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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.

Interactive lab

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

Time

0.00 s / 6.00 sLiveSpeed, pressure, and height sweeps stay parameter-based while the time rail inspects live tracer motion through the same Bernoulli pipe.

0.00 s6.00 s

Bernoulli's Principle

Change the flow rate, the throat width, the throat height, or the entry pressure. One ideal-flow pipe keeps continuity, static pressure, and the Bernoulli energy trade on the same compact bench.

Graphs

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

Section speed vs throat area

Change the throat width while keeping the entry state fixed.

Throat area A_B (m^2): 0.04 to 0.12Speed (m/s): 0 to 8

Section A speedThroat speed

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

Controls

Adjust the physical parameters and watch the motion respond.

Section A pressure

P_{A}

32 kPa

Sets the static pressure at section A.

Volume flow rate

Q

0.18 m^3/s

Changes how much fluid volume moves through the pipe each second.

Section A area

A_{A}

0.1 m^2

Adjusts the wider entry section.

Throat area

A_{B}

0.05 m^2

Adjusts the throat width. The throat can narrow, match, or widen relative to the entry section.

More tools

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

Show

Throat rise

Δ y

0.25 m

Raises or lowers the throat relative to section A.

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 the throat smaller. The same

Q

stays in both sections, the throat speed arrow grows, and the throat pressure gauge drops because the kinetic term is taking a bigger share.

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.

P_{A}

Section A pressure

32 kPa

Raises or lowers the whole Bernoulli budget at the entry. A larger starting pressure lifts the throat pressure too, but it does not erase the speed or height trade.

Graph: Pressure vs throat areaGraph: Pressure vs flow rateGraph: Pressure vs throat heightOverlay: Pressure gaugesOverlay: Bernoulli budget bars

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 trade in view at a time so speed, pressure, and height stay tied to the same flowing pipe.

ObservationPrompt 1 of 2

Graph: Section speed vs throat area

Pull the throat smaller. The same

Q

stays in both sections, the throat speed arrow grows, and the throat pressure gauge drops because the kinetic term is taking a bigger share.

Control: Throat areaGraph: Section speed vs throat areaGraph: Pressure vs throat areaOverlay: Continuity bridgeOverlay: Speed arrowsOverlay: Pressure gaugesEquation

A_{B}

Guided overlays

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

4 visible

Overlay focus

Continuity bridge

Marks sections A and B and keeps the same $Q$ attached to both.

What to notice

The same flow rate label stays attached to both sections even when their speeds differ.

Why it matters

It keeps Bernoulli tied to the continuity bookkeeping rather than treating speed as arbitrary.

Control: Volume flow rateControl: Section A areaControl: Throat area

Challenge mode

Use the same pipe for direct Bernoulli targets and compare matches. The checks read the live speeds, throat pressure, and geometry from the same bounded model.

0/2 solved

MatchStretch

1 of 9 checks

Same entry state, wider B recovers pressure

Start from Baseline venturi, switch to compare mode, leave Setup A alone, and tune Setup B until it keeps the same entry pressure and flow rate but recovers the throat pressure by widening only the throat.

Compare modeGraph-linkedGuided start

Suggested start

Setup A stays as the baseline narrow throat. Edit only Setup B.

Pending

Stay in compare mode while editing Setup B.

Explore

Pending

Open the pressure-vs-throat-area graph.

Section speed vs throat area

Matched

Keep the pressure gauges visible.

Pending

Keep Setup B at the same entry pressure.

Pending

Keep Setup B at the same flow rate.

Pending

Keep the same raised throat height in Setup B.

Pending

Widen Setup B's throat to about 0.09 m^2.

Pending

Recover the throat pressure to about 29.2 kPa.

Pending

Bring Setup B's throat speed down near 2 m/s.

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 from section A with area 0.1 m^2 and speed 1.8 m/s into a throat with area 0.05 m^2, height rise 0.25 m, speed 3.6 m/s, and static pressure 24.69 kPa. The throat pressure is lower because more of the same total Bernoulli budget now sits in speed and height terms. Continuity sets the speed change; Bernoulli tells you how the pressure changes with it in this bounded ideal-flow bench.

Equation detailsDeeper interpretation, notes, and worked variable context.

Continuity for the same stream

The same steady incompressible flow rate passes section A and throat B.

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

Q

Volume flow rate 0.18 m^3/s

A_{A}

Section A area 0.1 m^2

A_{B}

Throat area 0.05 m^2

Bernoulli's principle

Along one ideal steady streamline, static pressure, kinetic term, and height term trade within one conserved budget.

P + \frac{1}{2} ρ v^{2} + ρ g y = constant

P_{A}

Section A pressure 32 kPa

Q

Volume flow rate 0.18 m^3/s

Δ y

Throat rise 0.25 m

Throat pressure from section A

If the throat is faster or higher, the static pressure left at B can be lower.

P_{B} = P_{A} + \frac{1}{2} ρ (v_{A}^{2} - v_{B}^{2}) - ρ g (y_{B} - y_{A})

P_{A}

Section A pressure 32 kPa

Q

Volume flow rate 0.18 m^3/s

A_{B}

Throat area 0.05 m^2

Δ y

Throat rise 0.25 m

Speed ratio from continuity

The speed change still comes from the area change before Bernoulli translates it into a pressure change.

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

A_{A}

Section A area 0.1 m^2

A_{B}

Throat area 0.05 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 3 of 50 / 5 complete

Fluid and Pressure

Earlier steps still set up Bernoulli's Principle.

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

Previous step: Continuity Equation.

