Equation of Continuity

Introduction

The equation of continuity is a simple statement that states that if a normal pipe with no leakage is considered and water is passed through it, the mass of the water (or any other substance) at one end of the pipe will be the same as the mass of the same substance at the other end from where the water finally comes out (of the pipe) under the condition that no other substance is added to the water or in the pipeline. Therefore, in the end, the mass flow rate of the substance before they enter the pipe will be the same as the mass flow rate of the same substance after coming out of the pipe. Mass flow rate refers to the mass per minute time (mass/time). 

Key takeaways from the Equation 

1. As per the equation, Av= constant (Here, A= Cross-section area and v= Velocity of the flow of the fluids).
2. Even if the cross-section area of the pipe is non-uniform, the rate of flow of any liquid inside the pipe will remain constant throughout the streamline of the aforementioned pipeline.
3. If pressure is applied, the volume of the fluid inside the pipeline does not change and the density remains the same. Therefore, the liquid is incompressible i.e., the liquid cannot be compressed and the density cannot be altered by any means. 
4. The equation of continuity is based on the principles of mass conservation. For these reasons, the equation is also known as ‘The conservation of mass of incompressible fluids.  

Significance of Equation of Continuity

Considered as the fundamental rule of Bernoulli’s Principle, the Equation of Continuity has massive relevance in the fields of Aerodynamics, Electromagnetism, Quantum Mechanics, and Hydrodynamics. To add to this, the equation can be also used on other applications like Noether’s Theorem, General Theory of Relativity, and Special Theory of Relativity.  

Apart from this, to check the consistency of the Schrodinger Equation, the Equation of Continuity is applied. 

The formula of the Equation of Continuity

The equation attempts to prove the fact that the amount of mass of any fluid which enters one point of a pipe (entry time) will stay in the same amount once the fluid comes out from the other end of the pipe (exit). The fluid in this case can be any liquid that flows. Usually, fluids are considered to be gases as well but, in most cases, liquids are concerned as they can flow in a pipe due to the slope due to gravitational pull. These fluids are incompressible in nature and thus, their density remains unaltered as well. Thus, this is essentially a statement that confirms the law of conservation of mass. Therefore, the explicit formula can be stated as;

ρ1​A1​v1​=ρ2​A2​v2 

where ρ represents the density, A is the cross-sectional area, and v stands for velocity (flow of the fluid). Subscripts 1 and 2 denote the two different points of the given pipeline.

Conclusion

The equation of continuity supports the statement of the law of conservation of mass. In the topic, the concerned fluid is considered to be a liquid that flows inside the pipe due to an existing downward slope fueled by gravitational force. The rate of flow of the fluid will remain the same throughout its flow inside the pipe and if other things remain constant and nothing more is added, the more the cross-sectional area of the pipe, the rate of flow will be less due to the presence of a more area inside the pipe and vice-versa. If the cross-section area of the pipe is non-uniform, the volume flux or rate of flow of the fluid inside the pipeline will remain constant throughout the streamline of the aforementioned pipeline.