Derivation of Continuity Equation

What is Continuity Equation? 

The continuity equation describes the nature of the movement of physical quantities. The continuity equation is usually applied to the conserved quantities, but it can also be generalized for the extensive quantities. Quantities like mass, momentum, energy, and electric charge are some major conserved quantities. The continuity equation can be applied to these quantities to describe nature and other physical phenomena.


The continuity equation plays a significant role while studying the movement of fluids, especially when fluid is passed through a tube of varying diameter. Normally the fluids which are taken into consideration have a constant density and are incompressible. This concept can be related to the human body in several aspects.


For example, the blood vessels or arteries are divided into several capillaries, which then join to form a vein. The continuity equation can calculate the speed of the blood flowing through the blood vessels. Since the blood vessels are elastic in nature, several other factors are to be applied with the continuity equation precisely to make the proper calculation. This includes the elasticity and the diameter of the blood vessels.


The understand continuity equation; let's consider the flow rate f first:

f=Av

Where,

f = flow rate

A = the cross-sectional area of a point in the pipe

v = the average speed at which a fluid is moving inside the pipe.


The flow rate is the amount of liquid that passes from a particular point in a unit time. For example, the amount of water (in volume) coming out from a pipe per minute. The unit of flow rate is usually calculated in terms of milliliters per second.


The application of the continuity equation can be seen while calculating the amount of blood that the heart pumps into the vessels, thus determining a person's health condition. This process is also helpful in determining whether a blood vessel is clogged, and taking further measures against heart issues.


Derivation of Continuity Equation Assumption

The following points are the assumptions of continuity equation:

  • The tube, which is taken into consideration, has a single entry and a single exit.

  • The fluid that flows in the tube is non-viscous fluid.

  • The fluid is incompressible.

  • Fluid flow is steady.


Derivation of Continuity Equation 

Let us consider the following diagram:

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Let us consider that the fluid flows in the tube for a short duration Δt. During this time, the fluid will cover a distance of Δ\[x_{1}\], with a velocity of \[v_{1}\]  in the lower part of the pipe.

The distance covered by the fluid with speed \[v_{1}\] in time Δt will be given by,

Δ\[x_{1}\] = \[v_{1}\]Δt


Now, in the lower part of the pipe, the volume of fluid flows into the pipe is,

V = \[A_{1}\] Δ\[x_{1}\] = \[A_{1}\] \[v_{1}\] Δt


We know that mass (m) = Density (ρ) × Volume (V). So, the mass of fluid in region Δ\[x_{1}\]  will be:

Δm1= Density × Volume


⇒ Δm1 = \[ρ_{1}\]\[A_{1}\]\[v_{1}\]Δt ——–(Equation 1)


Now, we have to calculate the mass flux at the lower part of the pipe. Mass flux is the total defined mass of the fluid that flows through the given cross-sectional area per unit time. For the lower part of the pipe, with the lower end of pipe having a cross-sectional area \[A_{1}\], the mass flux will be given by,


Δm1/Δt = \[ρ_{1}\]\[A_{1}\]\[v_{1}\] ——–(Equation 2)


Similarly, the mass flux of the fluid at the upper end of the pipe will be:


Δm2/Δt = \[ρ_{2}\]\[A_{2}\]\[v_{2}\] ——–(Equation 3)


Where,

\[v_{2}\] = velocity of the fluid flowing in the upper end of the pipe.


Δ\[x_{2}\]  = distance traveled by the fluid.


Δt = time, and


\[A_{2}\] = area of a cross-section of the upper end of the pipe.


It is assumed that the density of the fluid in the lower end of the pipe is the same as that of the upper end. Thus, the fluid flow is said to be streamlined. Thus, the mass flux at the bottom point of the pipe will also be equal to the mass flux at the upper end of the pipe. Hence


Equation 2 = Equation 3.


Thus,

\[ρ_{1}\]\[A_{1}\]\[v_{1}\] = \[ρ_{2}\]\[A_{2}\]\[v_{2}\] ——–(Equation 4)


Based on equation 4 it can be stated that:


ρ A v = constant


The above equation helps to prove the law of conservation of mass in fluid dynamics. As the fluid is taken to be incompressible, the density of the fluid will be constant for steady flow.


So, \[ρ_{1}\] =\[ρ_{2}\].


Applying this to Equation 4; it can be written as:


\[A_{1}\] \[v_{1}\] = \[A_{2}\] \[v_{2}\]


The generalized form of this equation is:

A v = constant


Now, let's consider R as the volume flow rate, hence the equation can be expressed as:

R = A v = constant


This is the derivation of the continuity equation.

FAQ (Frequently Asked Questions)

1. What are the Applications of the Equation of Continuity?

Ans The primary application of the Equation of Continuity is seen in the field of Hydrodynamics, Electromagnetism, Aerodynamics, and Quantum Mechanics. The equation of continuity forms the fundamental rule of Bernoulli's Principle. It is also associated with the Aerodynamics principle and its applications.


The differential form of the equation of continuity is used to determine the consistency of Maxwell's Equation. Apart from it, the differential form of the equation of continuity is also used in Electromagnetism.


Equation of continuity is used to check the consistency of Schrodinger Equation.


General and Special Theory of Relativity, Noether's Theorem, also used the equation of continuity.

2. What is the Significance of Continuity Equation?

Ans The equation of continuity is based on the assumption that the fluid that flows in will be equal to the fluid that flows out. This is a useful tool to solve many properties of the fluid during its motion:

Since flow in = flow out, we can calculate some properties of a liquid under some conditions, and then we can apply the continuity equation to measure properties of that fluid under other conditions.

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Q1=Q2

This can be expressed as:

A1∗v1=A2∗v2A

Here, the equation of continuity does find its application to any incompressible fluid. Since the fluid is incompressible, the amount of fluid that flows in a surface must equal the amount of fluid that flows out of the surface.


Significance

We can observe the effect of the equation of continuity in our garden. Water flows through the pipe of our garden, and when it reaches the narrow end of the pipe or the nozzle, the speed of water increases. With the increase of speed of the fluid, the cross-sectional area decreases and with the decrease in speed of fluid decreases, the cross-sectional area increases. This is a consequence of the equation of continuity.