Derivation Of Continuity Equation - Byju's

Assumption of Continuity Equation

Following are the assumptions of continuity equation:

• The tube is having a single entry and single exit
• The fluid flowing in the tube is non-viscous
• The flow is incompressible
• The fluid flow is steady

Related Article: Bernoulli’s Principle

Derivation

Consider the following diagram:

Now, consider the fluid flows for a short interval of time in the tube. So, assume that short interval of time as Δt. In this time, the fluid will cover a distance of Δx1 with a velocity v1 at the lower end of the pipe.

At this time, the distance covered by the fluid will be:

Δx1 = v1Δt

Now, at the lower end of the pipe, the volume of the fluid that will flow into the pipe will be:

V = A1 Δx1 = A1 v1 Δt

It is known that mass (m) = Density (ρ) × Volume (V). So, the mass of the fluid in Δx1 region will be:

Δm1= Density × Volume

=> Δm1 = ρ1A1v1Δt ——–(Equation 1)

Now, the mass flux has to be calculated at the lower end. Mass flux is simply defined as the mass of the fluid per unit time passing through any cross-sectional area. For the lower end with cross-sectional area A1, mass flux will be:

Δm1/Δt = ρ1A1v1 ——–(Equation 2)

Similarly, the mass flux at the upper end will be:

Δm2/Δt = ρ2A2v2 ——–(Equation 3)

Here, v2 is the velocity of the fluid through the upper end of the pipe i.e. through Δx2 , in Δt time and A2, is the cross-sectional area of the upper end.

In this, the density of the fluid between the lower end of the pipe and the upper end of the pipe remains the same with time as the flow is steady. So, the mass flux at the lower end of the pipe is equal to the mass flux at the upper end of the pipe i.e. Equation 2 = Equation 3.

Thus,

ρ1A1v1 = ρ2A2v2 ——–(Equation 4)

This can be written in a more general form as:

ρ A v = constant

The equation proves the law of conservation of mass in fluid dynamics. Also, if the fluid is incompressible, the density will remain constant for steady flow. So, ρ1 =ρ2.

Thus, Equation 4 can be now written as:

A1 v1 = A2 v2

This equation can be written in general form as:

A v = constant

Now, if R is the volume flow rate, the above equation can be expressed as:

R = A v = constant

This was the derivation of continuity equation.

Watch the video and learn about the basics of continuity equation

Continuity Equation in Cylindrical Coordinates

Following is the continuity equation in cylindrical coordinates:

\(\begin{array}{l}\frac{\partial \rho }{\partial t}+\frac{1}{r}\frac{\partial r\rho u}{\partial r}+\frac{1}{r}\frac{\partial \rho v}{\partial \theta }+\frac{\partial \rho w}{\partial z}=0\end{array} \)

Incompressible Flow Continuity Equation

Following is the continuity equation for incompressible flow as the density, ρ = constant and is independent of space and time, we get:

∇.v = 0

Steady Flow Continuity Equation

Following is the continuity equation in cylindrical coordinates:

\(\begin{array}{l}\frac{\partial }{\partial x}(\rho u)+\frac{\partial}{\partial y}(\rho v)+\frac{\partial }{\partial z}(\rho w)=0\end{array} \)

Frequently Asked Questions – FAQs

Q1

Define continuity equation.

According to the continuity equation, the product of the cross-sectional area of the pipe and the velocity of the fluid at any given point along the pipe is constant. Q2

Give the continuity equation.

R = A v = constant Where,

R is the volume flow rate

A is the flow area

v is the flow velocity

Q3

State true or false: According to the assumptions of continuity equation, the fluid flow is steady

True. Q4

What is the continuity equation for incompressible flow as the density, ρ = constant and is independent of space and time.

∇.v = 0 Q5

In which field the continuity equation is used?

The continuity equation is used in fluid dynamics.

Stay tuned with BYJU’S to learn more interesting science topics with engaging videos!

Test your knowledge on Continuity equation derivation Q5

Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin!

Select the correct answer and click on the “Finish” buttonCheck your score and answers at the end of the quiz

Start Quiz

Congrats!

Visit BYJU’S for all Physics related queries and study materials

Your result is as below

0 out of 0 arewrong 0 out of 0 are correct 0 out of 0 are Unattempted View Quiz Answers and Analysis X Login To View Results Mobile Number* Send OTP Did not receive OTP?

Request OTP on Voice Call

Login To View Results Name* Email ID* Grade* Grade 1 2 3 4 5 6 7 8 9 10 11 12 City* View Result

Register with BYJU'S & Download Free PDFs

* Send OTP * * * * * Grade Class 1 Class 2 Class 3 Class 4 Class 5 Class 6 Class 7 Class 8 Class 9 Class 10 Class 11 Class 12 IAS CAT Bank Exam GATE Download Now

Register with BYJU'S & Watch Live Videos

* Send OTP * * * * * Grade Class 1 Class 2 Class 3 Class 4 Class 5 Class 6 Class 7 Class 8 Class 9 Class 10 Class 11 Class 12 IAS CAT Bank Exam GATE Watch Now