# Bernoulli Equation Calculator

URL: https://speedycalculator.com/calculators/physics/bernoulli-equation/
Last updated: 2026-10-30

> The Bernoulli equation calculator applies energy conservation along a streamline to find the downstream pressure or velocity from the upstream pressure, velocity and height, optionally computing velocities from pipe diameters and flow rate.

## Inputs

- Find
- Fluid density ρ (kg/m³, g/cm³, g/mL, lb/ft³, lb/in³)
- Velocities from
- Upstream pressure P₁ (kPa, Pa, bar, psi, atm, MPa)
- Upstream velocity v₁ (m/s, km/h, mph, ft/s, knots)
- Upstream height h₁ (m, cm, mm, km, in, ft, yd, mi)
- Volumetric flow rate Q (L/s, m³/s, m³/h, L/min, US gpm, ft³/s)
- Upstream diameter d₁ (mm, cm, m, in, µm)
- Downstream diameter d₂ (mm, cm, m, in, µm)
- Downstream pressure P₂ (kPa, Pa, bar, psi, atm, MPa)
- Downstream velocity v₂ (m/s, km/h, mph, ft/s, knots)
- Downstream height h₂ (m, cm, mm, km, in, ft, yd, mi)

## Results

- Downstream pressure P₂
- Downstream velocity v₂
- Upstream velocity
- Upstream dynamic pressure

## Formula

```
P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂,  v = Q/A
```

## Explanation

Where a fluid speeds up its pressure drops – the Bernoulli effect behind venturi meters, carburettors, atomisers and part of an aircraft wing's lift. The equation assumes steady, incompressible, frictionless flow.

### How to use the Bernoulli Equation Calculator

- **Find**
- **Fluid density ρ**
- **Velocities from**
- **Upstream pressure P₁**
- **Upstream velocity v₁**
- **Upstream height h₁**
- **Volumetric flow rate Q**
- **Upstream diameter d₁**
- **Downstream diameter d₂**
- **Downstream pressure P₂**
- **Downstream velocity v₂**
- **Downstream height h₂**

Results update as you type, every field has its own unit menu, and everything is calculated in your browser.

### How it works

Choose what to find. Velocities can come directly or from the flow rate and pipe diameters.

### Inputs explained

FieldWhat to enterFindChoose one: Downstream pressure P₂, Downstream velocity v₂Fluid density ρUnits: kg/m³, g/cm³, g/mL, lb/ft³, lb/in³Velocities fromChoose one: Velocities, Flow rate and diametersUpstream pressure P₁Units: kPa, Pa, bar, psi, atm, MPaUpstream velocity v₁Units: m/s, km/h, mph, ft/s, knotsUpstream height h₁Units: m, cm, mm, km, in, ft …Volumetric flow rate QUnits: L/s, m³/s, m³/h, L/min, US gpm, ft³/sUpstream diameter d₁Units: mm, cm, m, in, µmDownstream diameter d₂Units: mm, cm, m, in, µmDownstream pressure P₂Units: kPa, Pa, bar, psi, atm, MPaDownstream velocity v₂Units: m/s, km/h, mph, ft/s, knotsDownstream height h₂Units: m, cm, mm, km, in, ft …

### What each result means

- **Downstream pressure P₂** = [Upstream pressure P₁] + 0.5×[Fluid density ρ]×([v1]^2 - [v2]^2) + [Fluid density ρ]×9.80665×([Upstream height h₁] - [Downstream height h₂])
- **Downstream velocity v₂** = sqrt([v1]^2 + 2×([Upstream pressure P₁] - [Downstream pressure P₂])÷[Fluid density ρ] + 2×9.80665×([Upstream height h₁] - [Downstream height h₂]))
- **Upstream velocity** = [v1]
- **Upstream dynamic pressure** = 0.5×[Fluid density ρ]×[v1]^2

### Example values

How the main result changes when **Fluid density ρ** changes and everything else stays at the example values above:

Fluid density ρ (kg/m³)Downstream pressure P₂ (kPa)499.1189.865748.65184.7974998.2179.72991,247.75174.66241,497.3169.59491,996.4159.4598

## Worked example

Inputs — Find: Downstream pressure P₂; Fluid density ρ: 998.2 kg/m³; Velocities from: Velocities; Upstream pressure P₁: 200 kPa; Upstream velocity v₁: 2 m/s; Upstream height h₁: 0 m; Downstream velocity v₂: 5 m/s; Downstream height h₂: 1 m.

Results — **Downstream pressure P₂ = 179.7299 kPa**; Upstream velocity = 2 m/s; Upstream dynamic pressure = 1.9964 kPa.

## Key terms

- **Streamline**: Path followed by a fluid particle in steady flow.

## FAQ

**What does Bernoulli's equation say?**
Pressure, kinetic and potential energy per volume stay constant along a streamline.

**Why does pressure drop in a narrowing?**
The fluid speeds up, so its pressure falls.

**When is the equation invalid?**
With friction, pumps, turbines or compressible flow.

**Does it explain lift?**
Partly – lift is best explained by the flow turning around the wing.


## Sources

- [OpenStax – University Physics Volume 1](https://openstax.org/details/books/university-physics-volume-1)
- [The Engineering ToolBox](https://www.engineeringtoolbox.com/)