Bernoulli's principle

In a pipe of varying cross-section and height, continuity forces the fluid's speed to change. The acceleration needs a net force from the surrounding fluid, so the pressure must differ from region to region.

Daniel Bernoulli (1738) related the pressure difference between two points to the change in speed (kinetic energy) and the change in height (potential energy), using conservation of energy.

Work done on a slug of fluid of volume ΔV: P₁ΔV at the inlet end minus P₂ΔV at the outlet end, (P₁ − P₂)ΔV. It supplies ΔK = ½ρΔV(v₂² − v₁²) and ΔU = ρgΔV(h₂ − h₁).

Dividing by ΔV: P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂, or P + ½ρv² + ρgh = constant along a streamline.

In words: along a streamline, pressure + kinetic energy per unit volume (½ρv²) + potential energy per unit volume (ρgh) stays constant.

Assumptions: no energy lost to internal friction (zero viscosity), an incompressible fluid (no elastic energy), and steady flow. It fails for turbulent or non-steady flow, where speed and pressure fluctuate. It still works well for low-viscosity, incompressible fluids.

Fluid at rest (v = 0 everywhere): Bernoulli's equation reduces to P₁ − P₂ = ρg(h₂ − h₁), the same as the result for pressure with depth.

Horizontal pipe: where the fluid moves faster, its pressure is lower. A narrow throat therefore has lower pressure than the wide part feeding it.

Torricelli's law (efflux means outflow): tank of liquid, small side hole at height y₁, surface at y₂ with pressure P above it. If the tank area is much larger than the hole, the surface is nearly at rest, and v₁ = √[2gh + 2(P − P_a)/ρ] with h = y₂ − y₁.

When P ≫ P_a and 2gh can be ignored, the container pressure alone sets the outflow speed, as in rocket propulsion.

Tank open to the air (P = P_a): v₁ = √(2gh), the same speed as a body falling freely through h.