Bernoulli's principle states that the total mechanical energy of a flowing fluid—consisting of pressure energy, kinetic energy, and potential energy—remains constant along a streamline in frictionless, incompressible flow. Mathematically, E = P + ½ρv² + ρgh = constant. This elegant relationship reveals a profound trade-off: when fluid accelerates (v increases), pressure must decrease to maintain constant total energy. This counterintuitive phenomenon explains phenomena from aircraft lift to the behavior of water jets and explains why fast-moving fluids exert less pressure than slow-moving ones. Bernoulli's equation is valid for inviscid (frictionless), incompressible, steady-state flow along a single streamline.

The three components of Bernoulli's constant represent different forms of mechanical energy. Pressure energy (P) is work per unit volume done by pressure forces. Kinetic energy (½ρv²) represents motion energy proportional to density and velocity squared. Potential energy (ρgh) represents gravitational energy based on elevation. These three terms must sum to the same value at any point along a streamline. In real applications, friction losses (viscous effects) cause the Bernoulli constant to decrease slightly as fluid flows, requiring the extended Bernoulli equation with loss terms. Modern fluid dynamics uses Bernoulli's equation as a foundation for understanding aerodynamic lift, pump design, fluid flow measurement, and countless engineering applications. The principle fails for compressible flows (high-speed gases), turbulent flows with significant energy dissipation, and non-streamline conditions.