Venturi tube: speed up, pressure down.

• 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.

Teaching this idea? Put the simulation on your own page or LMS with this code.