What is Bernoulli's equation?

Bernoulli's equation states that for steady, incompressible, frictionless flow along a streamline, the total energy per unit weight is constant: p/(rho g) + v^2/(2g) + z = constant. The three terms are the pressure head, velocity head, and elevation head.

What are the assumptions of Bernoulli's equation?

The flow is steady, incompressible, non-viscous (frictionless), and along a single streamline, with no energy added or removed by pumps or turbines. Real flows include friction, so a head-loss term is added for engineering use.

Why does pressure drop where velocity increases?

Because total energy is conserved: where a fluid speeds up (higher velocity head), the pressure head must fall to keep the sum constant. This is the principle behind venturimeters, aerofoil lift, and the carburettor - faster flow means lower pressure.

Bernoulli's Equation — Derivation, Assumptions, Applications, and Worked Examples

Bernoulli's equation is the cornerstone of fluid mechanics — a statement of energy conservation for a flowing fluid. It explains how aircraft wings lift, how a venturimeter measures flow, and how water pressure changes through a pipe network. This article derives it carefully and applies it to numerical examples.

The Equation

p/(ρg) + v²/(2g) + z = constant   (along a streamline)

Term	Name	Represents
p/(ρg)	Pressure head	Energy due to pressure
v²/(2g)	Velocity (kinetic) head	Energy due to motion
z	Elevation (potential) head	Energy due to position

Each term has units of length (metres of fluid), and their sum — the total head — stays constant for ideal flow.

Derivation (Energy Approach)

Consider a fluid element moving along a streamline. Applying the work-energy principle (or integrating Euler's equation of motion along the streamline) between two points 1 and 2 for steady, incompressible, frictionless flow:

p₁/(ρg) + v₁²/(2g) + z₁ = p₂/(ρg) + v₂²/(2g) + z₂

The pressure work, kinetic energy change and potential energy change balance exactly because no energy is dissipated. This is Bernoulli's theorem.

Assumptions and Limitations

Steady flow (conditions do not change with time).
Incompressible fluid (constant density — valid for liquids and low-speed gas flow).
Non-viscous (no friction losses).
Flow along a single streamline; no energy added/removed by machines.

For real pipe flow we add a head-loss term h_f (and pump/turbine heads) to get the engineering energy equation:

p₁/ρg + v₁²/2g + z₁ + h_pump = p₂/ρg + v₂²/2g + z₂ + h_f

The Continuity Equation (Always Used Alongside)

A₁ v₁ = A₂ v₂ = Q   (conservation of mass for incompressible flow)

Worked Example 1 — Pipe Contraction

Water flows through a horizontal pipe that contracts from 100 mm to 50 mm diameter. At the 100 mm section the velocity is 2 m/s and the pressure is 200 kPa. Find the velocity and pressure at the 50 mm section. (ρ = 1000 kg/m³)

Continuity: v₂ = v₁ (A₁/A₂) = v₁ (d₁/d₂)² = 2 × (100/50)² = 2 × 4 = 8 m/s
Bernoulli (z₁ = z₂): p₂ = p₁ + ½ρ(v₁² − v₂²) = 200 000 + 500 × (4 − 64) = 200 000 − 30 000 = 170 kPa

The fluid speeds up and the pressure drops — exactly as Bernoulli predicts.

Worked Example 2 — Venturimeter Discharge

A horizontal venturimeter has inlet 200 mm and throat 100 mm. The pressure-head difference between inlet and throat is 1.5 m of water. Find the theoretical discharge. (C_d = 0.98)

A₁ = π/4 × 0.2² = 0.03142 m²; A₂ = π/4 × 0.1² = 0.007854 m²
Q = C_d · (A₁A₂)/√(A₁² − A₂²) · √(2g·Δh)
√(A₁² − A₂²) = √(9.87×10⁻⁴ − 6.17×10⁻⁵) = √(9.25×10⁻⁴) = 0.03042
√(2 × 9.81 × 1.5) = √29.43 = 5.425
Q = 0.98 × (0.03142 × 0.007854 / 0.03042) × 5.425 = 0.98 × 0.008113 × 5.425 = 0.0431 m³/s ≈ 43 L/s

Applications

Flow measurement: venturimeter, orifice meter, Pitot tube.
Aerofoil lift and the curveball effect.
Pipe network analysis (with head-loss terms).
Spillways, nozzles and jets.

Common Mistakes

Applying Bernoulli across a pump, turbine or region of strong turbulence without the extra energy/loss terms.
Forgetting to use continuity to relate the two velocities.
Mixing gauge and absolute pressures inconsistently between the two points.