The continuity equation is a physical law that states that the amount of mass or fluid entering a closed system is equal to the amount of mass or fluid leaving the system in the same period of time.
In mathematical terms, the continuity equation is expressed by the following formula:
A_{1} * v_{1} = A_{2} * v_{2}
Where:

A_{1} and A_{2} are the crosssectional areas of the conduit or pipe at points 1 and 2 respectively.

v_{1} and v_{2} are the velocities of the fluid at points 1 and 2 respectively.
According to the continuity equation, if the crosssectional area of the conduit or pipe through which the fluid flows is held constant, then the velocity of the fluid and the flow rate are inversely related. In other words, if the velocity of the fluid increases, the flow rate decreases and vice versa.
What is the continuity equation used for?
The continuity equation has multiple applications in physics and engineering, particularly in fluid mechanics. Here are some of its main applications:

Piping System Design – Used to calculate flow rate and fluid velocity at different points in the piping system, allowing the diameter and length of pipes to be sized to ensure constant and uniform flow.

Analysis of the flow in conduits and channels: it is applied to analyze the flow of liquids in conduits and channels, allowing the determination of speed and flow at different points of the system.

Optimization of the efficiency of hydraulic systems: it is used to optimize the efficiency of hydraulic systems, such as turbines and pumps, since it allows calculating the flow and velocity of the fluid at different points in the system and determining the optimal geometry of the system components.
For what type of fluids is it valid?
The continuity equation is valid for any type of fluid, as long as the fluid is incompressible and the flow is steady, that is, the speed and properties of the fluid at any point in the system do not vary with time.
An incompressible fluid is one that has a constant density and does not change its volume in response to the application of pressure.
Examples of use of the continuity equation
Here are some examples of its application:
Liquid flowing in a tube
A classic example of the application of the continuity equation is the flow of liquid in a tube.
Suppose a liquid flows through a tube of cross section A_{1} with a velocity v_{1} and then enters a tube of cross section A_{2} with a velocity v_{2} .
Using this equation we can size the tube sections to alter the flow velocity.
Water folwing in a river
The continuity equation also applies to the flow of water in a river.
This equation is used to calculate the velocity of the water at different points in the river. Therefore, the behavior of the river can be predicted in different conditions, such as when dams are built or engineering works for flood control are carried out.
Solved problems on the continuity equation in a fluid
Exercise 1
A pipe with a cross section of 0.02 m² transports water at a speed of 2 m/s. If the diameter of the tube is reduced to half its original value, what is the velocity of the water in the narrow tube?
Solution:
The continuity equation states that the volumetric flow rate of the fluid flowing through the tube is constant throughout the flow. Therefore, we can write:
A_{1·}v_{1} = A_{2·}v_{2}
Where A_{1} is the original cross section of the tube, v_{1} is the original velocity of the water, A_{2} is the cross section of the narrow tube, and v_{2} is the velocity of the water in the narrow tube.
We have A_{2} = A_{1} /4, since the diameter of the tube is reduced to half of its original value, therefore A_{2} = π(0.01 m)² = 0.000314 m².
Substituting the known values into the continuity equation, we get:
0.02 m² × 2 m/s = 0.000314 m² × v _{2}
v_{2} = (0.02 m² × 2 m/s) / 0.000314 m² = 127.39 m/s
Therefore, the speed of the water in the narrow tube is 127.39 m/s.
Exercise 2
A 0.1 m diameter pipe carries water at a velocity of 2 m/s. If two 0.05mdiameter pipes are added, what is the velocity of the water in each of the smaller pipes?
Solution:
The cross section of a 0.1 m diameter pipe is A_{1} = π(0.05 m)² = 0.00785 m². Therefore, the volumetric flow rate of water flowing through the 0.1 m pipe is:
Q = A_{1}·v_{1} = 0.00785 m² × 2 m/s = 0.0157 m³/s
The cross section of a 0.05 m diameter pipe is A _{2} = π(0.025 m)² = 0.0001963 m². Since there are two 0.05 m diameter pipes, the total area is A_{3} = 2·A_{2} = 0.0003926 m². Therefore, the volumetric flow rate of water flowing through the two 0.05 m pipes is:
Q = A_{3}v_{3}
v_{3} = Q / A_{3} = 0.0157 m³/s / 0.0003926 m² = 40.11 m/s
Therefore, the velocity of the water in each of the 0.05mdiameter pipes is 40.11 m/s.