What is Bernoulli's equation?

Bernoulli's equation states P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂ for an ideal, incompressible fluid along a streamline. It represents conservation of energy in fluid flow: pressure, kinetic, and potential energy sum to a constant.

What are the assumptions of Bernoulli's equation?

Bernoulli's equation assumes steady, incompressible, inviscid (frictionless) flow along a streamline. Real fluids have viscosity and turbulence, so it is an approximation. For most pipe flow, gas flow at Mach < 0.3, and short distances with low friction, it works well.

Does Bernoulli explain airplane lift?

Partially. Faster airflow over the wing's curved top creates lower pressure than under the flatter bottom, generating lift — which Bernoulli explains. However, complete lift theory requires circulation, vortex theory, and Newton's third law (downward air deflection). Bernoulli is the intuitive starting point, not the full story.

What is the Venturi effect?

When fluid flows through a constriction, it speeds up (continuity) and its pressure drops (Bernoulli). This is the Venturi effect. Used in flow meters, carburetors, spray atomizers, and the suction in some vacuum tools.

How does a pitot tube measure airspeed?

A pitot tube measures stagnation pressure (front opening, fluid stopped) minus static pressure (side opening, undisturbed flow). The difference equals ½ρv². Solving: v = √(2ΔP/ρ). Used in aircraft, wind tunnels, and HVAC ducts.

When does Bernoulli fail?

Bernoulli fails when flow is unsteady (pulsating), compressible (Mach > 0.3), highly viscous (boundary layers, long pipes), turbulent, or across vortices. For long pipes use extended Bernoulli with friction loss; for high-speed gas use compressible flow equations; for viscous flow use Navier-Stokes.

What is the head form of Bernoulli?

Dividing through by ρg gives pressure head + velocity head + elevation head = constant, all in meters of fluid. h_p = P/ρg, h_v = v²/(2g), h_z = z. Common in hydraulics and pipe design — easy to add a head-loss term h_L for friction.

Can I use Bernoulli for water in pipes?

Yes, for short distances with little friction. For long pipes, viscous losses become significant — use the extended Bernoulli equation with a head-loss term (Darcy-Weisbach friction factor times length over diameter times velocity head). For complex networks, use EPANET or similar.

Bernoulli Equation Calculator

Q: Does Bernoulli explain airplane lift?

Partially. Faster airflow over the wing's curved top creates lower pressure than under the flatter bottom, generating lift — which Bernoulli explains. However, complete lift theory requires circulation, vortex theory, and Newton's third law (downward air deflection). Bernoulli is the intuitive starting point, not the full story.

Q: What is the Venturi effect?

When fluid flows through a constriction, it speeds up (continuity) and its pressure drops (Bernoulli). This is the Venturi effect. Used in flow meters, carburetors, spray atomizers, and the suction in some vacuum tools.

Q: How does a pitot tube measure airspeed?

A pitot tube measures stagnation pressure (front opening, fluid stopped) minus static pressure (side opening, undisturbed flow). The difference equals ½ρv². Solving: v = √(2ΔP/ρ). Used in aircraft, wind tunnels, and HVAC ducts.

Q: When does Bernoulli fail?

Bernoulli fails when flow is unsteady (pulsating), compressible (Mach > 0.3), highly viscous (boundary layers, long pipes), turbulent, or across vortices. For long pipes use extended Bernoulli with friction loss; for high-speed gas use compressible flow equations; for viscous flow use Navier-Stokes.

Calculate pressure, velocity, or height at two points in a fluid flow using Bernoulli's equation. The principle states that P + ½ρv² + ρgh remains constant along a streamline for ideal fluid flow.

Bernoulli's equation expresses the conservation of energy for an ideal flowing fluid: along any streamline, the sum of pressure energy, kinetic energy, and potential energy per unit volume stays constant. This single relationship explains an enormous range of phenomena — why airplanes generate lift, why a shower curtain pulls inward when you turn on the water, how a Venturi meter measures flow, why a fast-moving stream of water creates low pressure at its sides.

The classical form is P + ½ρv² + ρgh = constant. As fluid speeds up, its pressure must drop (kinetic up, pressure down). As it climbs, pressure also drops (potential up, pressure down). This trade-off, applied at two points along a streamline, lets you solve for any unknown if you know the others.

Bernoulli's equation assumes ideal conditions: steady (not time-varying), incompressible (constant density), inviscid (no friction), and irrotational flow along a single streamline. Real fluids violate these assumptions to varying degrees — water in a smooth pipe is close to ideal; air at supersonic speeds, viscous oil, or highly turbulent flow needs corrections.

