Glossary term
Bernoulli Equation
An energy relation for steady flow that links pressure, velocity, and elevation along a streamline under defined assumptions.
Definition
modelAn energy relation for steady flow that links pressure, velocity, and elevation along a streamline under defined assumptions.
The Bernoulli equation is a simplified mechanical-energy balance for fluid flow. It is powerful for first estimates and instrumentation concepts, but it must be modified or replaced when viscosity, pumps, turbines, compressibility, heat transfer, unsteadiness, or losses are important.
The Bernoulli equation relates pressure, velocity, and elevation in a flowing fluid. A common head form for steady, incompressible, inviscid flow along a streamline is:
where p is pressure, \rho is density, v is flow speed, g is gravitational acceleration, and z is elevation. Each term can be interpreted as pressure head, velocity head, and elevation head.
Engineering role
The equation is widely used for first-pass analysis of pipes, nozzles, tanks, siphons, aircraft pitot-static systems, flow meters, and hydraulic systems. It helps explain why velocity rises when static pressure falls in a constriction, and why pressure changes with elevation. In practice, it is often combined with the continuity equation to relate area, velocity, and flow rate.
Assumptions
The basic Bernoulli equation assumes steady flow, negligible viscosity, no shaft work, no heat transfer, constant density, and evaluation along a streamline. These assumptions are restrictive. Real pipe systems need head-loss terms. Pumps and turbines add or remove energy. Compressible gas flow requires additional treatment. Unsteady flows, separated flows, shocks, and strong turbulence cannot be handled by the elementary equation alone.
Engineering extensions
For real fluid systems, engineers often write an extended energy equation with pump head, turbine head, and loss head. Venturi meters and orifice plates use Bernoulli-style pressure differences, but require discharge coefficients because viscosity, separation, and geometry affect the actual flow. In aerodynamic applications, Bernoulli reasoning is valid only where the flow assumptions are appropriate; it should not be used as a standalone explanation for all lift or compressible-flow effects.
Common mistakes
Common mistakes include applying Bernoulli across a pump, fan, turbine, or major loss without adding the relevant energy term; mixing gauge and absolute pressure incorrectly; and using the equation between points not connected by a streamline. Another error is ignoring velocity-profile and loss effects in small pipes, valves, elbows, and orifices where viscosity and separation dominate.