fluid is incompressible its density is constant along the tube, therefore:
This relation is called the continuity equation.
The continuity equation explains why when water reaches the nozzle of a hose, narrower that the hose itself, its velocity increases. Or why the blood speed is different in arteries than in capillaries.