13.18:
Bernoulli’s Equation
Imagine an incompressible fluid passing through a tube. If the tube's cross-sectional area narrows in a particular region, the fluid’s flow speeds up.
This suggests that the kinetic energy of the fluid in this region is more. Since the principle of conservation of energy holds true, it implies that net work is done on this part of the fluid. That is, a force acts on the fluid to speed it up.
The force arises from the pressure difference across regions with different cross-sectional areas. Moreover, if the height of the two regions is different, the gravitational force also does work on the fluid.
Bernoulli's Equation is obtained by applying the work-energy theorem to an incompressible fluid in a laminar, steady flow. It states that the sum of pressure, kinetic energy density, and gravitational energy density is constant over time and over a streamline.
13.18:
Bernoulli’s Equation
In the middle of the nineteenth century, it was observed that two trains passing each other at a high relative speed get pulled towards each other. The same occurs when two cars pass each other at a high relative speed. The reason is that the fluid pressure drops in the region where the fluid speeds up. As the air between the trains or the cars increases in speed, its pressure reduces. The pressure on the outer parts of the vehicles is still the atmospheric pressure, while the resultant pressure difference creates a net inward force on the vehicles.
Bernoulli’s equation describes the relationship between fluid pressure and its speed. It is named after David Bernoulli (1700–1782), who published his studies on fluid motion in the book Hydrodynamica (1738). The equation is deduced by applying the principle of conservation of energy to a frictionless, steady, laminar fluid flow. Although it describes an ideal condition that fluids do not practically exhibit in real life, it almost holds true in many real-life situations and helps analyze them.
The three pressure terms in Bernoulli's equation are the fluid pressure, the kinetic energy of the fluid per unit volume, and the gravitational potential energy per unit volume. The latter two have the dimensions of pressure and are also called kinetic and gravitational energy densities. The equation states that the total pressure of the fluid, including the energy densities, is constant. As the fluid flows, the fluid pressure changes to accommodate the kinetic and gravitational energy contributions.
This text is adapted from Openstax, University Physics Volume 1, Section 14.6: Bernoulli's Equation.