Start from track beginning

Short explanation

What the system is doing

Bernoulli's principle is the bounded steady-flow energy story for one streamline in an ideal incompressible fluid. If viscosity and pumps are left out, the same flow keeps trading among static pressure, kinetic energy per volume, and height. That is why the shorthand is $P + \frac{1}{2} ρ v^{2} + ρ g y = constant$ .

This page keeps that idea compact and honest. The same pipe shows a section A entry and a raised throat B. Continuity still decides where the speed changes, because the same volume flow rate has to pass both sections. Bernoulli then tells you what happens to the static pressure when that same flow gets faster, climbs higher, or does both at once.

The point is not that fast always means low pressure in every fluid situation. The point is narrower and higher parts of this one bounded stream must spend more of the same Bernoulli budget on speed and height, so the static pressure left over there can be lower.

Key ideas

01For one steady incompressible streamline in this ideal-flow bench, $P + \tfrac{1}{2}\rho v^2 + \rho g y$ stays nearly constant from section A to throat B.

02Continuity supplies the speed link first: if the throat area is smaller, the throat speed must be larger for the same $Q$.

03If the throat is faster, some of the same Bernoulli budget shifts into the kinetic term, so the static pressure can drop there.

04If the throat is also higher, the height term takes another share of the budget, so the throat static pressure drops further.

05Bernoulli complements continuity. Continuity says where the speed change lives; Bernoulli says how pressure and height fit that same change.

Live Bernoulli checks

Solve the exact state on screen.

Use the live bench. The current entry pressure, flow rate, throat width, and throat height drive both worked examples.

Live valuesFollowing current parameters

For the current bench with $P_{A} = 32 kPa$ , $Q = 0.18 m^{3} /s$ , $A_{A} = 0.1 m^{2}$ , $A_{B} = 0.05 m^{2}$ , and $Δ y = 0.25 m$ , what throat speed and throat pressure does the bounded Bernoulli model predict?

P_{A}

Section A pressure

32 kPa

Q

Volume flow rate

0.18 m^3/s

A_{A}

Section A area

0.1 m^2

A_{B}

Throat area

0.05 m^2

Δ y

Throat rise

0.25 m

1. Use continuity to get the throat speed

The current areas give

v_{A} = 1.8 m/s

and

v_{B} = 3.6 m/s

, so the throat is 2 times as fast as section A.

2. Read the speed-driven pressure share

Moving from section A to throat B shifts about 4.86\,\mathrm{kPa}$ of static pressure into the kinetic term.

3. Include the height term

Lifting the throat by 0.25\,\mathrm{m}

cos t s an o t h er 2.45 kPa

in the height term.

4. Read the static pressure left over

So the throat pressure is

P_{B} = 24.69 kPa

, with a total drop of 7.31\,\mathrm{kPa}$. Most of the throat pressure drop here is being spent on extra speed through the narrower throat.

Current throat state

v_{B} = 3.6 m/s, P_{B} = 24.69 kPa

Most of the throat pressure drop here is being spent on extra speed through the narrower throat.

Bernoulli checkpoint

A steady stream keeps the same entry pressure and flow rate, but the throat becomes narrower and also rises higher. What must happen to the throat's static pressure in this bounded Bernoulli model?

Prediction prompt

Decide whether the throat pressure goes up, stays the same, or drops.

Check your reasoning

It drops.

Continuity makes the narrower throat faster, so the kinetic term grows there. Raising the throat also adds a height term. Both of those leave less of the same Bernoulli budget available as static pressure.

Common misconception

Bernoulli means fast flow magically causes low pressure everywhere.

On this page the lower throat pressure comes from a specific bounded steady-flow model where the same streamline keeps one Bernoulli budget.

Continuity and the pipe geometry matter first. The speed changes because the same $Q$ passes through different areas, and Bernoulli explains the matching pressure trade inside that same ideal-flow setup.

Quick test

Variable effect

Question 1 of 3

Answer from the same pipe, not from a slogan.

At the same entry pressure, flow rate, and height, what happens when the throat area gets smaller?

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 pipe with a section A entry and a raised throat B. The pipe thickness represents cross-sectional area, animated tracer dots show the flow moving through the pipe, and the speed arrows show which section is faster.

Static pressure appears as compact gauges near section A and throat B. The Bernoulli budget bars split each state into pressure, kinetic, and height shares of the same total so the pressure trade stays visible.

Compare mode ghosts an alternate pipe state so two Bernoulli setups can be read on the same bench without creating a second disconnected model.

Graph summary

The speed-throat-area graph isolates the continuity speed change that Bernoulli builds on.

The pressure graphs isolate how throat width, flow rate, and throat height reshape the throat pressure while the same bounded Bernoulli model stays in force.

Carry this through 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.

Pressure differences in fluidsFluids

Bernoulli's Principle

See each variable before you move it.

Continuity bridge

Same entry state, wider B recovers pressure

Fluid and Pressure

What the system is doing

Solve the exact state on screen.

For the current bench with PA​=32kPa, Q=0.18m3/s, AA​=0.1m2, AB​=0.05m2, and Δy=0.25m, what throat speed and throat pressure does the bounded Bernoulli model predict?

At the same entry pressure, flow rate, and height, what happens when the throat area gets smaller?

Keep moving through the concept map.

Buoyancy and Archimedes' Principle

Pressure and Hydrostatic Pressure

Ideal Gas Law and Kinetic Theory

For the current bench with $P_{A} = 32 kPa$ , $Q = 0.18 m^{3} /s$ , $A_{A} = 0.1 m^{2}$ , $A_{B} = 0.05 m^{2}$ , and $Δ y = 0.25 m$ , what throat speed and throat pressure does the bounded Bernoulli model predict?