Physics · Fluids

Bernoulli's Principle

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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: Buoyancy and Archimedes' PrinciplePressure differences in fluids

Key takeaway

Continuity sets the throat speed from the entry and throat areas before Bernoulli explains the pressure change.
A faster throat spends more of the same budget on kinetic energy per volume, so throat static pressure can drop.
A raised throat spends an additional share on height, which can lower throat pressure even when the speed story is unchanged.
Widening the throat reverses the speed trade and can recover pressure in the same ideal-flow model.

Common misconception

Bernoulli means any fast flow automatically has low pressure.

On this page, the lower throat pressure comes from one specific model: steady incompressible flow along the same streamline, with pumps and viscosity ignored.

Carry this through the fluids branch

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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.

Review static pressureFluids

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.

Contrast pressure from particle motionThermodynamics

Ideal Gas Law and Kinetic Theory

Connect pressure, volume, temperature, and particle number on one bounded particle box, then read the same pressure changes back as changes in particle speed and wall-collision rate.

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Equation snapshotContinuity for the same stream · Bernoulli's principleShow 3 equations

Use continuity first to get the two speeds, then use Bernoulli to split the same budget among pressure, kinetic, and height terms.

Continuity for the same stream
$Q = A_{A} v_{A} = A_{B} v_{B}$
The same steady incompressible flow rate passes section A and throat B.
Bernoulli's principle
$P + \frac{1}{2} ρ v^{2} + ρ g y = constant$
Along one ideal steady streamline, static pressure, kinetic term, and height term share the same total.
Throat pressure from section A
$P_{B} = P_{A} + \frac{1}{2} ρ (v_{A}^{2} - v_{B}^{2}) - ρ g (y_{B} - y_{A})$
A faster or higher throat can have lower static pressure because more of the same total is tied up in speed or height.

Why it behaves this way

Explanation

Track the same fluid from section A to throat B. In this ideal steady incompressible-flow model, no pumps or viscosity are adding or removing energy along that streamline, so pressure, speed, and height trade with one another. That is the idea behind $P + f r a c 12 h o v^{2} + h o g y = e x t co n s t an t$ .

Continuity still comes first. The same volume flow rate must pass section A and throat B, so a narrower throat has a larger speed. Bernoulli then tells you what that larger speed and any extra height do to the static pressure.

This does not mean fast flow always gives low pressure in every fluid situation. On this page, lower throat pressure happens for a specific reason: in the same ideal-flow budget, more of the total is tied up in speed and, if the throat is raised, height, so less is left as static pressure.

Key ideas

01Along one steady incompressible streamline in this ideal-flow bench,

P + f r a c 12 h o v^{2} + h o g y

stays nearly constant from section A to throat B.

02Continuity sets the speed first: if the throat area is smaller, the throat speed must be larger for the same

Q

03A faster throat means more of the same total sits in the kinetic term, so the throat's static pressure can be lower.

04If the throat is also higher, the height term takes another share of the same total, so the throat pressure can drop further.

05Continuity and Bernoulli do different jobs together: continuity links area to speed, and Bernoulli links speed and height to pressure.

Worked examples

Bernoulli worked examples

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Frozen walkthrough

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Frozen walkthrough

Use the live bench values. Each example starts from the current pipe state and connects the calculation to the gauges, arrows, and graphs.

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Example 1 of 2

Frozen valuesUsing frozen 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 does continuity give, and what throat pressure does Bernoulli predict?

P_{A}

Section A pressure

32 kPa

Q

Volume flow rate

0.18 m³/s

A_{A}

Section A area

0.1 m²

A_{B}

Throat area

0.05 m²

Δ y

Throat rise

0.25 m

1. Use continuity to find 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. Work out how much pressure shifts into the kinetic term

Moving from section A to throat B shifts about

4.86 kPa

of static pressure into the kinetic term.

3. Include the pressure change from the throat's height

Lifting the throat by

0.25 m

costs another

2.45 kPa

in the height term.

4. Read the throat pressure that remains

So the throat pressure is

P_{B} = 24.69 kPa

, with a total drop of

7.31 kPa

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

Predicted 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.

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 pipe with section A at the entry and throat B farther along. Pipe thickness represents cross-sectional area, tracer dots show the flow, and speed arrows show which section is moving faster.

Pressure gauges show the static pressure at section A and throat B. Energy bars split each state into pressure, kinetic, and height terms so the trade between them stays visible.

Compare mode overlays a second pipe state on the same bench so you can compare two Bernoulli setups without switching to a different page.

Graph summary

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

The pressure graphs show how throat width, flow rate, and throat height change the throat pressure within the same Bernoulli model.

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Fluid and Pressure

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