Fluid mechanics · Fluid dynamics

Bernoulli equation calculator

Calculate pressure at either of two points from pressure, speed and elevation at the other point. The balance includes static pressure, kinetic energy per volume and gravitational energy per volume.

Created Oct 3, 2026v1.0

By Thibaut Grzelak, Mechanical Analysis Engineer

Fluid dynamics

Points 1 and 2 show their relative elevations. Pressure values share one reference; the pipe geometry is schematic.

Quantity to calculate

Fluid density

Density of the fluid, strictly positive.

Gravity

Positive local gravitational acceleration.

Pressure P1

Use the same gauge or absolute reference at both points.

Speed v1

Local mean speed, ≥ 0.

Elevation h1

Signed elevation relative to a common datum.

Pressure P2

Use the same gauge or absolute reference at both points.

Speed v2

Local mean speed, ≥ 0.

Elevation h2

Signed elevation relative to a common datum.

Results

Pressure at section 2

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Same pressure reference as the input; a negative gauge value is possible.

Pressure at section 1

—

Same pressure reference as the input.

Relation and numerical substitutionP₂ = P₁ + 0.5 × ρ × (v1² − v2²) + ρ × g × (h1 − h2)

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Steady, incompressible, inviscid flow on one streamline; negligible losses and no pump or turbine between the points.

01Formulas and symbols

Formulas used

P2P₂ = P₁ + 0.5 × ρ × (v1² − v2²) + ρ × g × (h1 − h2)Same pressure reference as the input; a negative gauge value is possible.

P1P₁ = P₂ + 0.5 × ρ × (v2² − v1²) + ρ × g × (h2 − h1)Same pressure reference as the input.

02Assumptions and limits

Scope of validity

Steady, incompressible, inviscid flow on one streamline; negligible losses and no pump or turbine between the points.

03Validation example

Reference numerical case

ρ = 1000 kg/m³, P₁ = 200000 Pa, v₁ = 2 m/s, v₂ = 4 m/s, h₁ = h₂ = 0: P₂ = 200000 + 500(4 − 16) = 194000 Pa.

04References

Technical references

University Physics 1, §14.6 Bernoulli’s equation — OpenStax

FAQ

Can I use gauge pressure?

Yes, if both pressure values use the same reference. No atmospheric correction is added automatically.

Does this include pipe friction?

No. Use the pressure-drop tool to estimate losses; this ideal Bernoulli balance neglects them.

05Model and conventions

Calculate pressure at either of two points from pressure, speed and elevation at the other point. The balance includes static pressure, kinetic energy per volume and gravitational energy per volume.

06Limits of this model

Real pipe losses require a separate pressure-drop analysis. Negative gauge pressure is possible; if an absolute-pressure result is negative or approaches vapour pressure, the assumed liquid-flow model is not valid.

07Common mistakes

Use the same pressure reference and elevation datum at both points. Do not confuse pressure units with a liquid head in metres.