Derivation of Continuity Equation From Substantial Derivative

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.

In this session, we will learn about the derivation of the continuity equation.

Table of Contents:

• Continuity Equation Derivation
• Assumption of Continuity Equation
• Derivation
• Continuity Equation in Cylindrical Coordinates
• Incompressible Flow Continuity Equation
• Steady Flow Continuity Equation
• Frequently Asked Questions – FAQs

Continuity Equation Derivation

Continuity equation represents that the product of cross-sectional area of the pipe and the fluid speed at any point along the pipe is always constant. This product is equal to the volume flow per second or simply the flow rate.  The continuity equation is given as:

Where,

• R is the volume flow rate
• A is the flow area
• v is the flow velocity

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:

Δm_{1/Δt =} ρ₁A₁v₁ ——–(Equation 2)

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

Δm_2/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,

ρ₁A₁v_{1 =} 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 v_{1 =} 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.

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:

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

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.

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

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

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

In which field the continuity equation is used?

The continuity equation is used in fluid dynamics.

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