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
| Field | What to enter |
|---|---|
| Find | Choose one: Downstream pressure P₂, Downstream velocity v₂ |
| Fluid density ρ | Units: kg/m³, g/cm³, g/mL, lb/ft³, lb/in³ |
| Velocities from | Choose one: Velocities, Flow rate and diameters |
| Upstream pressure P₁ | Units: kPa, Pa, bar, psi, atm, MPa |
| Upstream velocity v₁ | Units: m/s, km/h, mph, ft/s, knots |
| Upstream height h₁ | Units: m, cm, mm, km, in, ft … |
| Volumetric flow rate Q | Units: L/s, m³/s, m³/h, L/min, US gpm, ft³/s |
| Upstream diameter d₁ | Units: mm, cm, m, in, µm |
| Downstream diameter d₂ | Units: mm, cm, m, in, µm |
| Downstream pressure P₂ | Units: kPa, Pa, bar, psi, atm, MPa |
| Downstream velocity v₂ | Units: m/s, km/h, mph, ft/s, knots |
| Downstream height h₂ | Units: 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) |
|---|---|
| 499.1 | 189.865 |
| 748.65 | 184.7974 |
| 998.2 | 179.7299 |
| 1,247.75 | 174.6624 |
| 1,497.3 | 169.5949 |
| 1,996.4 | 159.4598 |
Formula
P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂, v = Q/A
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.
Frequently asked questions
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.