Bernoulli's Equation Calculator

Enter 6 of 7 variables and the calculator solves for the missing one

By Drew Budwin · Last updated July 2026 · Methodology

Leave exactly one field blank. It will be solved automatically.

Point 1

Leave blank to solve for v₁

Leave blank to solve for P₁

Leave blank to solve for h₁

Point 2

Leave blank to solve for v₂

Leave blank to solve for P₂

Leave blank to solve for h₂

Leave blank to solve for ρ

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How We Calculate This

Speed and Pressure Trade Off

In a flowing fluid, faster-moving regions have lower pressure. That's why airplane wings generate lift (air moves faster over the curved top surface), why a shower curtain gets pulled inward, and why standing close to a passing train is dangerous. The equation holds only for ideal, incompressible, steady flow along a single streamline.

Worked Example: Pipe Narrowing at the Same Elevation

Water (ρ = 1,000 kg/m³) flows horizontally. Point 1: P₁ = 200,000 Pa, v₁ = 2 m/s. Point 2: v₂ = 6 m/s. P₂ = 200,000 + ½ × 1,000 × (4 − 36) = 184,000 Pa. Velocity tripled; pressure dropped 16 kPa. The energy moved from pressure to kinetic.

Bernoulli's Equation Calculator Formula

Bernoulli's equation expresses conservation of mechanical energy along a streamline in steady, incompressible, inviscid flow.

P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂

P = static pressure, ρ = fluid density, v = fluid velocity, g = 9.80665 m/s² (standard gravity), h = elevation above datum. Subscripts 1 and 2 denote the two points along the streamline.

How to Use This Calculator

  1. Leave exactly one of the seven fields blank. That is the variable to solve for.
  2. Enter values for the remaining six fields. Use the unit dropdown beside each field to choose your preferred unit. You can mix and match freely (e.g., pressure in psi, velocity in ft/s, height in m).
  3. Common fluid densities: water ≈ 998 kg/m³ (0.998 g/cm³ or 1.937 slug/ft³); air at sea level ≈ 1.225 kg/m³.
  4. Click Calculate to solve for the missing variable.
  5. The result is shown in multiple unit equivalents. The Bernoulli Component Breakdown table (always in Pa) confirms energy conservation: the total at Point 1 and Point 2 must be equal.

Frequently Asked Questions

What is Bernoulli's equation?

Bernoulli's equation states that along a streamline in steady, incompressible, inviscid flow the sum of static pressure, dynamic pressure (½ρv²), and elevation pressure (ρgh) is constant. Where fluid speeds up, pressure drops, and vice versa. This principle explains how airplane wings generate lift, how venturi meters measure flow rate, and how a garden hose nozzle increases water speed.

Which variable should I leave blank?

Leave blank the variable you want to find. For example, to find the fluid speed at a downstream nozzle exit, leave v₂ blank and fill in P₁, v₁, h₁, P₂, h₂, and ρ.

Can I mix metric and imperial units?

Yes. Each field has its own unit dropdown. You can enter pressure in psi, velocity in ft/s, and height in metres all at the same time. The calculator converts every input to SI internally before solving, then converts the result back to all equivalent units. For another physics calculator that solves for a missing variable, see the Ohm's law calculator.

What are the assumptions behind Bernoulli's equation?

The equation applies to steady (time-invariant), incompressible (constant density), inviscid (frictionless) flow along a single streamline. It does not account for viscous losses, turbulence, heat transfer, or compressibility at high Mach numbers.

What density should I use for common fluids?

Water at 20 °C: approximately 998 kg/m³ (0.998 g/cm³, 1.937 slug/ft³). Air at sea level and 15 °C: approximately 1.225 kg/m³ (0.001225 g/cm³, 0.00237 slug/ft³). For other fluids, consult an engineering reference or fluid properties table.

What is the venturi effect?

The venturi effect is a direct application of Bernoulli's principle: when fluid flows through a constricted section of pipe, its velocity increases and its pressure decreases. This is used in carburetors, atomizers, and flow measurement devices like venturi meters.

Why does pressure drop when velocity increases?

Because energy is conserved along the streamline. Bernoulli's equation is essentially a statement of energy conservation: the sum of pressure energy, kinetic energy, and potential energy per unit volume is constant. When the fluid speeds up, it gains kinetic energy. That energy has to come from somewhere, and it comes from the pressure term. No energy is created or lost; it just shifts form.

What happens to the result when elevation changes between the two points?

The elevation term (ρgh) accounts for potential energy. If Point 2 is higher than Point 1, some energy goes into lifting the fluid, leaving less for pressure or velocity. For horizontal flow (h₁ = h₂), the elevation terms cancel and the equation simplifies to P₁ + ½ρv₁² = P₂ + ½ρv₂². Set both elevation fields to the same value if you're analyzing horizontal flow.

When does Bernoulli's equation break down?

Bernoulli's equation loses accuracy when flow is turbulent, the fluid is compressible (air at high speeds), viscosity is significant (thick oils, flow near walls), or there is heat transfer. For engineering applications involving any of these, Bernoulli gives a useful first estimate but not a final answer. Real pipe flow analysis typically adds a friction loss term.

Further Reading

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