✈️ Bernoulli's Principle — Airfoil Lift & Flow

Static pressure is the pressure exerted by a fluid at rest relative to the measurement surface — the force per unit area due to molecular impacts. Dynamic pressure (q = 1/2 rho v^2) is the kinetic energy per unit volume of the moving fluid; it represents the pressure rise that occurs when a moving fluid is brought to rest (stagnated). A Pitot tube measures total (stagnation) pressure P + q, while a static port alongside the aircraft skin measures P; the difference gives the dynamic pressure, from which airspeed is calculated. All aircraft airspeed indicators work on this principle.

The Magnus effect is the lateral force on a spinning object moving through a fluid, caused by asymmetric boundary-layer growth: the side spinning into the flow sees increased relative velocity and lower pressure (Bernoulli), while the opposite side sees reduced velocity and higher pressure. The pressure difference pushes the object toward the low-pressure side. The effect causes topspin tennis balls to dip, footballs to swerve in free-kicks, and golf balls to follow curved trajectories. Flettner rotors — tall spinning cylinders used on cargo ships — also exploit the Magnus effect to provide auxiliary propulsion, cutting fuel consumption by up to 25%.

Stall occurs when the angle of attack of a wing exceeds the critical angle (typically 15–20° for most aerofoils) at which the smooth, attached airflow over the upper surface separates into turbulent, reversed-flow eddies. This dramatically reduces lift (coefficient of lift falls sharply) and increases drag. In an aircraft, stall means the wings can no longer support the aircraft's weight. Recovery requires reducing angle of attack — often by pushing the nose down, which feels counterintuitive when close to the ground. Accelerated stalls can occur at any speed if g-loading increases angle of attack suddenly, such as in a tight turn.

The Coanda effect is the tendency of a fluid jet to adhere to and follow a nearby curved surface rather than continuing in a straight line; the entrainment of surrounding fluid by the jet creates a low-pressure region between the jet and the surface, drawing it in. While Bernoulli describes pressure variations along a streamline for attached, streamlined flow, the Coanda effect involves the entrainment and redirection of the jet by surface curvature. Aircraft high-lift devices (blown flaps, circulation control wings) exploit the Coanda effect to delay separation and increase maximum lift coefficient significantly beyond what wing shape alone can achieve.

Torricelli's theorem states that the speed of fluid exiting a small hole in the side of a large tank is v = sqrt(2 g h), where h is the depth of the hole below the free surface. It is a direct application of Bernoulli's equation between the still surface (v ≈ 0, height h) and the exit point (height 0, velocity v): 0 + rho g h = 1/2 rho v^2, giving v = sqrt(2 g h). This means a hole 1 metre below a tank's waterline produces an exit velocity of about 4.4 m/s, regardless of the tank's cross-section.

A running shower creates a spray that drags air inward and downward, reducing air pressure inside the shower compared to outside. This pressure difference (a Bernoulli effect from the moving air-water mixture rather than the water jet directly) pushes the lightweight curtain toward the lower-pressure interior. A secondary effect is the entrainment of room air by the spray, which further reduces interior pressure. Engineers designing curtain replacements occasionally use curved weighted rods or "hurricane" style inward-curving curtains to increase the gap and reduce the inward billowing force.

Bernoulli's equation assumes steady, incompressible, inviscid flow along a single streamline — assumptions that break down in many real situations. Viscosity causes the boundary layer to develop along surfaces, and when flow separates (as in stall), Bernoulli does not apply in the separated region. At speeds above Mach 0.3 (about 370 km/h in air at sea level), compressibility effects become significant and the equation must be replaced with the compressible form. For gases in nozzles and jet engines, or for supersonic flows, Euler's equations or the full Navier-Stokes equations are required for accurate analysis.