What happens to water when a pipe suddenly becomes narrower? Does some of the water disappear? Does it pile up inside the pipe?

For a fluid such as water flowing steadily through a closed pipe, the answer is no. The fluid that enters one section must continue through the next section.

This idea is called the principle of continuity.

It connects three important quantities:

flow rate, cross-sectional area, and fluid velocity.

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What Is Continuity of Flow?

The principle of continuity is based on conservation of mass.

For steady flow:

The mass of fluid entering a section of a system per unit time equals the mass of fluid leaving it per unit time.

For a liquid such as water, which is approximately incompressible, its density stays nearly constant.

Therefore, the same volume of liquid per second must pass through each section of a pipe.

In simple terms:

What flows in must flow out.

If 5 L of water per second enter a pipe, then under steady conditions approximately 5 L/s must continue through each section.

Imagine a Pipe

Consider a pipe with one wide section and one narrow section.

       WIDE                     NARROW

  → → → → → → → →             → → → →
 ┌─────────────────┐          ┌───────┐
 │                 │──────────│       │
 │      WATER      │──────────│ WATER │
 │                 │──────────│       │
 └─────────────────┘          └───────┘

   Lower speed                 Higher speed
   Larger area                 Smaller area

If water is flowing steadily, the same volume must pass each section every second.

But the narrow section has less space available.

Therefore, the water must move faster through it.

So:

larger area → lower velocity

smaller area → higher velocity

for the same steady volume flow rate.

Cross-Sectional Area

The cross-sectional area of a pipe is the area you would see if you cut straight across the pipe.

For a circular pipe:

A = πr²

where:

• A = cross-sectional area
• r = radius

Because radius is squared, changing the radius can cause a large change in area.

For example, doubling the radius does not double the area.

It increases the area by a factor of:

2² = 4

This becomes very important in fluid-flow calculations.

Flow Rate

Recall from the previous topic that volume flow rate is:

Q = V/t

where:

• Q = flow rate
• V = volume
• t = time

Flow rate can also be calculated using:

Q = Av

where:

• A = cross-sectional area
• v = average fluid velocity

Therefore:

Flow rate = cross-sectional area × fluid velocity

For example, a wide pipe with slow-moving water can have the same flow rate as a narrow pipe containing faster-moving water.

Why Is Q = Av?

Imagine water travelling through a pipe during 1 second.

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If the water moves a distance d, then the volume that passes a point is approximately:

Volume = Area × distance

or:

V = Ad

But:

velocity = distance ÷ time

so:

d = vt

Substitute into the volume equation:

V = A(vt)

Therefore:

V = Avt

Divide by time:

V/t = Av

Since:

Q = V/t

we obtain:

Q = Av

The Continuity Equation

For steady flow of an incompressible fluid:

Q₁ = Q₂

Since:

Q = Av

we can write:

A₁v₁ = A₂v₂

This is the continuity equation.

where:

• A₁ = cross-sectional area at position 1
• v₁ = fluid velocity at position 1
• A₂ = cross-sectional area at position 2
• v₂ = fluid velocity at position 2

This equation tells us how fluid velocity changes when the pipe's area changes.

Understanding the Relationship

Suppose:

A₁ > A₂

The second section is narrower.

Because:

A₁v₁ = A₂v₂

the smaller area must be balanced by a larger velocity.

Therefore:

v₂ > v₁

So:

Fluid moves faster through the narrower section.

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An Everyday Analogy

Imagine ten students walking through a wide hallway.

They approach a narrow doorway.

If the same number of students must pass a line each second without building up in front of the door, they must move through the narrower space more quickly.

A fluid behaves similarly under steady-flow conditions.

The analogy is not perfect, but it helps illustrate the main idea:

smaller available area → greater required speed

when the flow rate stays constant.

Worked Example 1: Pipe Narrows by Half

Water flows through a pipe.

