Where fluid speeds up, pressure drops. The
Bernoulli equation P + ½ρv² + ρgh = const is
conservation of energy per unit volume. Combined with the
continuity equation A₁v₁ = A₂v₂, it explains
carburettors, aeroplane lift, and perfume atomisers.
Flow
Results
v₁2.00 m/s
v₂5.00 m/s
P₁101.3 kPa
P₂88.7 kPa
ΔP12.6 kPa
Flow Q—
Fluid
How it works
The continuity equation demands that fluid mass is
conserved: A₁v₁ = A₂v₂. So where the pipe narrows (A₂ < A₁),
velocity must increase. Bernoulli then tells us pressure must fall to
keep total energy constant: P₂ = P₁ + ½ρ(v₁² − v₂²).
The Venturi meter exploits this to measure flow rate:
by measuring the pressure difference ΔP between the wide and narrow
sections, the flow rate Q = A₁ · √(2ΔP / ρ(1/R²−1)) is known
precisely.
Real-world applications
Carburettors (air speeds up → fuel drawn in), Pitot tubes (aircraft
speed), aeroplane wings (air faster over curved top → lower pressure →
lift), perfume atomisers, and Bunsen burner air intake all use
Bernoulli's principle.
About the Bernoulli Equation
Bernoulli's equation is one of the most elegant results in fluid dynamics, derived by Daniel Bernoulli in 1738. It states that for an inviscid, incompressible fluid in steady flow, the total mechanical energy per unit volume remains constant along a streamline: P + ½ρv² + ρgh = constant, where P is static pressure, ½ρv² is dynamic pressure (kinetic energy density), and ρgh is hydrostatic pressure (potential energy density). This principle explains lift on aircraft wings, the curve of a spinning football, and why a fast-flowing river runs shallow while a slow river runs deep.
This simulation models fluid flow through a pipe with a constriction (Venturi tube). You can adjust inlet pressure, inlet velocity, and the narrowing ratio of the pipe to observe in real time how pressure drops as velocity increases through the constriction — and recover as the pipe widens again.
Frequently Asked Questions
What does Bernoulli's equation state, and what are its assumptions?
Bernoulli's equation P + ½ρv² + ρgh = const states that the sum of static pressure, dynamic pressure, and hydrostatic pressure is constant along a streamline. The key assumptions are: the fluid is inviscid (no viscosity), incompressible (constant density), steady (flow does not change with time), and irrotational (no turbulence). Real fluids violate these assumptions to varying degrees — viscosity causes energy loss, and turbulence dissipates kinetic energy as heat.
Why does pressure decrease when fluid speeds up in a constriction?
By conservation of mass (continuity equation), A₁v₁ = A₂v₂: a smaller cross-section requires higher velocity to carry the same volumetric flow rate. By conservation of energy (Bernoulli), the total energy per unit volume is constant, so the kinetic energy increase (½ρv²) must be offset by a pressure decrease. This is directly exploited in Venturi meters, carburettors, and aircraft Pitot tubes to measure flow velocity from pressure differences.
How does Bernoulli's principle explain lift on aircraft wings?
An aerofoil is shaped so that air flows faster over the curved upper surface than the flatter lower surface. By Bernoulli's equation, faster flow corresponds to lower pressure, so the upper surface has lower pressure than the lower surface. This pressure difference creates a net upward force — lift — on the wing. For a Boeing 747 at cruise, the pressure difference is about 200 Pa, acting on ~500 m² of wing area to produce ~100 kN of lift per wing.
What is the Venturi effect and how is it used?
The Venturi effect is the pressure drop that occurs when a fluid passes through a constriction. It is exploited in Venturi meters (measuring gas or liquid flow rates), carburettors (drawing fuel into an air stream by creating low pressure), Venturi scrubbers (drawing pollutant-laden air into a water spray), and medical nebulisers (creating fine droplets of medication). The Venturi tube was invented by Giovanni Venturi in 1797 and popularised by Clemens Herschel in the 1880s for water flow measurement.
What is the continuity equation for fluids?
The continuity equation expresses conservation of mass in fluid flow: for an incompressible fluid, the volumetric flow rate Q = Av is constant throughout the flow, where A is the cross-sectional area and v is the flow velocity. If a pipe narrows from area A₁ to A₂, the velocity increases from v₁ = Q/A₁ to v₂ = Q/A₂. For compressible flows (e.g., air at high speed), the equation becomes ρAv = constant, where density ρ also changes.
What is the Magnus effect?
The Magnus effect is the sideways force on a spinning object moving through a fluid. A spinning football curves because the spin drags air with it, accelerating it on one side (lower pressure) and decelerating it on the other (higher pressure), producing a pressure difference and sideways force. A football kicked at 70 mph with 600 rpm of spin can curve by over 1 metre. The same principle steers tennis balls, curve balls in baseball, and is being explored for wind-powered cargo ships (Flettner rotors).
What is dynamic pressure?
Dynamic pressure q = ½ρv² represents the kinetic energy of the fluid per unit volume. It is the pressure that would be needed to bring the fluid from velocity v to rest at a stagnation point. Pitot tubes measure dynamic pressure by pointing directly into the flow: the stagnation pressure at the tube tip minus the static pressure measured at a side port equals the dynamic pressure, giving airspeed v = √(2q/ρ). This is how aircraft airspeed indicators and meteorological anemometers work.
Where does Bernoulli's equation break down?
Bernoulli breaks down whenever any assumption fails significantly: in turbulent flow (kinetic energy is dissipated as heat by eddies), in viscous flow through narrow tubes (Hagen-Poiseuille flow replaces Bernoulli, with pressure dropping linearly with length), at high speeds where compressibility matters (above Mach ~0.3 for air), and near solid boundaries where the no-slip condition creates a viscous boundary layer. In these regimes, the Navier-Stokes equations must be solved instead.
How does Bernoulli's principle explain the curve ball in cricket?
A cricket ball with a polished side and rough side will swing in flight. The rough side creates turbulent boundary layer separation at a smaller angle, while the polished side maintains laminar flow further around the ball. This creates a pressure difference perpendicular to the flight path — late swing — curving the ball by up to 30 cm. At speeds above about 85 mph, the flow transitions to turbulent on both sides and the ball stops swinging; below about 65 mph it doesn't swing either, creating the "swing window" around 70–80 mph.
What is cavitation?
Cavitation occurs when the local pressure in a flowing liquid drops below the vapour pressure, causing the liquid to boil locally and form bubbles of vapour. When these bubbles collapse in a higher-pressure region downstream, they implode violently, producing intense shock waves and noise. Cavitation erodes ship propellers, pump impellers, and hydraulic turbines — pitting the metal surface — and must be avoided by keeping flow velocities below cavitation-inducing limits. In medical ultrasound, controlled cavitation can break up kidney stones (lithotripsy).