Thursday, January 31, 2019

Bernoulli's Theorem: Statement, Derivation And Application



Bernoulli's Theorem

Daniel Bernoulli, a Swiss mathematician and physicist stated a theorem which gives the interaction between the pressure acting at a point on the surface of the liquid and the velocity of its particles. Bernoulli's theorem states that total energy of a small amount of an incompressible liquid flowing from one point to another remains constant throughout the displacement.


Derivation: Consider a fluid moves through a tube of an area of cross section A1 and A2 respectively.

The volume of water entering A1 per second = A1V1
The volume of water leaving A2 per second = A2V2

Therefore, the mass of liquid entering per second at A1 = p1A1V1
The mass of liquid leaving per second at A2 = p2A2V2

Assuming, there is no loss of fluid in the tube that each,

Therefore,
or                                AV = Constant

this is the equation of continuity.

When fluid flows through a section of pipe with one end having a smaller cross-sectional area than the pipe at the other end. The velocity of the fluid in the constricted end must be greater than the velocity at the larger end. Bernoulli's equation applies conservation of energy to formalize this observation.

Consider a tube AB of varying cross-section A1 and A2 and a different heights H1 and H2. A liquid is flowing from A to B,

                           P1 > P2

Here,                   A1 > A2
So,                      V1 < V2

The force on the liquid at A = P1A1
The force on the liquid at B = P2A2

Now, the work done per second on the liquid at section A = P1V1
The work done per second on the liquid at section B =  P2V2

                       V1 = V2 = V

The equation of continuity.

Net work done =  P1V - P2V
The net work done per second is equals increases the potential energy and kinetic energy per second from A to B.

According to the law of conservation of energy,

Applications Of Bernoulli's Theorem

Motion of a ping-pong ball

Place a ping-pong ball in an upward stream of water as shown here we observe that when a continuous stream of water gushes out of the nozzle the ball rises to a certain height above the nozzle and stays at that height against the gravitational pull and starts spinning.

We also observe that in case the ball moves to one side, it is automatically brought back to the original position.

How does this happen?

Suppose, the ball moves towards the left the ball continues to spin due to the liquid on the right side. The direction of spin of the ball is in such a way that the fluid velocity is more on its right side and less on the left side of the ball. According to Bernoulli's theorem, the pressure of the fluid on the right is less than that on the left. Due to this difference in pressure, the ball returns to its original position.

The rise of oil in a narrow tube

Shown here is a horizontal tube AB dipped in oil. The area of a cross-section of the tube at C is much smaller than that at A and B. Observe the movement of water when it is flowing from A to B. At the narrow end C the velocity of the water increases and hence the pressure at that point decreases below the atmospheric pressure. Due to the difference in pressure the oil Rises into the vertical tube.

Atomizer

An atomizer is an instrument used form spraying perfumes and water. The experiment shown here gives the principle underlying the working of an atomizer. Displayed here is the arrangement used in an atomizer. When the rubber bulb B is squeezed the air flows through the tube T and comes out of the narrow aperture O of the tube. As the aperture is narrow the velocity of the air increases as it comes out. This reduces the pressure in the vicinity of O. Due to the difference in pressure the liquid rises up into the vertical tube but the stream of air coming out of the tube T blows the liquid into a fine spray. So next time you use perfume or deodorant remember this principle.

Bunsen burner

One of the most commonly used appliances based on Bernoulli's theorem is the Bunsen burner. In a Bunsen burner when the gas emerges out of the nozzle the velocity of the gas increases and as a result a region of low pressure is created around the nozzle. As a result, the atmospheric air flows towards the burner and the air necessary for combustion is sucked through the holes present on the burner.