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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.

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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

Why it behaves this way

Explanation

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.

Frozen walkthrough

Step through the frozen example

Frozen walkthrough

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

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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 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?

Make a prediction before you reveal the next step.

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

Check your reasoning against the live bench.

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?

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

Accessibility

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

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Pressure differences in fluidsFluids

Bernoulli's Principle

Fluid and Pressure

Explanation

Step through the frozen example

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 this idea moving

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?