Bernoulli's Equation
Simple Explanation
Bernoulli's equation — P+½ρv²+ρgh=constant — says that along a flow, pressure energy, kinetic energy, and gravitational potential energy (all per unit volume) trade off against each other, but their total stays the same.
Why Do We Need It?
This is the central law of fluid dynamics — it explains why fast-flowing fluid has lower pressure, and underlies everything from aircraft lift to how a drinking straw works.
See It
A pipe that starts wide and narrows to a smaller cross-section, with P1 v1 labelled in the wide, slower, higher-pressure section, and P2 v2 labelled in the narrow, faster, lower-pressure section
Formula
Bernoulli's Equation
P + ½ρv² + ρgh = constant
Along a streamline in a flowing, incompressible, non-viscous fluid, the sum of pressure energy, kinetic energy per volume, and gravitational potential energy per volume stays constant.
- P
- — pressure at a point in the fluid, in pascals (Pa)
- ρ
- — fluid density, in kilograms per cubic metre (kg/m³)
- v
- — flow speed at that point, in metres per second (m/s)
- h
- — height of that point above a reference level, in metres (m)
- g
- — acceleration due to gravity, 9.8 m/s²
When to use it: Whenever relating pressure, speed, and height at two different points along the same flow.
Worked Example
Find pressure using Bernoulli's equation
Water (ρ=1000 kg/m³) flows through a horizontal pipe (constant height) at P₁=200000 Pa and v₁=2 m/s in a wide section, speeding up to v₂=8 m/s in a narrow section. Find P₂.
Why Does This Work?
Bernoulli's equation is really a statement of energy conservation applied to a flowing fluid — as the fluid speeds up (gaining kinetic energy) in the narrow section, it must lose an equal amount of pressure energy to keep the total constant, exactly as the narrow section's lower P₂ shows here.
Real-Life Example
Why a shower curtain gets pulled inward
Running a shower on a windy day, the shower curtain sometimes billows inward toward the fast-moving stream of water and air.
The fast-moving air inside the shower stream has lower pressure than the still air outside the curtain, by Bernoulli's equation — the higher outside pressure pushes the curtain inward.
Practice
Water (ρ=1000 kg/m³) flows through a horizontal pipe at P₁=180000 Pa and v₁=1 m/s, speeding up to v₂=5 m/s. Find P₂.
HardCommon mistake
Forgetting that Bernoulli's equation (in this simple form) assumes non-viscous, incompressible, steady flow along a single streamline — applying it across turbulent or highly viscous flow gives an inaccurate result.
Quick Review
- P+½ρv²+ρgh = constant along a streamline.
- Faster flow ⟹ lower pressure (at the same height).
- A statement of energy conservation for a flowing fluid.