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Concept module

Continuity Equation

Keep one steady stream tube on screen and use Q = Av to connect cross-sectional area, flow speed, and the same volume flow rate through narrow and wide sections.

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Starter track

Step 2 of 50 / 5 complete

Fluid and Pressure

Earlier steps still set up Continuity Equation.

1. Pressure and Hydrostatic Pressure2. Continuity Equation3. Bernoulli's Principle4. Buoyancy and Archimedes' Principle+1 more steps

Previous step: Pressure and Hydrostatic Pressure.

Start from track beginning

Why it behaves this way

Explanation

The continuity equation is the bookkeeping rule for steady incompressible flow. If fluid is not piling up or leaving gaps inside one pipe, then each cross section must pass the same volume every second. That shared volume flow rate is written as $Q = A v$ .

A smaller section does not create more fluid each second. It gives the stream less area to move through, so the speed there has to rise to keep the same $Q$ . A wider section does the opposite: the same flow rate can move more slowly there.

This page stays bounded on purpose. The stage is one changing pipe with two labeled sections, animated tracers, and compact response graphs. It is not a full fluid-dynamics engine, but it keeps the speed-area story honest and sets up the later Bernoulli bridge.

Key ideas

01For steady incompressible flow, the same volume flow rate passes every cross section each second, so $Q = A_A v_A = A_B v_B$.

02If one section has a smaller area, the fluid speed there must be larger to keep the same flow rate.

03If the area gets larger, the same flow rate can move more slowly through that section.

04Continuity is the speed-and-area bookkeeping layer that Bernoulli later combines with pressure changes in moving fluids.

Frozen walkthrough

Step through the frozen example

Frozen walkthrough

Use the live pipe. The current flow rate and section areas drive both worked examples.

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Frozen valuesUsing frozen parameters

For the current stream with $Q = 0.18 m^{3} /s$ , $A_{A} = 0.24 m^{2}$ , and $A_{B} = 0.12 m^{2}$ , what speeds does continuity predict in sections A and B?

Q

Volume flow rate

0.18 m^3/s

A_{A}

Section A area

0.24 m^2

A_{B}

Section B area

0.12 m^2

1. Use $v = Q/A$ in section A

The entry speed is

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

2. Use the same $Q$ in section B

The middle speed is

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

3. Compare the two sections

v_{B} / v_{A} = 2

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

Current 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.

Continuity checkpoint

A steady incompressible stream enters a pipe section whose area becomes half as large. What must happen to the speed in that smaller section?

Make a prediction before you reveal the next step.

Decide whether the speed stays the same, doubles, or drops when

Q

stays fixed.

Check your reasoning against the live bench.

The speed doubles.

Continuity says

Q = A v

. If

Q

stays fixed and the area becomes half as large, the speed has to become twice as large.

Common misconception

A narrow section carries more fluid each second because the fluid shoots through it faster.

In steady incompressible flow, the volume per second stays the same through every section of the pipe.

The speed changes because the area changes. Faster does not mean a bigger $Q$ unless the area stays fixed.

Quick test

Variable effect

Question 1 of 3

Answer from the pipe, not from a slogan.

At the same volume flow rate, what happens to the speed in section B if section B's area is cut in half?

Use the live bench to test the result before moving on.

Accessibility

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.

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

Continuity Equation

Fluid and Pressure

Explanation

Step through the frozen example

For the current stream with Q=0.18m3/s, AA​=0.24m2, and AB​=0.12m2, what speeds does continuity predict in sections A and B?

At the same volume flow rate, what happens to the speed in section B if section B's area is cut in half?

Keep this idea moving

Bernoulli's Principle

Pressure and Hydrostatic Pressure

Buoyancy and Archimedes' Principle

For the current stream with $Q = 0.18 m^{3} /s$ , $A_{A} = 0.24 m^{2}$ , and $A_{B} = 0.12 m^{2}$ , what speeds does continuity predict in sections A and B?