Bernoulli's Equation

Bernoulli's Equation states that the sum of pressure, kinetic energy density, and gravitational energy density remains constant along a streamline in an incompressible, laminar, steady fluid flow. It describes the relationship between fluid pressure and its speed, showing how total pressure accommodates changes in kinetic and gravitational energy as fluid flows through regions of varying cross-sectional area and height.

When a tube's cross-sectional area narrows, the fluid must speed up to maintain continuous flow. This acceleration increases the fluid's kinetic energy. A pressure difference across regions with different cross-sectional areas provides the force necessary to accelerate the fluid, demonstrating how energy conservation and pressure differences work together in fluid motion.

Fluid pressure drops in regions where fluid speed increases. As the fluid accelerates, more of its total energy is converted to kinetic energy, leaving less available as pressure energy. This inverse relationship between pressure and velocity explains phenomena like trains being pulled together at high relative speeds, where reduced air pressure between them creates a net inward force.

Bernoulli's Equation is derived by applying the work-energy theorem to an incompressible fluid in laminar, steady flow. The theorem accounts for work done by pressure forces and gravitational forces on the fluid, connecting these forces to changes in kinetic energy and establishing that total pressure remains constant along a streamline.

The three terms are fluid pressure, kinetic energy density (kinetic energy per unit volume), and gravitational potential energy density (gravitational potential energy per unit volume). The latter two have pressure dimensions and represent the energy contributions of fluid motion and elevation. Together, these three terms sum to a constant value along a streamline.

Although Bernoulli's Equation describes an ideal condition assuming frictionless, laminar flow that real fluids do not perfectly exhibit, it holds true in many practical situations and helps analyze them. The equation provides valuable insights into real fluid behavior despite its idealized assumptions about friction and flow characteristics.

When two regions of fluid flow are at different heights, gravitational force does work on the fluid, affecting its energy distribution. Bernoulli's Equation accounts for this gravitational potential energy density as one of its three terms, ensuring that total pressure remains constant even when elevation changes along the flow path.