The Bernoulli equation calculator solves P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂ for any one missing variable. Enter conditions at two points in an ideal flow — the calculator finds the unknown pressure, velocity, or height.
Setup
1 Point 1
2 Point 2
Results
How to Use the Bernoulli Equation Calculator
Bernoulli's equation is an energy conservation law for ideal fluids. At any two points along a streamline, the sum of pressure energy, kinetic energy, and potential energy per unit volume remains constant: P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂.
Example: Water Pipe Narrowing (Venturi Effect)
A water pipe narrows from a large section to a smaller one. At Point 1: P₁ = 200,000 Pa, v₁ = 2 m/s, h₁ = 0 m. At Point 2: v₂ = 5 m/s, h₂ = 0 m. Solving for P₂: P₂ = P₁ + ½ρ(v₁² − v₂²) = 200,000 + ½ × 1000 × (4 − 25) = 200,000 − 10,500 = 189,500 Pa. Pressure drops as velocity increases — this is the Venturi effect.
How to Use the Solve-For Dropdown
Select which variable to calculate from the "Solve For" dropdown. The corresponding input field becomes read-only (labeled "SOLVING"). Enter the five known values and click Calculate. The calculator rearranges Bernoulli's equation algebraically to find the unknown.
Energy Breakdown Interpretation
The results show the three energy terms at each point: pressure term (P), kinetic term (½ρv²), and potential term (ρgh). All three should sum to the same total — confirming energy conservation. In pipe flow at constant height, pressure energy converts to kinetic energy as velocity increases.
Limitations
Bernoulli's equation assumes inviscid (no-friction), incompressible, steady flow. Real flows have viscous losses — use the extended Bernoulli equation with head loss terms for pipelines with significant length. For compressible flows (high-speed gases), use the compressible flow equations instead.
FAQ
What is Bernoulli's principle?
Bernoulli's principle states that for an inviscid, incompressible fluid flowing along a streamline, the total mechanical energy remains constant: P + ½ρv² + ρgh = constant. This means when velocity increases, pressure decreases (and vice versa). It explains airplane lift, carburetor function, and why a fast-moving river is shallow.
What are the assumptions of Bernoulli's equation?
Bernoulli's equation assumes: (1) steady, incompressible flow with constant density, (2) inviscid flow (no viscosity/friction), (3) along a single streamline, and (4) no energy addition from pumps or removal from turbines between the two points. For real pipes with friction, use the extended Bernoulli equation with head loss terms.
How does Bernoulli's equation explain airplane lift?
An airplane wing (airfoil) is shaped so air travels faster over the top than the bottom. Faster flow means lower pressure (Bernoulli). The pressure difference between bottom (higher) and top (lower) creates an upward net force — lift. At cruise: pressure difference ≈ 1,000–2,000 Pa creates enough force to support the aircraft weight.
What is the Venturi effect?
The Venturi effect is the velocity increase (and corresponding pressure drop) when a fluid flows through a narrowing pipe. From continuity equation A₁v₁ = A₂v₂: if pipe narrows by 50%, velocity doubles. Bernoulli gives the pressure drop: ΔP = ½ρ(v₂² - v₁²). Venturi meters use this to measure flow rates.
Is this calculator free?
Yes, completely free with no signup required. All calculations run in your browser.
Is my data private?
Yes. All calculations run locally in your browser. Nothing is transmitted to any server.
Can Bernoulli's equation be used for compressible flows like air?
The standard incompressible form applies to air at low speeds (Mach < 0.3). For aircraft at typical cruise speed (Mach 0.8), compressibility corrections become significant. The incompressible form gives errors under 5% for Mach < 0.3, which covers most everyday fluid problems including HVAC, plumbing, and low-speed aerodynamics.