Bernoulli's Equation Calculator — Fluid Pressure & Velocity
Apply Bernoulli's principle to find the pressure at any point in a steady, incompressible, inviscid flow — a core tool in fluid mechanics for analysing pipe flows, nozzles, aerofoils and the Venturi effect.
Fluid
Pa
m/s
m
m/s
m
Pressure change ΔP = -5.99 kPa
- 1
Dynamic pressure at point 1 (½ρv₁²)
½ × 998.2 × 2² = 1,996.4 Pa - 2
Dynamic pressure at point 2 (½ρv₂²)
½ × 998.2 × 4² = 7,985.6 Pa - 3
Total head H = P₁ + ½ρv₁² + ρgz₁
200,000 + 1,996.4 + 0 = 201,996.4 PaBernoulli constant — conserved along the streamline. - 4
P₂ = H − ½ρv₂² − ρgz₂
(201,996.4 − 7,985.6 − 0) ÷ 1000 = 194.011 kPa
How does this calculator work?
Bernoulli's equation: P₁ + ½ρv₁² + ρgz₁ = P₂ + ½ρv₂² + ρgz₂. Rearranged: P₂ = P₁ + ½ρ(v₁² − v₂²) + ρg(z₁ − z₂). Enter upstream pressure, both velocities, elevations and fluid density to find downstream pressure. Assumes steady, incompressible, frictionless flow along a streamline.
Formula
How this is calculated
Bernoulli's equation expresses the conservation of mechanical energy per unit volume along a streamline in an ideal fluid. It states that the sum of static pressure (P), dynamic pressure (½ρv²) and hydrostatic head (ρgz) is constant: P + ½ρv² + ρgz = constant. This constant is sometimes called the total head or stagnation pressure.
To find the pressure at a downstream point (2), rearrange the equation: P₂ = P₁ + ½ρ(v₁² − v₂²) + ρg(z₁ − z₂). When a fluid accelerates (v₂ > v₁) — as in a constricted pipe section (Venturi tube) or under an aerofoil — the dynamic pressure increases and, at the same elevation, the static pressure must fall. This pressure–velocity trade-off is the Venturi effect and underlies carburettors, atomisers, aircraft lift and Pitot tubes.
Assumptions and limitations: Bernoulli's equation applies to steady (time-independent), incompressible (ρ = constant), inviscid (no viscosity/friction) flow along a single streamline. Real flows deviate due to friction losses, turbulence, compressibility (important at Mach > 0.3 for gases) and flow separation. For engineering design, use the extended Bernoulli equation with head-loss terms. Density values shown are approximate 2025 references and are editable.
Frequently asked questions
Conservation of energy: the total mechanical energy per unit volume (P + ½ρv² + ρgz) is constant along a streamline. If kinetic energy (½ρv²) increases because velocity rises, the static pressure P must decrease to keep the total constant. This is the Venturi effect — exploited in carburettors, Venturi meters and the generation of aerodynamic lift.
It assumes ideal (inviscid, incompressible) steady flow along a streamline. Real fluids have viscosity, so friction losses reduce pressure below the Bernoulli prediction. The equation also fails for compressible flows (gases at high speed), turbulent flows and across rotating machinery. Engineers use the Darcy–Weisbach or extended Bernoulli equations to account for head losses.
Pressures are in Pascals (Pa) or kilopascals (kPa), velocities in m/s, elevations in metres and density in kg/m³. The gravitational acceleration used is 9.81 m/s². Convert: 1 bar = 100,000 Pa; 1 atm ≈ 101,325 Pa; 1 psi ≈ 6,895 Pa.
Also known as
TG we-Calculate Editorial Team. (2026). Bernoulli's Equation Calculator — Fluid Pressure & Velocity [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/bernoulli-equation-calculator
TG we-Calculate Editorial Team. "Bernoulli's Equation Calculator — Fluid Pressure & Velocity." TG we-Calculate. 2026. https://we-calculate.com/calculator/bernoulli-equation-calculator.
TG we-Calculate Editorial Team, "Bernoulli's Equation Calculator — Fluid Pressure & Velocity," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/bernoulli-equation-calculator
@misc{wecalculate_bernoulli_equation_calculator, title = {Bernoulli's Equation Calculator — Fluid Pressure & Velocity}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/bernoulli-equation-calculator}}, year = {2026}, note = {TG we-Calculate} }
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