At Section 1:

A₁ = 0.040 m²

v₁ = 2.0 m/s

The pipe narrows to:

A₂ = 0.020 m²

Calculate the new velocity.

Use:

A₁v₁ = A₂v₂

Substitute:

0.040 × 2.0 = 0.020 × v₂

0.080 = 0.020v₂

Therefore:

v₂ = 4.0 m/s

Answer

The fluid velocity increases from:

2.0 m/s → 4.0 m/s

The area was halved, so the velocity doubled.

A Useful Shortcut

If the cross-sectional area changes by a simple factor, we can often predict the answer without doing a full calculation.

If area becomes:

½ as large → velocity becomes 2 times larger

If area becomes:

⅓ as large → velocity becomes 3 times larger

If area becomes:

2 times larger → velocity becomes ½ as large

If area becomes:

4 times larger → velocity becomes ¼ as large

This is because area and velocity are inversely related when flow rate remains constant.

Area and Velocity Graph

For a constant flow rate:

v = Q/A

Therefore, velocity decreases as cross-sectional area increases.

This is an inverse relationship, not a straight-line relationship.

A very small area requires a relatively high velocity to maintain the same flow rate.

Worked Example 2: Pipe Gets Wider

Water moves through a narrow pipe at:

6.0 m/s

The cross-sectional area is:

0.010 m²

The pipe widens to:

0.030 m²

Calculate the new velocity.

Use:

A₁v₁ = A₂v₂

Substitute:

0.010 × 6.0 = 0.030 × v₂

0.060 = 0.030v₂

Therefore:

v₂ = 2.0 m/s

Answer

The water slows from:

6.0 m/s → 2.0 m/s

because the pipe becomes three times wider in cross-sectional area.

Interpreting Flow Diagrams

You may be given a diagram rather than numerical data.

For example:

Section A             Section B              Section C

██████████              ████                ████████████
→  →  →  →              →→→→→→              →  →  →
██████████              ████                ████████████

   Wide                 Narrow                 Widest

For the same steady flow:

Section B has the smallest area → greatest velocity

Section C has the largest area → lowest velocity

So:

vB > vA > vC

Being able to make this comparison without calculations is an important fluid-flow skill.

Worked Example 3: Three Sections

A pipe has three sections:

• Section A: 0.060 m²
• Section B: 0.020 m²
• Section C: 0.040 m²

Rank the fluid velocities from greatest to smallest.

For constant flow:

smaller area → greater velocity

The smallest area is B.

Then C.

The largest area is A.

Therefore:

vB > vC > vA

Using Diameter Instead of Area

Some questions give the diameter of a pipe rather than its cross-sectional area.

Be careful.

The continuity equation uses area, not diameter.

For a circular pipe:

A = πr²

and:

r = d/2

Therefore:

A = π(d/2)²

or:

A = πd²/4

This means area is proportional to:

diameter²

Why Diameter Can Be Tricky

Suppose the diameter of a pipe decreases by half.

A common mistake is to assume the area also decreases by half.

It does not.

Because:

A ∝ d²

if:

d₂ = ½d₁

then:

A₂ = (½)²A₁

so:

A₂ = ¼A₁

The cross-sectional area becomes four times smaller.

Therefore, for constant flow:

v₂ = 4v₁

The fluid moves four times faster.

Worked Example 4: Changing Diameter

Water flows at:

2 m/s

through a pipe with diameter:

8 cm

The pipe narrows to:

4 cm

Calculate the new speed.

Because diameter is halved:

Area becomes ¼ as large.

Therefore, to maintain the same flow rate:

velocity becomes 4 times larger.

So:

v₂ = 4 × 2

v₂ = 8 m/s

Answer

The water moves at:

8 m/s

in the narrower section.

Solving It Using Areas

We can verify the previous answer mathematically.

