Bernoulli's equation is a simple relation that can give useful insight into the balance among fluid pressure, flow speed, and elevation. It applies exclusively to ideal fluids with steady flow, that is, fluids with a constant density and no internal friction forces, whose flow patterns do not change with time. Despite its limitations, however, Bernoulli's equation is an essential tool in understanding the behavior of fluids in many practical applications, from plumbing systems to the flight of airplanes.
For a fluid element of density ρρ that flows along a streamline, Bernoulli's equation states that
p1+ρgh1+12ρv21=p2+ρgh2+12ρv22p1+ρgh1+12ρv12=p2+ρgh2+12ρv22,
whereppis the pressure,vvis the flow speed,hhis the height,ggis the acceleration due to gravity, and subscripts 1 and 2 refer to any two points along the streamline. The physical interpretation of Bernoulli's equation becomes clearer if we rearrange the terms of the equation as follows:
p1−p2=ρg(h2−h1)+12ρ(v22−v21)p1−p2=ρg(h2−h1)+12ρ(v22−v12).
The term p1−p2p1−p2 on the left-hand side represents the total work done on a unit volume of fluid by the pressure forces of the surrounding fluid to move that volume of fluid from point 1 to point 2. The two terms on the right-hand side represent, respectively, the change in potential energy, ρg(h2−h1)ρg(h2−h1), and the change in kinetic energy, 12ρ(v22−v21)12ρ(v22−v12), of the unit volume during its flow from point 1 to point 2. In other words, Bernoulli's equation states that the work done on a unit volume of fluid by the surrounding fluid is equal to the sum of the change in potential and kinetic energy per unit volume that occurs during the flow. This is nothing more than the statement of conservation of mechanical energy for an ideal fluid flowing along a streamline