HomeConcept libraryFluidsFluid and Pressure

FluidsIntroStarter track

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.

Interactive lab

Keep the stage, graph, and immediate control feedback in one working view.

Time

0.00 s / 6.00 sLiveArea and flow-rate sweeps stay parameter-based while the time rail inspects live tracer motion through the changing pipe.

0.00 s6.00 s

Continuity Equation

Change the entry area, middle area, or volume flow rate. The pipe shape, tracer speed, dye slices, and response graphs stay tied to one bounded continuity model.

Graphs

Switch graph views without breaking the live stage and time link.

Section speed vs entry area

Change section A while keeping section B and the flow rate fixed.

Entry area A_A (m^2): 0.16 to 0.32Speed (m/s): 0 to 2

Section A speedMiddle speed

Hover or scrub to link the graph back to the stage.Entry area A_A (m^2) / Speed (m/s)

Controls

Adjust the physical parameters and watch the motion respond.

Volume flow rate

Q

0.18 m^3/s

Changes how much volume the pipe carries each second.

Section A area

A_{A}

0.24 m^2

Adjusts the wider entry section.

Section B area

A_{B}

0.12 m^2

Adjusts the middle section so it can narrow or widen relative to section A.

More tools

Secondary controls, alternate presets, and less-used toggles stay nearby without crowding the main bench.

Show

More presets

Presets

Predict -> manipulate -> observe

Keep the active prompt next to the controls so each change has an immediate visible consequence.

ObservationPrompt 1 of 2

Pull section B smaller. The flow-rate labels stay matched, but the middle speed arrow gets longer because

v = Q / A

grows there.

Equation map

See each variable before you move it.

Select a symbol to highlight the matching control and the graph or overlay it most directly changes.

Q

Volume flow rate

0.18 m^3/s

Raises or lowers how much volume the pipe carries each second. Increasing $Q$ raises both section speeds together.

Graph: Section speed vs flow rateGraph: Volume flow rate by sectionOverlay: Flow-rate equalityOverlay: Same-time slices

Equations in play

Choose an equation to sync the active symbol, control highlight, and related graph mapping.

More tools

Detailed noticing prompts, guided overlays, and challenge tasks stay available without taking over the main bench.

Hide

What to notice

Keep one flow question in view at a time so the pipe shape, speeds, and flow-rate readout stay tied together.

ObservationPrompt 1 of 2

Pull section B smaller. The flow-rate labels stay matched, but the middle speed arrow gets longer because

v = Q / A

grows there.

Control: Section B areaGraph: Section speed vs middle areaOverlay: Speed arrowsOverlay: Flow-rate equalityEquation

A_{B}

Guided overlays

Focus one overlay at a time to see what it represents and what to notice in the live motion.

3 visible

Overlay focus

Area markers

Marks the two cross sections used in the continuity equation.

What to notice

Section B can be smaller or larger than section A, and the speed change follows that area change inversely.

Why it matters

It keeps the equation tied to visible geometry.

Control: Section A areaControl: Section B area

Challenge mode

Use the same pipe for direct continuity targets and compare matches. The checks read the live areas, speeds, and flow rate.

0/2 solved

MatchStretch

1 of 8 checks

Same flow, slower wide section

Start from Baseline stream, switch to compare mode, leave Setup A alone, and tune Setup B until it keeps the same flow rate but slows section B down by widening that middle section.

Compare modeGraph-linkedGuided start

Suggested start

Setup A stays as the baseline narrow middle section. Edit only Setup B.

Pending

Stay in compare mode while editing Setup B.

Explore

Pending

Open the flow-balance graph.

Section speed vs entry area

Matched

Keep the flow-rate equality overlay visible.

Pending

Keep Setup B at the same flow rate.

Pending

Keep Setup B near the same section A area.

Pending

Widen Setup B's middle section.

Pending

Make Setup B's middle speed lower than section A's.

Pending

Keep Setup B carrying the same flow rate.

The checklist updates from the live simulation state, active graph, overlays, inspect time, and compare setup.

At t = 0 s, the stream carries Q = 0.18 m^3/s through section A with area 0.24 m^2 and speed 0.75 m/s, then through the middle section with area 0.12 m^2 and speed 1.5 m/s. The middle section is narrower than section A, so the flow speeds up there. Both sections still carry the same flow rate because Q = Av in this bounded steady-flow model.

Equation detailsDeeper interpretation, notes, and worked variable context.

Volume flow rate

The stream carries a certain volume past a cross section each second.

Q = \frac{Δ V}{Δ t}

Q

Volume flow rate 0.18 m^3/s

Continuity equation

For steady incompressible flow, each section of the same pipe carries the same flow rate.

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

Speed from flow rate and area

At a fixed flow rate, a smaller area requires a larger speed.

v = \frac{Q}{A}

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

Speed ratio from area ratio

The speed change is the inverse of the area change between two sections.

\frac{v _{B}}{v _{A}} = \frac{A _{A}}{A _{B}}

A_{A}

Section A area 0.24 m^2

A_{B}

Section B area 0.12 m^2

Progress

Not startedMastery: NewLocal-first

Start exploring and Open Model Lab will keep this concept's progress on this browser first. Challenge mode has 2 compact tasks ready. No finished quick test, solved challenge, or completion mark is saved yet.

Let the live model runChange one real controlOpen What to notice

Try this setup

Copy the live bench state and reopen this concept with the same controls, graph, overlays, and compare context.

Stable links

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

Short explanation

What the system is doing

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.

Live flow checks

Solve the exact state on screen.

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

Live valuesFollowing current 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?

Prediction prompt

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

Q

stays fixed.

Check your reasoning

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?

Choose one answer to reveal feedback, then test the idea in the live system if a guided example is available.

Accessible description

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 moving through the concept map.

These suggestions come from the concept registry, so the reason label reflects either curated guidance or the fallback progression logic.

Turn speed changes into pressure changesFluids

Continuity Equation

See each variable before you move it.

Area markers

Same flow, slower wide section

Fluid and Pressure

What the system is doing

Solve the exact state on screen.

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 moving through the concept map.

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?