Section 1

Diameter:

8 cm

Radius:

4 cm

Area:

A₁ = π(4)²

A₁ = 16π cm²

Section 2

Diameter:

4 cm

Radius:

2 cm

Area:

A₂ = π(2)²

A₂ = 4π cm²

Now:

A₁v₁ = A₂v₂

16π × 2 = 4π × v₂

Cancel π:

32 = 4v₂

Therefore:

v₂ = 8 m/s

Same answer.

A Garden Hose

The continuity principle helps explain what happens at a hose nozzle.

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The hose has a relatively large cross-sectional area.

The nozzle provides a smaller exit area.

For a given flow rate, water passing through the smaller opening has a greater exit velocity.

This creates a faster, narrower jet.

However, real hose systems also experience pressure losses, and partially closing a nozzle may change the overall flow rate. The continuity equation describes how area and velocity relate for the flow rate that actually occurs.

Nozzles

Nozzles deliberately change cross-sectional area to control fluid velocity.

Examples include:

• garden hoses
• fire hoses
• spray bottles
• irrigation systems
• industrial jets

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A narrower outlet can produce a faster jet for a given flow rate.

This can allow the fluid to travel farther or strike a target with greater speed.

Rivers and Continuity

The principle of continuity can also help us understand rivers.

Suppose approximately the same volume of water passes through different parts of a river each second.

If the river channel becomes narrower:

cross-sectional area decreases

so:

average water velocity tends to increase

However, river cross-sectional area depends on both:

• width
• depth

So a narrower river does not automatically mean a smaller cross-sectional area if it also becomes much deeper.

Real rivers also gain and lose water, so the flow rate may not always remain constant.

Blood Flow

The continuity principle also applies to blood flowing through the circulatory system.

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If the same volume flow rate passes through a locally narrowed section of a blood vessel, the average blood velocity through that section increases.

This is an application of:

Q = Av

However, the human circulatory system is complex because vessels branch, stretch, pulse and change diameter.

One Vessel vs Many Vessels

There is an important distinction when discussing blood vessels.

A single capillary is extremely narrow.

But the body contains an enormous number of capillaries arranged in parallel.

The total cross-sectional area of all the capillaries together is very large.

Therefore, average blood velocity through the capillary network is relatively low.

This slower movement helps provide time for exchange of:

• oxygen
• carbon dioxide
• nutrients
• wastes

between blood and tissues.

Ventilation Systems

The continuity equation is also important in ventilation.

If air moves through a duct that narrows, its average velocity generally increases for the same volume flow rate.

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Engineers use these relationships when designing:

• air-conditioning systems
• ventilation ducts
• exhaust systems
• air intakes
• industrial ventilation

Controlling air speed can help manage:

• noise
• comfort
• energy use
• distribution of air

Fire Hoses

Firefighters need to deliver water over significant distances.

A nozzle can reduce the outlet area and produce a high-speed stream.

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The continuity relationship helps explain why the fluid velocity at the nozzle can be greater than in the wider hose.

However, pumps must provide sufficient pressure and flow because real hoses also experience resistance and energy losses.

Irrigation Systems

Agricultural irrigation systems use carefully designed pipes and nozzles.

The size of an outlet affects the velocity and amount of water delivered.

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Engineers must consider both:

flow rate

and:

exit velocity

A system may need to provide enough water while also producing a spray that reaches the required area.

Mass Continuity

So far, we have focused on liquids such as water.

For an incompressible fluid:

A₁v₁ = A₂v₂

But the more general idea is conservation of mass.

The mass flow rate is:

mass flow rate = ρAv

For steady flow:

ρ₁A₁v₁ = ρ₂A₂v₂

For an incompressible liquid:

ρ₁ ≈ ρ₂

so density cancels:

A₁v₁ = A₂v₂

This explains why the simple continuity equation works well for many liquid-flow problems.

What About Gases?

Gases can be compressed much more easily than liquids.

Therefore, their density can change significantly as they flow.

For simple situations with small density changes, we may still use:

A₁v₁ ≈ A₂v₂

as an approximation.

