Table of Contents
What is Bernoulli's Equation?
Bernoulli's equation is one of the most fundamental principles in fluid dynamics. Named after Swiss mathematician Daniel Bernoulli, it describes the conservation of energy in a flowing fluid. The equation states that for an ideal, incompressible fluid flowing along a streamline, the sum of pressure energy, kinetic energy, and potential energy remains constant.
This principle explains many everyday phenomena: why airplane wings generate lift, why a shower curtain gets pulled inward when the water is running, and why a curveball curves. When fluid velocity increases, pressure decreases, and vice versa. This inverse relationship between velocity and pressure is the core insight of Bernoulli's principle.
The Bernoulli Equation
Where P is static pressure, ρ is fluid density, v is flow velocity, g is gravitational acceleration (9.81 m/s²), and h is the elevation above a reference point.
Assumptions and Limitations
- The fluid must be incompressible (constant density) - valid for liquids and low-speed gas flows.
- The flow must be steady (not changing with time at any fixed point).
- The flow must be along a streamline.
- There must be no friction (inviscid flow) - real fluids have viscosity that causes energy losses.
- No energy is added or removed (no pumps, fans, or turbines in the flow path).
Real-World Applications
| Application | How Bernoulli Applies |
|---|---|
| Airplane lift | Faster air over curved wing top creates lower pressure than below |
| Venturi meter | Flow speeds up through narrow section, pressure drop measures flow rate |
| Pitot tube | Stagnation pressure minus static pressure gives dynamic pressure for airspeed |
| Carburetor | Air speeds through venturi, low pressure draws fuel into airstream |
Frequently Asked Questions
Can Bernoulli's equation be used for compressible flow?
The standard Bernoulli equation assumes incompressible flow. For compressible flows (high-speed gases above about Mach 0.3), you must use a modified version that accounts for changes in density. The compressible form involves specific heat ratios and is considerably more complex.
What is the difference between static and dynamic pressure?
Static pressure is the pressure exerted by the fluid at rest, while dynamic pressure (1/2 rho v squared) represents the kinetic energy per unit volume of the flowing fluid. The sum of static and dynamic pressure is called total or stagnation pressure. A pitot tube measures total pressure, and the difference from static pressure gives the dynamic pressure, from which velocity can be calculated.
Why does Bernoulli not apply to viscous flows?
Bernoulli's equation is derived from conservation of energy along a streamline with no energy losses. Viscous fluids experience internal friction that converts kinetic energy into heat, causing energy loss. For viscous flows, engineers use the modified Bernoulli equation with a head loss term, or use the full Navier-Stokes equations for accurate results.