PhysicsChapter 6: Fluid Mechanics

Equation of Continuity

Overview

The continuity equation is a mathematical statement of conservation of mass in fluid dynamics. It is used for pipes, tubes, ducts, rivers, blood vessels and power-plant systems.

Learning Outcomes

Derive continuity from conservation of mass.
Distinguish mass flow rate from volume flow rate.
Use the general and incompressible forms.
Explain the inverse area-speed relation.
Solve single-pipe and branched-flow problems.
Apply continuity to water jets, hoses, rivers and blood vessels.

Intermediate Level

Derivation from Conservation of Mass

Consider steady flow through a tube whose cross-sectional area changes. During the same short time Δt, fluid at sections 1 and 2 moves distances Δx₁ and Δx₂.

Key equation

Δ x₁=v₁Δ t,Δ x₂=v₂Δ t

Δx: distance moved in time Δt (m)

Each moving portion is a short cylinder. Since volume equals cross-sectional area multiplied by length:

Key equation

Δ V₁=A₁v₁Δ t,Δ V₂=A₂v₂Δ t

ΔV: volume crossing a section (m³)

Using mass = density × volume, the masses crossing the sections are:

Key equation

Δ m₁=ρ₁A₁v₁Δ t,Δ m₂=ρ₂A₂v₂Δ t

ρ: fluid density (kg m⁻³)

For steady flow with one entrance, one exit and no source or sink, mass cannot accumulate. Therefore Δm₁ = Δm₂. Cancelling the common time interval gives:

Key equation

ρ₁A₁v₁=ρ₂A₂v₂

Key equation

Δ mΔ t=ρ Av

Incompressible Form

For an incompressible fluid, density remains constant, so ρ₁ = ρ₂. The densities cancel from the general equation.

Key equation

A₁v₁=A₂v₂

Key equation

Av=constant

Volume Flow Rate

Volume flow rate Q is the volume passing a cross-section per unit time. A fluid column of length vΔt and area A has volume AvΔt, so:

Key equation

Q=Δ VΔ t=Av

Q: volume flow rate (m³ s⁻¹)

Quantity	Expression	SI unit
Volume flow rate	Q = Av	m³ s⁻¹
Mass flow rate	ρAv	kg s⁻¹
Steady incompressible condition	A₁v₁ = A₂v₂	same Q at both sections

Area-Speed Relation

When Q is constant, speed varies inversely with cross-sectional area. A narrow section has greater speed; a wide section has smaller speed.

Key equation

v∝1A

Key equation

v₂v₁=A₁A₂

Applications

A freely falling water stream accelerates, so its cross-sectional area decreases.
Squeezing a rubber hose reduces its opening and increases water speed.
For the same river discharge, a deep or wide region has larger area and slower average flow.
Blood from the aorta divides among many arteries; their combined area controls the new speed.
Pipe and duct systems use continuity at changing cross-sections.

Solved Examples

Example 1: Narrow Pipe

Water flows at 2.0 m s⁻¹ through area 6.0 cm². Find its speed where area becomes 3.0 cm².

Given

A₁ = 6.0 cm²
v₁ = 2.0 m s⁻¹
A₂ = 3.0 cm²

Working

A₁v₁ = A₂v₂

Step 2

v₂ = A₁v₁/A₂

Step 3

v₂ = 6.0 × 2.03.0

Final answer: v₂ = 4.0 m s⁻¹

Example 2: Volume Flow Rate

A pipe of radius 2.0 cm carries water at 1.5 m s⁻¹. Calculate Q.

Given

r = 2.0 × 10⁻² m
v = 1.5 m s⁻¹

Working

A = πr² = π(2.0 × 10⁻²)²

Step 2

A = 1.26 × 10⁻³ m²

Step 3

Q = Av

Step 4

Q = (1.26 × 10⁻³)(1.5)

Final answer: Q = 1.88 × 10⁻³ m³ s⁻¹

Example 3: Textbook Blood-Flow Example

Blood moves through an aorta of radius 0.8 cm at 0.33 m s⁻¹ and divides equally into 30 arteries of radius 0.4 cm. Find the speed in each artery.

Given

r₁ = 0.8 cm
v₁ = 0.33 m s⁻¹
N = 30
r₂ = 0.4 cm

Working

A₁v₁ = 30A₂v₂

Step 2

Use A = πr² and cancel π.

Step 3

v₂ = (130)(r₁r₂)²v₁

Step 4

v₂ = (130)(0.80.4)²(0.33)

Final answer: v₂ = 0.044 m s⁻¹

Example 4: Deep and Shallow River

Explanation: The comparison assumes the same volume flow rate.

Branched Flow

At a junction, total inflow equals total outflow. If one inlet divides into N identical outlets:

Key equation

A₁v₁=NA₂v₂

N: number of identical outlets (none)

Common Mistakes

Using A₁v₁ = A₂v₂ when density changes significantly.
Forgetting to include every outlet branch.
Using radius directly instead of A = πr².
Mixing cm² with m² when calculating Q.
Saying speed is inversely proportional to radius instead of area.
Assuming continuity alone determines pressure.

Short-Question Preparation

Q. What principle gives continuity?

A. Conservation of mass.

Q. State the general equation.

A. ρ₁A₁v₁ = ρ₂A₂v₂.

Q. State it for an incompressible fluid.

A. A₁v₁ = A₂v₂, or Av = constant.

Q. Define volume flow rate.

A. Q = ΔVΔt = Av.

Q. Why does water speed up at a squeezed hose end?

A. Area decreases, so speed increases to keep Av constant.

Long-Question Preparation

Q. Derive the equation of continuity.

A. 1. Take two sections during the same Δt. 2. Write Δx₁ = v₁Δt and Δx₂ = v₂Δt. 3. Volumes are A₁v₁Δt and A₂v₂Δt. 4. Masses are ρ₁A₁v₁Δt and ρ₂A₂v₂Δt. 5. Conserve mass and cancel Δt. 6. Obtain ρ₁A₁v₁ = ρ₂A₂v₂. 7. For constant density, obtain A₁v₁ = A₂v₂ = constant. 8. Identify Av as Q and explain v ∝ 1A.

Sticky Notes

Continuity = conservation of mass.
General: ρ₁A₁v₁ = ρ₂A₂v₂.
Incompressible: A₁v₁ = A₂v₂.
Q = Av.
Mass flow rate = ρAv.
Circular area = πr².
Smaller area → greater speed.

60-Second Revision

Key equation

ρ₁A₁v₁=ρ₂A₂v₂

Key equation

A₁v₁=A₂v₂=Q

Area change	Speed change
decreases	increases
increases	decreases
halves	doubles
becomes one-fourth	becomes four times

Summary

Continuity expresses conservation of mass. The mass flow rate ρAv remains constant during steady flow. If density is constant, Av and Q remain constant, producing the inverse relation between area and speed.

Ask-the-Tutor Modes

Explain continuity simply.
Show the full derivation.
Help me choose the correct form.
Solve an area-speed numerical.
Solve a branched-flow problem.
Explain an application.
Test me with an MCQ.

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