But when gas density changes significantly, the more general mass-flow equation is needed:

ρ₁A₁v₁ = ρ₂A₂v₂

This is important in advanced applications involving:

• aircraft engines
• compressors
• turbines
• high-speed gas flow

Continuity Does Not Mean Pressure Stays Constant

A very important point:

Continuity tells us about flow rate, area and velocity.

It does not say that pressure remains constant.

As a fluid changes speed, its pressure may also change.

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The relationship between fluid speed and pressure leads us toward another important idea in fluid dynamics: Bernoulli's Principle.

For now, remember:

continuity → conservation of flow/mass

while:

pressure changes require additional fluid-dynamics ideas.

A Step-by-Step Problem-Solving Method

For continuity problems, use this method.

Step 1: Identify the two sections

Label them:

Section 1

and:

Section 2

Step 2: Write down the known values

Identify:

• A₁
• v₁
• A₂
• v₂

Step 3: Write the continuity equation

A₁v₁ = A₂v₂

Step 4: Rearrange

For example, to find v₂:

v₂ = A₁v₁/A₂

Step 5: Substitute values

Include units.

Step 6: Check whether the answer makes sense

Ask:

Did the pipe become narrower?

Then velocity should increase.

Did the pipe become wider?

Then velocity should decrease.

This simple check can catch many calculation errors.

Worked Example 5: Finding Area

Water moves through a pipe at:

v₁ = 3 m/s

with:

A₁ = 0.040 m²

The water then slows to:

v₂ = 2 m/s

Find the new cross-sectional area.

Use:

A₁v₁ = A₂v₂

Substitute:

0.040 × 3 = A₂ × 2

0.120 = 2A₂

Therefore:

A₂ = 0.060 m²

Answer

The new area is:

0.060 m²

This makes sense because the water slowed down, so the pipe must have become wider.

Worked Example 6: Finding the Flow Rate

Water moves through a pipe with:

A = 0.025 m²

and:

v = 4 m/s

Calculate the volume flow rate.

Use:

Q = Av

Q = 0.025 × 4

Q = 0.100 m³/s

Answer

The flow rate is:

0.100 m³/s

Since:

1 m³ = 1000 L

this is also:

100 L/s

Worked Example 7: Challenge Problem

A pipe has a diameter of 12 cm.

Water moves through it at 1.5 m/s.

The pipe narrows to a diameter of 6 cm.

Find the new velocity.

The diameter is halved.

Therefore, area becomes:

(½)² = ¼

of its original value.

To keep:

Av = constant

velocity must become four times larger.

Therefore:

v₂ = 4 × 1.5

v₂ = 6.0 m/s

Answer

The water velocity in the narrow section is:

6.0 m/s

Interpreting Arrow Diagrams

Fluid-flow diagrams often use arrows to represent velocity.

Longer arrows may indicate greater speed.

For example:

WIDE                  NARROW                  WIDE

→  →  →             →→→→→→→→             →  →  →
██████████            █████                ██████████
██████████            █████                ██████████

You should be able to identify that:

• velocity increases entering the narrow section
• velocity decreases when the pipe widens again
• the steady volume flow rate remains the same through each section

This type of reasoning is often more important than calculation.

Common Mistakes

Mistake 1: Saying narrow pipes always have greater flow rates

For one steady incompressible flow:

flow rate stays the same

while:

velocity changes

The narrow section has greater velocity, not necessarily a greater volume flow rate.

Mistake 2: Confusing area and diameter

The continuity equation uses cross-sectional area.

If diameter halves, area becomes one-quarter, not one-half.

Remember:

A ∝ d²

Mistake 3: Saying water is "squeezed" into a smaller volume

Liquid water is approximately incompressible under ordinary conditions.

It does not need to become significantly compressed.

Instead, it moves faster through the smaller area.

