Notes : Derivation of the Equation of Continuity - Physics Kund

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Equation of Continuity | Fluid Mechanics | Class 11 Physics | NCERT Explained in Simple English 

Equation of Continuity

The Equation of Continuity is one of the most important equations in fluid dynamics. It is based on the Law of Conservation of Mass, which states that mass can neither be created nor destroyed. Therefore, during the steady flow of a fluid through a pipe, the mass entering one end of the pipe per second is equal to the mass leaving the other end per second, provided there is no source or sink of fluid between the two sections.

Law Expressed by the Equation of Continuity

The equation of continuity expresses the Law of Conservation of Mass.

Statement of the Equation of Continuity

For the steady flow of a fluid through a tube of varying cross-sectional area, the mass flow rate remains constant at every cross-section of the tube.

Assumptions

  • The fluid flow is steady (streamline flow).
  • There is no leakage or addition of fluid between two sections.
  • The flow takes place through a closed tube.
  • For an incompressible fluid, density remains constant.

Derivation of the Equation of Continuity

Consider a tube of varying cross-sectional area through which a fluid is flowing steadily.

  • At section A:
    • Cross-sectional area = \(A_1\)
    • Velocity = \(v_1\)
    • Density = \(\rho_1\)
  • At section B:
    • Cross-sectional area = \(A_2\)
    • Velocity = \(v_2\)
    • Density = \(\rho_2\)

Step 1: Distance Travelled in Time \( \Delta t \)

During a small time interval \( \Delta t \), the liquid at section A travels a distance

\[ x_1=v_1\Delta t \]

Similarly, at section B,

\[ x_2=v_2\Delta t \]

Step 2: Volume of Fluid Crossing Section A

Volume is equal to area multiplied by distance.

\[ \Delta V_1=A_1v_1\Delta t \]

Similarly,

\[ \Delta V_2=A_2v_2\Delta t \]

Step 3: Mass of Fluid Crossing Each Section

Mass is equal to density multiplied by volume.

For section A,

\[ \Delta m_1=\rho_1A_1v_1\Delta t \]

Dividing both sides by \( \Delta t \),

\[ \frac{\Delta m_1}{\Delta t}=\rho_1A_1v_1 \]

This quantity is called the Mass Flow Rate (or Mass Flux).

Similarly, for section B,

\[ \frac{\Delta m_2}{\Delta t}=\rho_2A_2v_2 \]

Step 4: Applying the Law of Conservation of Mass

Since no fluid is entering or leaving the pipe between sections A and B, the mass entering per second must be equal to the mass leaving per second.

Therefore,

\[ \frac{\Delta m_1}{\Delta t}=\frac{\Delta m_2}{\Delta t} \]

Substituting the expressions for mass flow rate,

\[ \boxed{\rho_1A_1v_1=\rho_2A_2v_2} \]

This is the general equation of continuity and is valid for all fluids, whether compressible or incompressible.

Equation of Continuity for an Incompressible Fluid

For liquids such as water, the density remains constant.

\[ \rho_1=\rho_2=\rho \]

Hence,

\[ \rho A_1v_1=\rho A_2v_2 \]

Cancelling \( \rho \),

\[ \boxed{A_1v_1=A_2v_2} \]

or

\[ \boxed{Av=\text{Constant}} \]

Physical Meaning of the Equation

The equation \(Av=\text{Constant}\) shows that the velocity of a fluid is inversely proportional to the cross-sectional area of the tube.

Mathematically,

\[ v\propto\frac{1}{A} \]

  • If the cross-sectional area decreases, the velocity increases.
  • If the cross-sectional area increases, the velocity decreases.

Thus, fluids move faster through narrow sections and slower through wider sections of a pipe.

Volume Flow Rate (Discharge)

The volume of fluid crossing any cross-section of a tube per unit time is called the Volume Flow Rate or Discharge.

Mathematically,

\[ Q=\frac{dV}{dt}=Av \]

where,

  • \(Q\) = Volume flow rate
  • \(A\) = Cross-sectional area of the tube
  • \(v\) = Velocity of the fluid

SI Unit: \(m^3s^{-1}\)

Dimension: \(L^3T^{-1}\)

Mass Flow Rate

The mass of fluid crossing any cross-section of a tube per unit time is called the Mass Flow Rate.

