Bernoulli Equation Calculator

The Bernoulli equation relates the pressure, velocity, and elevation of a fluid at two points along a streamline. It states that the sum of static pressure, dynamic pressure (½ρv²), and hydrostatic pressure (ρgh) remains constant for an incompressible, steady, non-viscous fluid. By knowing five of the six variables, you can solve for the sixth.

The Bernoulli equation is: P + ½ρv² + ρgh = constant, where P is static pressure (Pa), ρ is fluid density (kg/m³), v is flow velocity (m/s), g is gravitational acceleration (m/s²), and h is elevation (m). When comparing two points: P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂.

The four key assumptions are: (1) the fluid is incompressible, meaning its density is constant; (2) the flow is steady and does not change with time; (3) the fluid is inviscid, meaning viscosity (internal friction) is negligible; and (4) the equation applies along a single streamline within the flow.

The Bernoulli equation is used widely in engineering and physics to analyze pipe flow, calculate pressure drops in venturi meters and nozzles, explain lift on airplane wings, design hydraulic systems, and study wind behavior around structures. It is a foundational principle in fluid mechanics and aerodynamics.

Airplane wings (airfoils) are shaped so that air travels faster over the curved top surface than the flatter bottom surface. According to Bernoulli's principle, faster-moving air exerts lower pressure. This pressure difference between the lower (higher pressure) and upper (lower pressure) surfaces of the wing generates an upward lift force that keeps the aircraft airborne.

Dynamic pressure is the kinetic energy component of the fluid per unit volume, given by ½ρv². It represents the pressure exerted by the fluid due to its motion. When velocity increases, dynamic pressure increases, and static pressure decreases proportionally so that the total Bernoulli constant remains the same.

The standard Bernoulli equation applies to incompressible fluids (like water) where density remains constant regardless of pressure changes. For compressible fluids (like gases at high speeds), density changes with pressure, and a modified compressible form of Bernoulli's equation must be used, which accounts for changes in fluid density along the flow.

In SI units, pressure is measured in Pascals (Pa), density in kilograms per cubic metre (kg/m³), velocity in metres per second (m/s), elevation in metres (m), and gravitational acceleration in m/s². Each term in the Bernoulli equation has units of Pascals (Pa), representing energy per unit volume.