Physics · Fluids

Continuity Equation

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Wrap-up

What you learned

Recommended next: Open concept testCheck whether the core ideas are ready without leaving this concept.
Read next: Bernoulli's PrincipleAdd pressure to the speed change

Key takeaway

The same steady incompressible stream carries one shared volume flow rate through sections A and B.
At fixed Q, shrinking a cross-sectional area raises the local speed, while widening it lowers the local speed.
Same-time slices make the compensation visible: longer fast slices and shorter slow slices can represent the same volume per second.
Continuity supplies the speed-area link that Bernoulli uses when pressure joins the story.

Common misconception

A narrow section carries more volume per second just because the fluid moves faster there.

In steady incompressible flow, one section of the pipe does not carry more volume per second than another.

Carry this into Bernoulli and the fluids branch

Keep this idea moving

Open the next concept, route, or track only when you want the current model to widen into a larger branch.

Turn speed changes into pressure changesFluids

Bernoulli's Principle

Follow one steady ideal-flow pipe and see how pressure, speed, and height trade within the same Bernoulli budget while continuity keeps the flow-rate story honest.

Contrast static and moving fluidsFluids

Pressure and Hydrostatic Pressure

Use one piston-and-tank bench to connect force per area, pressure acting in all directions, and the way density, gravity, and depth build hydrostatic pressure.

Return to pressure differencesFluids

Buoyancy and Archimedes' Principle

Use one immersed-block bench to connect pressure difference, displaced fluid, and the density balance behind floating, sinking, and neutral buoyancy.

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Reference and support

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Equation snapshotContinuity equation · Speed from flow rate and areaShow 3 equations

Start with the shared flow rate Q, then use v = Q/A to see why the smaller cross section gets the larger speed.

Continuity equation
$Q = A_{A} v_{A} = A_{B} v_{B}$
For steady incompressible flow, each section of the same pipe carries the same flow rate.
Speed from flow rate and area
$v = \frac{Q}{A}$
At a fixed flow rate, a smaller area requires a larger speed.
Speed ratio from area ratio
$\frac{v _{B}}{v _{A}} = \frac{A _{A}}{A _{B}}$
The speed change is the inverse of the area change between two sections.

Why it behaves this way

Explanation

Imagine tracking one short chunk of fluid as it passes two cross sections of a pipe. In steady incompressible flow, fluid is not piling up or disappearing between those sections, so the same volume must pass each cross section each second. That shared volume flow rate is written as $Q = A v$ .

A narrow section does not create more flow each second. It gives the same flow less area to move through, so the speed there has to increase. A wider section does the opposite: the same $Q$ can pass at a lower speed.

Use the stage, speed arrows, same-time slices, and graphs as one connected explanation. When the area changes, the speed adjusts, while $Q$ stays matched across the pipe. This is the continuity part of fluid flow, before pressure enters later with Bernoulli.

Key ideas

01In steady incompressible flow, the same volume per second passes every cross section, so

Q = A_{A} v_{A} = A_{B} v_{B}

02A smaller area means the fluid must move faster there to keep the same

Q

03A larger area means the same

Q

can pass with a lower speed.

04The stage and graphs should agree on the same story: area changes cause the speed change, not a change in how much fluid per second is flowing.

05Continuity gives the speed-area link that Bernoulli later combines with pressure changes.

Worked examples

Continuity worked examples

Open examples when you want to see the same idea walked through step by step.

Frozen walkthrough

Step through the frozen example

Frozen walkthrough

Use the live pipe values. Each example starts from the current

Q

and areas, then links the calculation to what you see on the stage.

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Example 1 of 2

Frozen valuesUsing frozen parameters

Using the current values $Q = 0.18 m^{3} /s$ , $A_{A} = 0.24 m^{2}$ , and $A_{B} = 0.12 m^{2}$ , what speeds should continuity predict at sections A and B?

Q

Volume flow rate

0.18 m³/s

A_{A}

Section A area

0.24 m²

A_{B}

Section B area

0.12 m²

1. Calculate the speed at section A with $v = Q/A$

The entry speed is

v_{A} = Q / A_{A} = 0.75 m/s

2. Use the same $Q$ to calculate the speed at section B

The middle speed is

v_{B} = Q / A_{B} = 1.5 m/s

3. Compare the two speeds and connect them to the pipe shape

v_{B} / v_{A} = 2

. Because the middle area is smaller, the same flow rate must move faster there.

Predicted section speeds

v_{A} = 0.75 m/s, v_{B} = 1.5 m/s, v_{B} / v_{A} = 2

Because the middle area is smaller, the same flow rate must move faster there.

Quick test

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Accessibility

Open the text-first descriptions when you need the simulation and graph translated into words.

The simulation shows one steady stream tube with a labeled section A at the entry and a labeled section B in the middle. The pipe height represents cross-sectional area, the animated tracer dots show the flow moving through the pipe, and the section speed arrows show which region is faster or slower.

The readout card reports the volume flow rate, both section areas, both section speeds, and the speed ratio. Compare mode ghosts one alternate pipe shape so two different continuity states can be read on the same bench.

The same-time slice overlay marks how much pipe length a short equal-time fluid slice occupies in each section. It is a visual cue for the same conserved flow rate.

Graph summary

The section-speed graphs isolate how entry area, middle area, or flow rate changes the two section speeds.

The flow-balance graph keeps the two section flow-rate lines matched so continuity stays explicit while the speed adjustments happen elsewhere.

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Step 2 of 5

Fluid and Pressure

Continuity Equation appears later in this track, so it is cleaner to start from the beginning first.

Previous step: Pressure and Hydrostatic Pressure

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