Bernoulli Equation Calculator

Background.

The Bernoulli equation is the cornerstone of elementary fluid mechanics, relating pressure, velocity, and elevation for steady, incompressible, inviscid flow along a streamline. It was derived by Daniel Bernoulli in 1738 in his treatise Hydrodynamica, though the modern form was refined by Leonhard Euler. Engineers apply it to pipe flow metering, aircraft wing design, pump and turbine sizing, and the analysis of siphons, nozzles, and venturi tubes. The Bernoulli equation calculator solves for any one unknown among pressure, velocity, or elevation when the other quantities are specified, providing rapid verification of hand calculations and supporting iterative design workflows.

The canonical use case is the analysis of a venturi meter installed in a water pipeline. A venturi constricts the flow, raising the velocity and lowering the pressure in the throat relative to the upstream section. By measuring the pressure difference with a differential manometer, an engineer can compute the flow rate without inserting a mechanical obstruction. If the upstream pipe diameter is 100 mm, the throat diameter is 50 mm, and the pressure drop is 15 kPa, continuity gives v_throat = 4 × v_upstream. Applying Bernoulli between the upstream section and the throat yields the velocity, and multiplying by the cross-sectional area gives the volumetric flow rate. This calculation, repeated for multiple operating points, generates a calibration curve for the meter.

In aerodynamics, Bernoulli's principle explains lift generation at the subsonic speeds typical of general aviation. An airfoil is shaped so that air travelling over the curved upper surface moves faster than air beneath the flat lower surface. According to Bernoulli, the higher velocity on top corresponds to lower pressure, producing a net upward force. For a Cessna 172 at cruise, the pressure difference between the upper and lower surfaces is only a few kilopascals, but distributed over the wing area of 16 square metres this yields a lift force sufficient to support the aircraft mass of 1100 kg. The principle breaks down at transonic and supersonic speeds because compressibility becomes significant and shock waves form, but for incompressible flow it provides an excellent first approximation.

The physical content of the Bernoulli equation is the conservation of mechanical energy per unit mass along a streamline. The term P represents pressure work, ½v² represents kinetic energy, and gh represents gravitational potential energy. When the equation is divided by ρg, each term acquires dimensions of length and is interpreted as a head: pressure head, velocity head, and elevation head. Civil engineers prefer this form because pump curves and turbine specifications are given in metres of water head. The sum of the three heads is the total head, which is constant for ideal flow and decreases monotonically for real flow because of viscous dissipation and turbulence.

Strictly, the Bernoulli equation applies only under restrictive assumptions: steady flow, incompressible fluid, negligible viscosity, and along a single streamline. Real flows violate at least one of these conditions. Water in a pipe has viscosity, which creates wall shear and a parabolic velocity profile rather than the uniform velocity implied by a single v. Air at high speed is compressible, and its density changes with pressure. Unsteady flows, such as those in pulsating biological vessels or water hammer in pipelines, require the unsteady Bernoulli equation with an additional ∂φ/∂t term. Despite these limitations, the Bernoulli equation remains the first tool engineers reach for because it captures the essential trade-off between pressure and velocity and because its limitations are well understood.

What is bernoulli equation calculator?

The Bernoulli equation states that for an inviscid, incompressible fluid in steady flow, the quantity P + ½ρv² + ρgh is constant along any streamline. Each term represents energy per unit volume: pressure energy, kinetic energy, and gravitational potential energy. The equation is not an independent law of nature but a special case of the Navier-Stokes momentum equation integrated along a streamline under the assumptions of zero viscosity and constant density.

The SI units of each term are pascals (N/m²), which are dimensionally equivalent to joules per cubic metre. When divided by ρg, the terms become metres of fluid head. The equation is invalid across streamlines unless the flow is irrotational, and it is invalid through regions of significant viscosity, such as boundary layers, wakes, and fully developed pipe flow. It is also invalid for compressible flows where Mach number exceeds approximately 0.3, because density variations couple pressure and velocity nonlinearly. Despite these restrictions, the equation provides the conceptual foundation for venturi meters, pitot-static tubes, and pump head calculations in hydraulic engineering. Stagnation pressure, the sum of static and dynamic pressure, is measured by a Pitot tube aligned parallel to the flow and is the key quantity used to compute airspeed in aviation.

How to use this calculator.

1. 01Select the fluid from the dropdown or enter a custom density in kg/m³.
2. 02Enter the pressure, velocity, and elevation at the upstream point (point 1).
3. 03Enter the known quantities at the downstream point (point 2).
4. 04Select the unknown variable to solve for: P₂, v₂, or h₂.
5. 05Verify that gravitational acceleration is set correctly for your location (default 9.80665 m/s²).
6. 06Click Calculate to obtain the solved variable and the total head at both points.
7. 07If total head at point 2 is significantly lower than at point 1, viscous losses dominate and the ideal-flow assumption may be invalid.

The formula.

The Bernoulli equation can be derived from Newton's second law applied to a fluid particle. Consider a small cylindrical fluid element of cross-sectional area dA and length ds aligned with a streamline. The pressure forces on the two ends are P dA and (P + dP) dA, the gravitational force is ρg dA ds cosθ, and the acceleration is dv/dt. For steady flow, dv/dt = v dv/ds. Summing forces along the streamline and dividing by dA ds yields the Euler equation along a streamline: (1/ρ) dP/ds + v dv/ds + g dh/ds = 0. Integrating with respect to s from point 1 to point 2, and assuming constant density, gives P/ρ + v²/2 + gh = constant, which is equivalent to the Bernoulli equation after multiplication by ρ.

The restriction to inviscid flow is severe. Viscous fluids obey the Navier-Stokes equations, which include shear stress terms that dissipate mechanical energy into heat. In pipe flow, the Bernoulli equation is modified by adding a head-loss term h_L: P₁/ρg + v₁²/2g + h₁ = P₂/ρg + v₂²/2g + h₂ + h_L. The Darcy-Weisbach equation gives h_L = f (L/D) (v²/2g), where f is the Moody friction factor, L is pipe length, and D is diameter. The calculator does not compute h_L; it solves the ideal Bernoulli equation. Users analysing real pipe systems should add the Darcy-Weisbach loss separately or use a dedicated pipe-flow calculator.

For compressible flows, the assumption ρ = constant fails. The isentropic Bernoulli equation for an ideal gas replaces P/ρ with ∫ dP/ρ(P), which for isentropic flow evaluates to γ/(γ−1) × P/ρ, where γ is the heat capacity ratio. At Mach numbers below 0.3, the density variation is less than 5 percent and the incompressible form is acceptable. Above Mach 0.3, compressibility corrections or full computational fluid dynamics are required. The calculator is restricted to incompressible flow and warns the user if velocity exceeds the compressibility threshold for the selected fluid.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Bernoulli, D. (1738). Hydrodynamica, sive de Viribus et Motibus Fluidorum Commentarii. Dulsecker.
2. [2]Batchelor, G.K. (1967). An Introduction to Fluid Dynamics. Cambridge University Press. ISBN 978-0-521-66396-0
3. [3]White, F.M. (2016). Fluid Mechanics, 8th ed. McGraw-Hill. ISBN 978-1-259-69587-0
4. [4]Anderson, J.D. (2017). Fundamentals of Aerodynamics, 6th ed. McGraw-Hill. ISBN 978-1-259-12991-9
5. [5]CODATA (2018). Recommended Values of the Fundamental Physical Constants.