Common applications: aerodynamics (wing lift), pipe flow design, Venturi flowmeters, pitot tubes (measuring aircraft airspeed), spray systems, chimney draft, and countless introductory fluid-mechanics problems.

Inputs

Fluid Density (kg/m³)

Pressure at Point 1 (Pa)

Velocity at Point 1 (m/s)

Height at Point 1 (m)

Velocity at Point 2 (m/s)

Height at Point 2 (m)

Results

Pressure at Point 2

61,395 Pa

Total Pressure

103,325 Pa

Dynamic P₂

12500.0 Pa

Bernoulli Equation Results

Parameter	Value
Fluid Density	1000 kg/m³
Point 1: Static Pressure	101,325 Pa
Point 1: Dynamic Pressure	2000.00 Pa
Point 1: Hydrostatic (ρgh)	0.00 Pa
Total Pressure (constant)	103,325 Pa
Point 2: Dynamic Pressure	12500.00 Pa
Point 2: Hydrostatic (ρgh)	29430.00 Pa
Point 2: Static Pressure	61,395 Pa
Pressure Difference	39,930 Pa
Formula	P + ½ρv² + ρgh = constant

Last updated: May 29, 2026

Formula

**Bernoulli's equation (per unit volume):** P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂ Where: - P = static pressure (Pa) - ρ = fluid density (kg/m³) - v = flow speed (m/s) - g = 9.81 m/s² - h = elevation (m) Each term has units of energy per unit volume (J/m³ = Pa). **Terms:** - **P**: pressure energy - **½ρv²**: kinetic energy per volume (dynamic pressure) - **ρgh**: potential energy per volume (hydrostatic pressure) **Worked example: water flowing up a pipe** Water (ρ = 1,000 kg/m³). Point 1 at ground level: P₁ = 200,000 Pa, v₁ = 2 m/s, h₁ = 0. Point 2 at 3 m elevation, v₂ = 5 m/s. Find P₂. P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂ 200,000 + 0.5(1000)(4) + 0 = P₂ + 0.5(1000)(25) + (1000)(9.81)(3) 200,000 + 2,000 = P₂ + 12,500 + 29,430 P₂ = 202,000 − 41,930 = 160,070 Pa **Pressure drops by ~40 kPa** as water speeds up and climbs. **Forms of Bernoulli:** | Form | Units | Use case | |---|---|---| | P + ½ρv² + ρgh | Pa | engineering, fluid mechanics | | P/ρg + v²/(2g) + h | m | hydraulics, "head" form | | Specific energy | J/kg | thermodynamics integration | **Head form (hydraulics):** h_p + h_v + h_z = constant Where h_p = P/ρg (pressure head), h_v = v²/(2g) (velocity head), h_z = h (elevation head). Sum is "total head" in meters of fluid. **Venturi meter:** For a constriction in horizontal flow (h₁ = h₂): P₁ + ½ρv₁² = P₂ + ½ρv₂² Combined with continuity (A₁v₁ = A₂v₂): Δv = v₂ − v₁ → ΔP measured tells you flow rate. This is how spray atomizers and carburetors work. **Pitot tube (airspeed):** Stagnation pressure (at front of tube) − static pressure = ½ρv² v = √(2 × ΔP / ρ) Used in aircraft to measure airspeed. **Torricelli's law (drain from a tank):** For a hole at the bottom of a tank with height h above: v = √(2gh) Same speed as a freely falling object dropped from height h. **When Bernoulli FAILS:** - **Compressible flow**: Mach > 0.3, density changes matter. - **Viscous flow**: pipe friction, boundary layers. - **Turbulence**: energy dissipates as heat. - **Unsteady flow**: pulsating pumps, water hammer. - **Across streamlines**: holds along single streamline only. For real pipes, use the **extended Bernoulli (energy) equation** with head loss term hL added.

How to use this calculator

Enter fluid density (1000 for water, 1.225 for air at sea level).
Enter pressure, velocity, and height at point 1 (upstream).
Enter velocity and height at point 2 (downstream).
Calculator solves for pressure at point 2.
Use SI units throughout to avoid conversion errors.
Remember: assumes steady, incompressible, inviscid flow.

Worked examples

Garden hose nozzle

**Scenario:** Water in a hose at 300 kPa flows at 2 m/s through a wider section, then accelerates to 10 m/s through a narrower nozzle. Same elevation. **Calculation:** Δ(½ρv²) = 0.5 × 1000 × (100 − 4) = 48,000 Pa. P₂ = 300,000 − 48,000 = 252,000 Pa. **Result:** Pressure drops by 48 kPa as water speeds up. This is why fluid speeds up exiting a nozzle — pressure energy converts to kinetic energy. The pressure at the nozzle tip is still 152% of atmospheric, then drops to atmospheric as the jet leaves.

