Derivation of continuity equation is one of the most important derivations in fluid dynamics.

*The continuity equation is defined as the product of cross-sectional area of the pipe and the velocity of the fluid at any given point along the pipe is constant.*

## 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:

R = A v = constant |

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

**with a velocity**

*Δx*_{1}**at the lower end of the pipe.**

*v*_{1}At this time, the distance covered by the fluid will be:

*Δx _{1 }= v_{1}Δt*

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

*V = A _{1 }Δx_{1 }= A_{1} v_{1 }Δt*

It is known that** mass (m) = Density (ρ) × Volume (V). **So, the mass of the fluid in

**region will be:**

*Δx*_{1 }*Δm _{1}= Density × Volume*

*=> Δm _{1 }= ρ_{1}A_{1}v_{1}Δt ——–(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 ** A_{1},** mass flux will be:

*Δm _{1/Δt = }ρ_{1}A_{1}v_{1 }——–(Equation 2)*

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

*Δm _{2/Δt }= ρ_{2}A_{2}v_{2 }——–(Equation 3)*

Here, ** v_{2}** is the velocity of the fluid through the upper end of the pipe i.e. through

*Δx*_{2 }*,*in

*time and*

**Δt***is the cross-sectional area of the upper end.*

**A**_{2},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,

*ρ _{1}A_{1}v_{1 = }ρ_{2}A_{2}v_{2 }*

*——–(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,

*ρ*_{1 }=ρ_{2}.Thus, ** Equation 4 **can be now written as:

*A _{1 }v_{1 = }A_{2 }v_{2}*

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:

\(\frac{\partial \rho }{\partial t}+\frac{1}{r}\frac{\partial r\rho u}{\partial r}+\frac{1}{r}\frac{\partial \rho v}{\partial \theta }+\frac{\partial \rho w}{\partial z}=0\) |

### 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:

\(\frac{\partial }{\partial x}(\rho u)+\frac{\partial}{\partial y}(\rho v)+\frac{\partial }{\partial z}(\rho w)=0\) |