Continuity Equation

PhysicsMotion

Introduction

Praveen and Shubham are good friends. Praveen purchased a bike and went for a ride with Shubham. They went to the petrol pump and asked the vendor to fill 10 liters of petrol in his bike. The vendor started filling, meanwhile Shubham asked Praveen, “how do you know that we will get the same amount of petrol in our bike tank?” Then Praveen replied, “Do you know about the equation of continuity? Shubham said no, can you explain this?” Then Praveen said that the Continuity equation tells us that the law of conservation of mass in fluid. It means that the amount of fluid exiting from the reservoir of the petrol pump is the same amount entered in the tank of the bike. After that they went to a park, where the gardener is giving water to the plants by using a pipe. Shubham again asked why the gardener is pressing and expanding the mouth of the pipe by his hand. Praveen replied that it is also related to the concept of Continuity Equation. When the gardener is pressing the mouth of the pipe, then he is decreasing the cross-sectional area of the pipe, which leads to an increase in the velocity of the water coming through the pipe and when he is again expanding it, the velocity of outgoing water decreases. By doing this he can water the farther and nearer plants just by standing at one place. Praveen continued that the velocity of the water is inversely proportional to the area of the pipe. Then Shubham asked Praveen to explain the Continuity equation in detail when they will reach home.

Derivation of Continuity Equation

Continuity equation tells us that for an incompressible fluid, the product of cross-sectional area and velocity is always constant. But from where this statement comes from and under which conditions. So the continuity equation is valid for ideal fluid. It means that the fluid should be incompressible, non-viscous and the flow should be steady.

Let us discuss these terminologies before entering into the core derivation -

• Incompressible fluid: In this case we have to assume that if we compress the fluid and its change in volume will be less than the 5 % of the original volume then the fluid will be treated as incompressible.

• Non-viscous fluid: All the fluids have multiple layers in-between. When the fluid flows then these layers will also flow over each other and will cause friction which is called viscosity. So here we have to assume that the viscosity is zero for the fluid which is taken for consideration.

• Steady flow: Steady flow means that the properties of the fluid will not change at a particular point with time. In the case of derivation of the continuity equation, properties of fluid mean the velocity of the fluid.

Let us start with the derivation of the continuity equation for a fluid which is incompressible, non-viscous and having steady flow.

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Figure 1: A pipe

Let us take a pipe as shown in Figure - 1. This pipe has two ends which are having the cross-section area $\mathrm{A_1}$ and $\mathrm{A_2}$ as shown in Figure- 1. Let us assume that apart from these two openings the pipe is not having any other opening or hole. Let the fluid enter from the end having the cross sectional area $\mathrm{A_1}$ with a velocity $\mathrm{V_1}$. Let us assume that after time ‘t’, the distance traveled by that part of fluid will be 𝚫x1. So we can write -

$\mathrm{\Rightarrow\:\Delta X 1\:=\:V1.t\:\:…..(1)}$

So the volume of fluid $\mathrm{(V_f)}$ at the inlet of pipe after time ‘t’will be -

$\mathrm{\Rightarrow\:V_f\:=\:A1.\Delta X 1}$

By using (1), we get -

$\mathrm{\Rightarrow\:V_f\:=\:A_1.V_1 t\:\:….. (2)}$

As we know that density can be written as -

$\mathrm{\rho\:=\:m/(Vf)\:\:\:…… (3)}$

Where m = mass and $\mathrm{\rho}$ = density of the fluid

So, the mass of water at the inlet of pipe till time ‘t’ will be -

$\mathrm{\Rightarrow\:m_1\:=\:\rho_1.A_1.V_1.t\:\:\:…… (4)}$

Now by using (4), we can write mass flow rate (ṁ) as -

$\mathrm{\Rightarrow\:m_1/t\:=\:m_1\:=\:\rho_1.A_1.V_1\:\:\:…… (5)}$

Now, similarly we can write the value of mass flow rate (ṁ) for the exit end -

$\mathrm{\Rightarrow\:m_2\:=\:\rho_2.A_2.V_2\:\:\:…… (6)}$

Now we know that the mass remains conserved by the law of conservation of mass. So the mass of fluid entering the pipe will be equal to the mass of the fluid exiting the pipe. Similarly, the mass flow rate will also remain the same for the same interval of time. So,

$$\mathrm{\Rightarrow\:m_1\:=\:m_2}$$

By using equation (5) and (6), we can say that -

$\mathrm{\Rightarrow\:\rho_1.A_1.V_1\:=\:\rho_2.A_2.V_2\:\:\:……(7)}$

So we can say that -

$\mathrm{\rho}$.A.V = Constant

This equation is valid for all kinds of fluids.

Now since we have assumed that the fluid is incompressible, the density remains the same at the inlet and exit. So,

$\mathrm{\Rightarrow\:\rho_1\:=\:\rho_2}$

So from the equation (7), we can write that -

$\mathrm{A_1.V_1 \:= \:A_2.V_2 }$

So, we can write that -

A.V = Constant

This is the Continuity equation for incompressible fluids.

Application of Continuity Equation

This continuity equation is useful in the various fields of Aerodynamics and Hydrodynamics. It is one of the most important equations for the derivation of Bernouli’s theorem. It will help in various calculations of measuring equipment like Venturimeter, Orifice Meter etc.

FAQs

Q1. What are conditions necessary for the derivations of Continuity equations?

Answer- The conditions for the derivation of continuity equation is that the fluid -

• Should be incompressible

• Should be non-viscous

• Should have steady flow

Q2. What do you mean by incompressible fluid?

Answer - Incompressible fluid means the fluid whose density changes will be negligible after compression. The maximum allowance is that the change in volume should not be greater than the 5% of the original volume.

Q3. What do you mean by steady flow?

Answer - Steady flow means that the properties of the fluid will not change with time at a particular position.

Q4. Please state the formula of continuity equation for compressible and incompressible fluids.

Answer - For compressible fluid, the continuity equation can be written as -

$\mathrm{\rho}$.A.V = Constant

For incompressible fluid, the continuity equation can be written as -

A.V = Constant

Q5. What are the applications of the continuity equation?

Answer- Continuity equations can be used for the derivation of Bernouli’s theorem. It is used in the calculation of many physical quantities with the help of equipment like Venturimeter, orifice meter, flowmeter etc. It is used in Aerodynamics and Hydrodynamics.