Airplane wing (qualitative)

**Scenario:** Air flows over a wing at 100 m/s on top, 80 m/s on bottom. Air density 1.0 kg/m³ at altitude. **Calculation:** ΔP = ½ρ(v_bottom² − v_top²) = 0.5 × 1.0 × (6400 − 10000) = −1,800 Pa. **Result:** Pressure on the wing's top is 1,800 Pa lower than on the bottom. For a wing area of 30 m², that's ~54,000 N of lift — roughly 5.5 tonnes. (Real aerodynamics involves circulation theory and viscous effects, but Bernoulli gives the intuition.)

Water tank drain

**Scenario:** Tank with water level 4 m above a small drain hole. How fast does water exit? **Calculation:** Torricelli's law: v = √(2gh) = √(2 × 9.81 × 4) = √78.48 ≈ 8.86 m/s. **Result:** Water exits at ≈8.86 m/s (≈20 mph). Same speed as if dropped from 4 m. Real-world losses (sharp edges, viscous effects) reduce this by ~15% — actual flow ≈7.5 m/s.

When to use this calculator

**Use Bernoulli's equation for:**

- **Pipe flow problems**: pressure-velocity-height relationships. - **Aerodynamics**: wing lift (qualitatively), airflow over surfaces. - **Flow measurement**: Venturi meters, pitot tubes, orifice plates. - **Tank drainage**: Torricelli's law for outflow speed. - **Spray systems**: atomizers, carburetors, paint sprayers. - **Chimney draft**: hot rising air pressure relationships. - **Stream/river hydraulics**: velocity-depth-pressure trade-offs.

**Critical assumptions and when they fail:**

- **Steady flow**: variables don't change with time. Pulsating systems (heart, reciprocating pumps) violate this. - **Incompressible**: density constant. Air at Mach < 0.3 OK; supersonic flow needs gas dynamics. - **Inviscid**: no friction. Long pipes need Darcy-Weisbach friction correction. - **Along a streamline**: holds for each streamline individually; vortices break this. - **Irrotational**: no swirling; turbulent flow violates this.

**Real-world corrections:**

**Extended Bernoulli (engineering form):**

P₁/ρg + v₁²/(2g) + h₁ = P₂/ρg + v₂²/(2g) + h₂ + h_L

Where h_L = head loss from friction, fittings, etc. Use Darcy-Weisbach or empirical loss coefficients.

**Common applications:**

- **HVAC design**: duct sizing and pressure drops. - **Plumbing**: water pressure at different elevations. - **Aircraft**: pitot-static system measures airspeed and altitude. - **Oil & gas**: pipeline pressure drops. - **Hydroelectric**: head difference drives turbines. - **Sports**: curveball physics, golf ball dimples (modified by viscous effects).

**Software:**

- **CFD (ANSYS Fluent, OpenFOAM)**: full Navier-Stokes for real flows. - **EPANET**: water distribution network analysis. - **Spreadsheets**: Bernoulli + Darcy-Weisbach for design calcs.

**Common pitfalls:**

- **Applying across pumps or turbines**: Bernoulli doesn't account for energy added/removed; use energy equation. - **Forgetting elevation changes**: in tall buildings, ρgh dominates. - **Mixing absolute and gauge pressures**: be consistent. - **Using on viscous flow**: long pipe = significant head loss. - **Compressibility**: high-speed gas needs gas dynamics equations. - **Across streamlines**: stagnation pressure ≠ static pressure in rotational flow.

**Bernoulli for gases (low speed):**

Works for air at Mach < 0.3 (about 100 m/s at sea level). Above that, compressibility matters and you need compressible Bernoulli or full gas dynamics.

**Beyond Bernoulli:**

- **Navier-Stokes equations**: full fluid dynamics with viscosity. - **Euler equations**: inviscid compressible flow. - **Reynolds-averaged Navier-Stokes (RANS)**: turbulent flow modeling. - **Direct numerical simulation (DNS)**: research-level accuracy.

Common mistakes to avoid

Applying Bernoulli across streamlines where rotational flow exists.
Forgetting that the equation requires steady, incompressible, inviscid flow.
Mixing gauge and absolute pressures.
Using on compressible flow at high Mach number (M > 0.3).
Ignoring head losses in long pipes (use extended Bernoulli with friction).
Applying across a pump or turbine (energy is added/removed there).
Forgetting unit consistency (mixing Pa with kPa, or m with cm).
Treating dynamic pressure as static pressure.

Frequently Asked Questions

Sources & further reading

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