Mistake 4: Assuming velocity increases when a pipe widens

It is the opposite for the same steady flow:

larger area → lower velocity

Mistake 5: Thinking continuity means velocity is constant

Continuity means the appropriate flow quantity is conserved.

Velocity can change when cross-sectional area changes.

Mistake 6: Assuming pressure is constant

The continuity equation does not tell us that pressure remains constant.

Pressure can change as fluid velocity and height change.

Mistake 7: Forgetting that real systems have resistance

The simple continuity equation describes conservation of mass.

Real pipes also have:

• friction
• turbulence
• bends
• valves
• pumps

These can affect the actual flow rate and pressure.

Mistake 8: Applying the simple equation to strongly compressed gases

The equation:

A₁v₁ = A₂v₂

assumes approximately constant density.

For significantly compressible gas flow, density must also be included.

Check Your Understanding

1. Recall

State the principle of continuity in your own words.

2. Predict

Water flows from a wide section of pipe into a narrower section.

What happens to its average velocity?

Explain why.

3. Compare

Section A has an area of:

0.050 m²

Section B has an area of:

0.010 m²

If the same steady water flow passes through both sections, which has the greater velocity?

4. Calculate

Water flows through:

A₁ = 0.040 m²

at:

v₁ = 3 m/s

The pipe narrows to:

A₂ = 0.020 m²

Calculate v₂.

5. Calculate

Water flows at:

8 m/s

through a pipe with area:

0.015 m²

The pipe widens to:

0.060 m²

Calculate the new velocity.

6. Flow Rate

Water moves through a pipe with:

A = 0.030 m²

v = 5 m/s

Calculate the volume flow rate.

7. Diameter

The diameter of a pipe decreases from:

10 cm → 5 cm

By what factor does its cross-sectional area change?

If the flow rate remains constant, by what factor does the velocity change?

8. Diagram

A pipe has three sections:

A → B → C

with:

AA > AC > AB

Rank the fluid velocities from greatest to smallest.

9. Apply

Explain how the continuity principle helps explain the high-speed stream leaving a fire-hose nozzle.

10. Challenge

A student says:

"When a pipe gets narrower, less water can fit through it, so the flow rate must decrease."

Explain why this is not necessarily true for steady flow through different sections of the same pipe.

Use the terms:

cross-sectional area, velocity, and flow rate.

Key Terms

• Continuity – principle based on conservation of mass during fluid flow
• Continuity equation – equation relating fluid velocity and cross-sectional area
• Cross-sectional area – area of a pipe perpendicular to the direction of flow
• Volume flow rate – volume of fluid passing a point per unit time
• Mass flow rate – mass of fluid passing a point per unit time
• Velocity – speed of fluid in a particular direction
• Incompressible fluid – fluid whose density changes very little under ordinary conditions
• Steady flow – flow in which conditions at a particular location do not change significantly with time
• Conservation of mass – principle that mass cannot simply appear or disappear
• Nozzle – device designed to control the direction or velocity of a flowing fluid

Key Takeaways

• The principle of continuity is based on conservation of mass.
• For steady flow, the amount of fluid entering a system must equal the amount leaving.
• For an approximately incompressible fluid, the volume flow rate remains constant through different sections of the same unbranched pipe.
• Volume flow rate is given by Q = Av.
• The continuity equation is A₁v₁ = A₂v₂.
• A smaller cross-sectional area produces greater fluid velocity for the same steady flow rate.
• A larger cross-sectional area produces lower fluid velocity.
• Area and velocity are inversely related when flow rate is constant.
• Pipe diameter and area are not the same; area is proportional to diameter squared.
• Continuity helps explain fluid behaviour in nozzles, hoses, rivers, blood vessels, ventilation systems and irrigation systems.
• Continuity does not mean pressure remains constant.
• For significantly compressible fluids, such as gases under large pressure changes, density must also be included using ρ₁A₁v₁ = ρ₂A₂v₂.