Mathematically,

\[ \dot{m}=\rho\frac{dV}{dt} \]

Since,

\[ \frac{dV}{dt}=Av \]

Therefore,

\[ \boxed{\dot{m}=\rho Av} \]

  • \(\dot{m}\) = Mass flow rate
  • \(\rho\) = Density of fluid
  • \(A\) = Cross-sectional area
  • \(v\) = Velocity of fluid

SI Unit: \(kg\,s^{-1}\)

Dimension: \(ML^0T^{-1}\)

Applications of the Equation of Continuity

1. Flow Through a Narrow Pipe

When the cross-sectional area of a pipe decreases, the velocity of the fluid increases. This is because the product \(Av\) remains constant.

2. Flow Through a Wide Pipe

When the cross-sectional area increases, the velocity decreases to maintain a constant flow rate.

3. Water Hose

When the nozzle of a water hose is partially closed, the outlet area decreases and water comes out with greater speed.

4. Deep Rivers Flow Slowly

Deep rivers generally have a larger cross-sectional area. According to the continuity equation, a larger area results in a lower velocity, so deep rivers flow more slowly.

5. Hydraulic Systems

The continuity equation is used in the design of pipelines, hydraulic machines, pumps, irrigation systems, and water supply networks.

Advantages of the Continuity Equation

  • Explains the relationship between area and velocity.
  • Based on the law of conservation of mass.
  • Useful in engineering and fluid mechanics.
  • Applicable to liquids and gases.
  • Helps determine unknown velocity or cross-sectional area.

Limitations

  • Applicable only for steady flow.
  • Assumes no leakage or addition of fluid.
  • For the simplified equation \(Av=\text{constant}\), the fluid must be incompressible.

Important Formulae

Quantity Formula
General Continuity Equation \(\rho_1A_1v_1=\rho_2A_2v_2\)
Incompressible Fluid \(A_1v_1=A_2v_2\)
Continuity Equation \(Av=\text{Constant}\)
Volume Flow Rate \(Q=Av=\frac{dV}{dt}\)
Mass Flow Rate \(\dot{m}=\rho Av\)

Key Points to Remember

  • The continuity equation is based on the Law of Conservation of Mass.
  • Mass flow rate remains constant throughout a flowing fluid.
  • Velocity is inversely proportional to the cross-sectional area.
  • Narrow pipe → Higher velocity.
  • Wide pipe → Lower velocity.
  • Volume flow rate is represented by \(Q=Av\).
  • Mass flow rate is represented by \(\dot{m}=\rho Av\).
  • The equation \(Av=\text{constant}\) is valid only for incompressible fluids.

Frequently Asked Questions (FAQs)

Q1. What is the equation of continuity?

It is the mathematical expression of the law of conservation of mass for flowing fluids.

Q2. Which law is represented by the continuity equation?

Law of Conservation of Mass.

Q3. Write the general continuity equation.

\[ \boxed{\rho_1A_1v_1=\rho_2A_2v_2} \]

Q4. Write the continuity equation for an incompressible fluid.

\[ \boxed{A_1v_1=A_2v_2} \]

Q5. Why does water flow faster through a narrow pipe?

Because the cross-sectional area decreases. According to the continuity equation, the velocity increases when the area decreases.

Q6. What is the SI unit of volume flow rate?

\(m^3/s\)

Q7. What is the SI unit of mass flow rate?

\(kg/s\)

Q8. What is volume flow rate?

\[ Q=\frac{dV}{dt}=Av \]

Q9. What is mass flow rate?

\[ \dot{m}=\rho Av \]

Q10. When is the equation \(Av=\text{constant}\) applicable?

It is applicable only for incompressible fluids where density remains constant.

