Bernoulli's Equation Calculator
Solve Bernoulli's equation P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂ for the pressure at a second point in a flow, accounting for both speed changes and elevation changes. Two 3D diagrams show a Venturi tube constriction and an airfoil generating lift, with charts showing how pressure responds to velocity and height.
Reviewed by the ToolNestr Editorial Team — July 2026
Bernoulli's principle in action
1. A Venturi tube constriction
A pipe narrows in the middle — flow speeds up through the constriction, and pressure there drops, exactly as Bernoulli's equation predicts.
2. An airfoil generating lift
Air travels faster over the curved top of the wing than along the flatter bottom, creating a pressure difference that contributes to lift.
Bernoulli's equation graphs
How it works
The core idea in one line: Bernoulli's equation is really just conservation of energy applied to a flowing fluid — pressure, speed, and height are three different ways the same total energy can be stored, and increasing one at a fixed total means the others must give something back.
P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂
Bernoulli's equation — conservation of energy along a streamline
P₂ = P₁ + ½ρ(v₁² − v₂²) + ρg(h₁ − h₂)
rearranged to solve directly for the unknown pressure at point 2
A₁v₁ = A₂v₂
continuity equation — used alongside Bernoulli to relate velocity to pipe cross-sectional area
Each term in P + ½ρv² + ρgh represents an energy density along the flow: P is the fluid's own pressure energy, ½ρv² is its kinetic energy (from speed), and ρgh is its gravitational potential energy (from height). Because this total stays constant along a streamline for an idealized fluid, rearranging the equation to P₂ = P₁ + ½ρ(v₁²−v₂²) + ρg(h₁−h₂) shows precisely how a change in either speed or height at point 2 must be balanced by an opposite change in pressure.
Worked example 1 — a horizontal pipe constriction (Venturi effect)
Given: Water (ρ = 1000 kg/m³) flows through a horizontal pipe (h₁ = h₂). At the wide section, P₁ = 200,000 Pa and v₁ = 2 m/s. At a narrower section, the speed rises to v₂ = 5 m/s. Find P₂.
Pressure drops by 10,500 Pa in the faster, narrower section — the same Venturi effect used in carburetors and perfume atomizers to draw in a second fluid at the low-pressure point.
Worked example 2 — an oil pipeline rising in elevation
Given: Oil (ρ = 850 kg/m³) flows through a pipe of constant diameter (so v₁ = v₂ = 3 m/s). P₁ = 300,000 Pa at h₁ = 0 m, rising to h₂ = 10 m.
Even with no speed change, pressure drops by over 83 kPa just from climbing 10 m — the same hydrostatic penalty water pays when pumped up to a rooftop tank.
The Venturi effect — pipe narrowing vs pressure drop
Fixed inlet v₁ = 2 m/s, P₁ = 200,000 Pa, water (ρ = 1000 kg/m³), horizontal pipe — narrower area ratios force higher exit speeds via continuity, which drop the pressure further.
| Area ratio A₁/A₂ | Exit speed v₂ | Pressure P₂ |
|---|---|---|
| 1× (no change) | 2 m/s | 200,000 Pa |
| 2× narrower ★ | 4 m/s | 194,000 Pa |
| 3× narrower | 6 m/s | 184,000 Pa |
| 4× narrower | 8 m/s | 170,000 Pa |
★ Computed via continuity v₂ = v₁×(A₁/A₂), then Bernoulli P₂ = P₁ + ½ρ(v₁²−v₂²). Pressure keeps falling faster than the area ratio because kinetic energy depends on velocity squared.
Where Bernoulli's equation actually matters
✈️ Airplane wing lift
An airfoil's curved shape speeds up airflow over the top surface; by Bernoulli's principle, that faster air has lower pressure, contributing to the net upward lift force that keeps aircraft airborne.
🧪 Carburetors and atomizers
A Venturi constriction speeds up airflow, dropping local pressure enough to draw fuel or perfume up through a side tube and mix it into the fast-moving stream — a direct engineering application of the pressure-speed tradeoff.
