Bernoulli Equation & Fluid Flow Calculator

Bernoulli Equation Guide

Bernoulli's Equation

P + ½ρv² + ρgh = constant. P = static pressure (Pa), ½ρv² = dynamic pressure (Pa), ρgh = hydrostatic pressure (Pa). Between two points in steady, inviscid, incompressible flow: P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂. Key insight: as velocity increases, pressure decreases (and vice versa). Example: pipe narrows from 5cm to 2.5cm diameter. Continuity: A₁v₁ = A₂v₂. Flow at 2 m/s through large section: v₂ = v₁ × (A₁/A₂) = 2 × 4 = 8 m/s. Pressure drop: ΔP = ½ρ(v₂² - v₁²) = ½×1000×(64-4) = 30,000 Pa.

Torricelli's Theorem

A special case of Bernoulli: the speed of fluid exiting a small hole in a large tank. v_exit = √(2gh). Where h = height of fluid above the exit hole. Same formula as free-fall velocity — the fluid accelerates as if falling from height h. Example: tank with 5m of water above exit: v = √(2×9.81×5) = √98.1 = 9.9 m/s. This approximation assumes: the exit hole is small compared to the tank. The tank level drops slowly. Viscosity is negligible (reasonable for water through large holes). Real orifices

Venturi Meter

A venturi meter measures flow rate by creating a controlled narrowing. Mass flow rate: Q = A₁A₂√(2ΔP/(ρ(A₁²-A₂²))). Pressure drop ΔP = P₁ - P₂ measured by a manometer or differential pressure sensor. Applications: water supply measurement, natural gas metering, carburettors (the venturi principle draws fuel into the airstream). Aircraft pitot-static tube: measures dynamic pressure ½ρv² from which airspeed is calculated. Pitot tube: total pressure = static + dynamic. Static port: static pressure

Bernoulli and Lift

Aircraft wings (aerofoils) generate lift through a pressure difference — but the mechanism is more nuanced than the textbook explanation. Common (oversimplified) explanation: air travels faster over the curved top surface → lower pressure → upward lift (Bernoulli). More accurate: lift arises primarily from circulation of airflow around the wing (Kutta-Joukowski theorem). The wing deflects air downward — by Newton's 3rd law, the wing is pushed upward. Bernoulli pressure difference is a real effec