Multiple Choice Questions (MCQs)

  1. The continuity equation is based on:

    • (A) Conservation of Energy
    • (B) Conservation of Momentum
    • (C) Conservation of Mass ✓
    • (D) Newton's Second Law

  2. For an incompressible fluid, which quantity remains constant?

    • (A) Pressure
    • (B) \(Av\) ✓
    • (C) Velocity
    • (D) Area

  3. If the cross-sectional area of a pipe decreases, the velocity of the fluid:

    • (A) Decreases
    • (B) Increases ✓
    • (C) Remains constant
    • (D) Becomes zero

  4. The SI unit of volume flow rate is:

    • (A) kg/s
    • (B) m/s
    • (C) m³/s ✓
    • (D) N

  5. The SI unit of mass flow rate is:

    • (A) kg/s ✓
    • (B) m³/s
    • (C) kg
    • (D) N

  6. The general equation of continuity is:

    \[ \boxed{\rho_1A_1v_1=\rho_2A_2v_2} \]

  7. The continuity equation is applicable to:

    • (A) Steady fluid flow ✓
    • (B) Electric current
    • (C) Heat conduction
    • (D) Magnetic field

  8. Velocity is ______ proportional to the cross-sectional area.

    • (A) Directly
    • (B) Inversely ✓
    • (C) Equal
    • (D) None of these

True / False

State whether the following statements are True or False.

  1. The continuity equation is based on the law of conservation of mass. (True)
  2. Velocity decreases when the cross-sectional area decreases. (False)
  3. For incompressible fluids, density remains constant. (True)
  4. Mass flow rate remains constant during steady flow. (True)
  5. The SI unit of volume flow rate is m³/s. (True)
  6. The SI unit of mass flow rate is kg/s. (True)
  7. Water flows faster through a narrow pipe. (True)
  8. \(Av=\text{Constant}\) for incompressible fluids. (True)
  9. Velocity is directly proportional to area. (False)
  10. The continuity equation is derived from the conservation of energy. (False)

Fill in the Blanks

  1. The continuity equation is based on the law of Conservation of Mass.
  2. For an incompressible fluid, density remains constant.
  3. The volume flow rate is given by \(Q=Av\).
  4. The mass flow rate is given by \(\dot{m}=\rho Av\).
  5. The SI unit of volume flow rate is m³/s.
  6. The SI unit of mass flow rate is kg/s.
  7. If the area decreases, the velocity increases.
  8. The product \(Av\) remains constant for incompressible flow.
  9. The continuity equation represents the conservation of mass.
  10. Velocity is inversely proportional to cross-sectional area.

Very Short Answer Questions (1 Mark)

  1. What is the equation of continuity?
  2. Which law is represented by the continuity equation?
  3. Write the SI unit of volume flow rate.
  4. Write the SI unit of mass flow rate.
  5. Write the formula for volume flow rate.
  6. Write the formula for mass flow rate.
  7. Define mass flow rate.
  8. Define volume flow rate.
  9. What happens to the velocity when the area decreases?
  10. What is an incompressible fluid?

Short Answer Questions (2–3 Marks)

  1. State and explain the equation of continuity.
  2. Derive the continuity equation for an incompressible fluid.
  3. Define volume flow rate and mass flow rate.
  4. Explain why water flows faster through a narrow pipe.
  5. State the assumptions of the continuity equation.
  6. Differentiate between volume flow rate and mass flow rate.
  7. Explain the relation between area and velocity using the continuity equation.
  8. State two practical applications of the continuity equation.

Long Answer Questions (5 Marks)

  1. State and derive the equation of continuity for the streamline flow of a fluid. Explain the physical significance of the equation.
  2. Derive the equation \[ \rho_1A_1v_1=\rho_2A_2v_2 \] and obtain \[ A_1v_1=A_2v_2 \] for an incompressible fluid.
  3. Explain the continuity equation with a neat diagram. Discuss its assumptions and applications.
  4. Define volume flow rate and mass flow rate. Derive the continuity equation and explain its importance in fluid mechanics.
  5. Using the continuity equation, explain why:
    • Water flows faster through a narrow pipe.
    • Deep rivers flow slowly.
    • A nozzle increases the speed of water coming out of a hose.

Chapter Summary

  • The continuity equation is based on the Law of Conservation of Mass.
  • Mass flow rate remains constant during steady flow.
  • The general continuity equation is: \[ \boxed{\rho_1A_1v_1=\rho_2A_2v_2} \]
  • For incompressible fluids: \[ \boxed{A_1v_1=A_2v_2} \]
  • Volume flow rate: \[ Q=Av \]
  • Mass flow rate: \[ \dot{m}=\rho Av \]
  • Velocity is inversely proportional to the cross-sectional area.
  • Narrow pipe → High velocity.
  • Wide pipe → Low velocity.
  • The continuity equation is widely used in fluid mechanics, hydraulic engineering, irrigation systems, pipelines, and water distribution networks.

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