🩺 Blood flow and arterial narrowing
Blood speeding up through a narrowed (stenotic) artery experiences a local pressure drop, described by the same Bernoulli relationship — a principle cardiologists use to help interpret blood flow measurements.
🚰 Water tower and plumbing design
Engineers use Bernoulli's equation to predict how pressure and flow speed trade off across elevation changes and pipe diameter changes in municipal water systems.
Common misconceptions
"Faster-moving fluid always has higher pressure."
It's the opposite for a horizontal flow — Bernoulli's equation shows that at the same elevation, faster fluid has lower pressure, not higher. This inverse relationship is the entire basis of the Venturi effect.
"Bernoulli's equation applies to any fluid flow, including turbulent or viscous flows."
The equation assumes an idealized, incompressible, non-viscous (inviscid) fluid in steady flow along a single streamline. Real flows with turbulence or significant viscosity (like thick oil through a rough pipe) deviate from the ideal prediction.
"Only speed affects pressure in a flowing fluid — elevation doesn't matter."
Elevation matters just as much through the ρgh term, which behaves exactly like ordinary hydrostatic pressure. Fluid flowing uphill loses pressure (or speed) even if its speed through the pipe never changes.
"Bernoulli's principle is the complete explanation of how airplane wings generate lift."
While Bernoulli's principle correctly describes a pressure difference from unequal airflow speeds, the complete modern explanation of lift also involves Newton's third law — the wing deflecting air downward and the air pushing back up on the wing in reaction.
Formula sources & further reading
The formulas here are standard, traceable to:
- • OpenStax, University Physics Volume 1 — Chapter 14, "Fluid Mechanics" (free, peer-reviewed). openstax.org
- • Halliday, Resnick & Walker, Fundamentals of Physics — Chapter 14, Fluids.
- • Serway & Jewett, Physics for Scientists and Engineers — Chapter 14, Fluid Mechanics.
P₂ = P₁ + ½ρ(v₁²−v₂²) + ρg(h₁−h₂), for an idealized incompressible, non-viscous, steady flow. Standard gravity g=9.81 m/s² used. Results are rounded for display.
How to use this calculator
Enter point-1 conditions
Provide the fluid density, and the pressure, velocity, and elevation at the first point.
Enter point-2 conditions
Provide the velocity and elevation at the second point (often from a continuity-equation calculation).
Read the result
Pressure at point 2 solves live, showing the combined effect of the speed and elevation change.
Related tools
Frequently asked questions
What is Bernoulli's equation?
Bernoulli's equation, P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂, expresses conservation of energy for an ideal, incompressible, non-viscous fluid in steady flow — it relates pressure, kinetic energy (from speed), and potential energy (from height) at any two points along a streamline.
Why does pressure drop where fluid speeds up?
In a horizontal flow (h₁ = h₂), Bernoulli's equation reduces to P₁ + ½ρv₁² = P₂ + ½ρv₂². Since the sum must stay constant, a faster speed at point 2 means a lower pressure at point 2 — this is the Venturi effect, used in carburetors, atomizers, and airflow measurement.
How does elevation affect pressure in Bernoulli's equation?
The ρgh term acts exactly like hydrostatic pressure — fluid flowing to a higher elevation loses pressure (or speed) to "pay for" the increase in gravitational potential energy, just as it would if it were standing still.
What assumptions does Bernoulli's equation make?
It assumes the fluid is incompressible (constant density), inviscid (no internal friction/viscosity), and the flow is steady along a single streamline. Real fluids like air and water deviate from this ideal, especially at high speeds or with turbulence, but the equation is an excellent approximation in many engineering situations.
How does Bernoulli's principle explain lift on an airplane wing?
An airfoil is shaped so air travels faster over the curved top surface than along the flatter bottom. By Bernoulli's principle, that faster-moving air over the top has lower pressure than the slower air below, creating a net upward force — lift (though modern aerodynamics shows this is only part of the full explanation, which also involves Newton's third law and airflow deflection).