Bernouli Theorem Definations, facts, derivation

 

The Bernoulli theorem, also known as Bernoulli's principle, is a fundamental concept in fluid dynamics that states that as the speed of a fluid (such as a gas or liquid) increases, its pressure decreases. The theorem is named after the Swiss mathematician and physicist Daniel Bernoulli, who first formulated it in his book "Hydrodynamica" published in 1738.

The basic idea behind the Bernoulli theorem is that fluids are considered as a collection of many small particles. When a fluid flows through a pipe or a channel, the particles at the center of the flow move faster than those near the walls, due to the restriction of the pipe or channel walls. The faster moving particles have less time to interact with the slower particles and therefore have less force to push against them. This decrease in force leads to a decrease in pressure, and the faster moving fluid particles create a lower pressure zone compared to the slower particles near the walls.

The Bernoulli theorem can be expressed mathematically as the equation of energy conservation: the sum of the fluid's internal energy, kinetic energy, and potential energy remains constant along a streamline in a steady flow. This equation is known as the Bernoulli equation and can be used to calculate the pressure difference between two points in a fluid flow.

One of the most common applications of the Bernoulli theorem is in the design of airfoils, such as airplane wings. When air flows over the curved surface of an airfoil, it moves faster over the curved upper surface and slower over the flat lower surface. This difference in speed creates a difference in pressure, with the lower pressure region being on top of the wing. This lower pressure region generates lift, which enables an airplane to fly.

Another example of the Bernoulli theorem in action is seen in fluid piping systems. When a fluid flows through a pipe with a constricted section, the speed of the fluid increases, which leads to a decrease in pressure. This pressure difference can be used to create a pumping action, which is the principle behind many types of pumps, such as centrifugal pumps and positive displacement pumps.

The Bernoulli theorem can also be used to explain the operation of hydraulic systems, such as those used in automobiles, construction equipment, and aircraft landing gear. In a hydraulic system, fluid is forced through a pipe or channel, and the pressure difference generated by the fluid flow can be used to produce a mechanical force. This force is proportional to the pressure difference and can be used to perform work, such as lifting a load or generating motion.

In addition to its applications in fluid mechanics and engineering, the Bernoulli theorem has numerous other applications in various fields, including meteorology, physiology, and even sports. For example, the Bernoulli theorem can be used to explain why a baseball thrown with spin will move differently than one without spin, or why a golf ball with dimples will travel farther than a smooth golf ball.

It's important to note that the Bernoulli theorem only applies to incompressible fluids, meaning fluids that do not change volume as their pressure changes. This means that gases cannot be described by the Bernoulli equation, as they are highly compressible and will change their volume as their pressure changes.

In conclusion, the Bernoulli theorem is a fundamental concept in fluid mechanics that states that as the speed of a fluid increases, its pressure decreases. It has numerous practical applications in fields such as engineering, meteorology, physiology, and sports, and can be used to calculate pressure differences in fluid flow and to explain the operation of various systems, such as airfoils, pumps, and hydraulic systems.