Bernoulli's equation relates the energy conservation in a fluid moving along a streamline. The equation applies to incompressible and inviscid fluids under steady flow. For such a flow, Newton's second law is applied to a small fluid element, which experiences forces due to pressure differences, gravity, and velocity variations. The force balance leads to the following form of Bernoulli's equation:

Here, P is the pressure, ρ is the density of the fluid, v is velocity, g denotes the acceleration due to gravity, and elevation is denoted by h. Each term in this equation represents energy per unit volume of the fluid.

Consider water flowing through a horizontal pipe with two sections of different diameters. At point 1, the pipe has a large diameter, while at point 2, the pipe narrows. Bernoulli's equation tells us that if the velocity of the water increases at the narrower section (point 2), the pressure must decrease to conserve energy along the streamline. Applying Bernoulli's equation to the pipe sections gives:

Bernoulli's principle is crucial in understanding and designing fluid flow systems. For instance, in water distribution networks, variations in pipe diameters cause pressure changes that affect flow efficiency. As the equation dictates, a narrower pipe increases velocity while reducing pressure, ensuring consistent supply across urban systems. In dam spillways, the fluid's potential energy due to elevation is converted into kinetic energy as water descends. This results in an increase in velocity and a corresponding drop in pressure, a phenomenon used to design spillways capable of safely managing varying discharge rates.