Principle of Continuity

The law of conservation of mass states that mass can neither be created nor be destroyed. However, as per Einstein’s mass-energy equation, mass can be converted into energy and vice versa. If body A has mass m and goes through any physical phenomenon, its shape, volume, etc., might change, but the mass m will be the same. If its mass decreases, some energy will be given out during the process. This same concept derives from the equation of continuity in fluid dynamics.

Assumptions for Principle of Continuity

Bernoulli’s flow equation indicates that when a fluid flows through a pipe, it exhibits both pressure and velocity. This type of flow can be considered laminar because no turbulence is there. As the flow is smooth, some assumptions need to be made.

• The fluid flowing through the pipe doesn’t have any viscous force acting along with the layers of the fluid.

• No frictional force will exist between the fluid and the inner surface of the pipe.

• The fluid needs to be incompressible, i.e., no external pressure will compress the fluid and change its volume.

Introduction to the equation of continuity

The principle of continuity states that the total volume of the fluid entering the pipe is equal to the sum of the total volume of the fluid leaving the pipe and the volume of fluid held back inside the pipe. But if we consider an incompressible fluid, no fluid will be held back inside the pipe due to density changes. In such a condition, we can derive the equation of continuity as the total mass of fluid entering the pipe equals the mass of the fluid leaving the pipe.

If m1 is the mass of the fluid entering the pipe, and m2 is the mass of the fluid leaving the pipe, then as per the continuity equation, we can say that:

m1 = m2

Another version of the continuity equation depends on the fluid velocity. Let us consider a pipe having two sections with different surface areas. When the fluid flows through the pipe, it exhibits a velocity that will vary inversely to its cross-sectional area. Therefore, the continuity equation states that the product of the velocity of the flowing fluid and the pipe’s cross-sectional area at any given point will be constant.

If V is the velocity of the flowing fluid and A is the cross-sectional area of the pipe, then we can derive the equation of continuity as:

AV = constant

Continuity equation for two different points of flow

Let us consider a laminar liquid flow with two sections having different cross-sectional areas- A1 and A2. When the fluid flows through A1, the velocity is V1, while at the A2 area, let’s consider the velocity V2.

According to the equation of continuity, we can derive the form as:

A1V1 = constant

A2V2 = constant

Combining these two equations, we can say that:

A1V1 = A2V2

Or, A1/A2 = V2/V1

Or, A ∝ 1/V

Evaluation of the continuity equation

From the above derivation, it’s clear that the pipe’s cross-sectional area through which the fluid is flowing is inversely proportional to the velocity of the fluid. Therefore, if the area increases, the velocity of the fluid will decrease and vice versa.

Applications of the continuity equation in fluid mechanics

The continuity equation is the basic principle of fluid mechanics, so it holds much importance. Following are some real-life examples where the equation of continuity is applicable.

• Water flowing through the garden hose

You have attached two garden hoses having different cross-sectional areas. The velocity of water at the inlet will be less due to the more cross-sectional area of the hose. As soon as it enters the lesser cross-sectional area of the pipe, the velocity increases, which is why water coming out of the pipe has so much speed that the throw distance is quite large.

• Coolant tubes of air conditioners

The coolant tubes in the air conditioners are also kept thinner to ensure that the cross-sectional area is less. As a result, the velocity with which the cold air is expelled from the outlet is huge so that the appliance can have a large throw distance.

Conclusion

The equation of continuity has two uses in terms of fluid dynamics. First, it defines that the volume of the fluid at the inlet will be similar to the volume of the fluid at the outlet. It also states that the rate of flow of the fluid at the inlet will be the same as at the outlet. The product of the pipe’s cross-sectional area and fluid velocity at that point of the pipe will be constant throughout the flow. This equation forms the very base of fluid dynamics. It also supports Bernoulli’s equation.