Bernoulli's Principle

Bernoulli's equation

For an inviscid, incompressible, steady-flow fluid along a streamline:

P + ½·ρ·v² + ρ·g·h = constant

where:

• P — pressure (Pa).
• ρ — fluid density (kg/m³).
• v — fluid velocity (m/s).
• g — gravitational acceleration (9.81 m/s²).
• h — height (m).

This is energy conservation per unit volume — pressure work + kinetic energy + gravitational PE.

Key insights

Change in flow	Bernoulli's prediction	Example
v increases	P decreases	Air over airplane wing
v decreases	P increases	Pitot tube tip (air comes to rest)
h increases	P decreases (in static fluid, h up = lower P)	Pressure decreases with altitude
Pipe narrows (continuity: v_2 = v_1·A_1/A_2)	v↑, P↓	Venturi meter

Continuity equation

For incompressible fluid in a pipe of varying cross-section:

A·v = constant (along the pipe)

Or for compressible: ρ·A·v = constant (mass flow rate).

This says fluid speeds up where pipes narrow. Combine with Bernoulli — narrower pipes have higher velocity AND lower pressure.

Real-world Bernoulli applications

System	Bernoulli at work
Airplane wing	Curved upper surface → faster air → lower pressure → lift
Pitot tube (airspeed)	Stagnation point at tube tip; pressure difference indicates speed
Venturi meter	Pipe constriction; pressure drop measures flow rate
Magnus effect	Spinning ball; airflow asymmetry causes pressure difference, sideways force
Carburetor (older cars)	Air rushing through narrow throat creates low pressure, sucking in fuel
Spray bottles	Fast air drawn across tube creates low pressure, pulls liquid up
Hurricane eye	Inward-rushing air → fast → low pressure at center
Chimneys	Wind across top creates low pressure, draws smoke out

Lift on an airplane wing

Wing shape (airfoil) curves more on top than bottom. Air over top travels a longer path in the same time → faster v. By Bernoulli, top side has LOWER P than bottom.

Pressure difference × wing area = lift force.

For a typical commercial airplane:

• Cruise speed ~250 m/s.
• Air over top ~280 m/s, bottom ~250 m/s.
• Pressure difference ~ ½·ρ·(280² - 250²) = ½ × 0.4 × 15,900 ≈ 3,200 Pa.
• Wing area ~500 m² for big jet.
• Lift ~ 3,200 × 500 = 1.6 MN — supports 160 tonnes.

Note: Modern aerodynamics also emphasizes Newton's 3rd law (wing deflects air downward, gets pushed up). Both views coexist.

JavaScript — Bernoulli calculations

// Bernoulli equation: solve for any unknown
function bernoulliPressure(P1, v1, h1, v2, h2, rho) {
  // P1 + ½ρv1² + ρgh1 = P2 + ½ρv2² + ρgh2
  // Solve for P2
  return P1 + 0.5 * rho * v1 * v1 + rho * 9.81 * h1
       - 0.5 * rho * v2 * v2 - rho * 9.81 * h2;
}

// Pressure drop in a pipe constriction (h1 = h2)
function venturi(P1, v1, A1, A2, rho = 1000) {
  // Continuity: v2 = v1 * A1/A2
  const v2 = v1 * A1 / A2;
  // Bernoulli: ΔP = ½ρ(v1² - v2²)
  return P1 + 0.5 * rho * (v1*v1 - v2*v2);
}

// Pipe narrows from 10 cm² to 5 cm² (factor of 2)
console.log(`Venturi: P drops to ${venturi(101325, 1, 0.001, 0.0005).toFixed(0)} Pa from 101325`);
// v2 = 2 m/s, ΔP = ½ × 1000 × (1 - 4) = -1500 Pa, so P2 = 99,825 Pa

// Airplane lift estimate
function wingLift(rho, v_top, v_bottom, area) {
  // Pressure difference
  const dP = 0.5 * rho * (v_top*v_top - v_bottom*v_bottom);
  return dP * area;  // Force is downward pressure × area on bottom, but top has lower; net lift = dP·A
}

console.log(`Boeing 747 cruise lift: ${(wingLift(0.4, 280, 250, 500) / 1000).toFixed(0)} kN`);
// ~3,180 kN supports ~325 tonnes

// Pitot tube — measure airspeed
function airspeedFromPitot(deltaP, rho = 1.225) {
  // Stagnation: ½ρv² = deltaP. So v = √(2·ΔP/ρ)
  return Math.sqrt(2 * deltaP / rho);
}

console.log(`200 Pa pitot: ${airspeedFromPitot(200).toFixed(1)} m/s`); // ~18 m/s
console.log(`5000 Pa pitot: ${airspeedFromPitot(5000).toFixed(1)} m/s`); // ~90 m/s

// Hurricane: very low pressure at center from fast inflow
function hurricaneCenter(P_outside, v_outside, v_center, rho = 1.225) {
  return P_outside + 0.5 * rho * (v_outside*v_outside - v_center*v_center);
}

// Outside winds 0 m/s, eyewall winds 70 m/s (Cat 4)
console.log(`Hurricane center: ${hurricaneCenter(101325, 0, 70).toFixed(0)} Pa`);
// 101325 - 0.5 × 1.225 × 70² = 101325 - 3,000 = 98,325 (~970 mbar; matches real hurricanes)

Where Bernoulli matters

• Aerodynamics. Aircraft, helicopters, kites, projectiles. Lift, drag, pressure distribution.
• Hydraulics. Pipe flow, water hammers, dam spillways, pumps.
• HVAC. Air conditioning, ventilation, smoke control in buildings.
• Sports. Magnus effect on spinning balls, swimming/sailing aerodynamics.
• Medicine. Spirometers, blood flow analysis, atherosclerosis (narrowed arteries → fast flow → low pressure → can collapse).
• Industry. Venturi meters, ejectors, jet pumps, atomizers.
• Meteorology. Hurricane low-pressure centers, jet stream dynamics, wind speed estimation.

Common mistakes

• Applying it across streamlines. Bernoulli is per streamline; comparing two regions only works if they're on the same flow path or in equivalent flow conditions.
• Forgetting it's for inviscid flow. Real fluids have viscosity. Bernoulli is approximate; gradually fails for high-viscosity fluids or large pipes.
• Using for compressible flow at high speed. At Mach > 0.3, density changes substantially. Need compressible-flow Bernoulli (more complex).
• Confusing static and dynamic pressure. Static — pressure on a wall (P). Dynamic — kinetic-energy-equivalent (½ρv²). Total = stagnation pressure (sum). Pitot tubes measure stagnation - static = dynamic, giving velocity.
• Wrong direction for lift on airplane. Pressure is HIGHER on the BOTTOM, LOWER on top. Net force is UP. Easy to confuse "low" with "no" pressure.
• Treating airplane lift as 100% Bernoulli. Modern view — angle of attack and Newton's 3rd are major contributors. Bernoulli alone gives ~half the lift; deflection theory gives the other half. Both views are correct; they're equivalent.