Bernoulli’s theorem

When air is blown over the top of a sheet of paper, the paper rises in the air stream. This happens because the pressure falls above the paper where the air is moving faster. We take a table tennis ball and place in a funnel and hold it with the mouth sloping upwards. When we blow it, we can blow the ball out. Similarly, two balls are suspended side by side and air is blown up through the space between them. As the air flows through the narrow space between the balloons, the pressure falls. The atmospheric pressure from the sides brings the balls together.

The above observations lead us to conclude that there is a relation between pressure and velocity of air. Bernoulli’s equation is a fundamental relation in fluid mechanics. It can be derived from the work-energy theorem. The work-energy theorem says that the work done by the resultant force acting on a system is equal to the change in kinetic energy of the system. Any moving liquid has three kinds of energies:

• Kinetic energy by virtue of its motion
• Potential energy by virtue of its position
• Pressure energy when it is subject to pressure

The work-energy theorem states that the work done by the resultant force acting on a system is equal to the change in kinetic energy of the system. Let m be mass of the liquid and v be its velocity in motion.

Then its kinetic energy will be

The kinetic energy per unit mass will be:

Similarly, Let h be the height of the liquid above the earth’s surface.

Then its potential energy = mgh

Potential energy per unit mass = gh

Similarly, Let P be the hydrostatic pressure exerted by a liquid, r be its density and V be its volume.

Then its pressure energy = PV

These three types of energies possessed by a liquid under flow are mutually convertible one into another. Bernoulli’s theorem says that the sum of the energies possessed by a flowing, non-viscous, incompressible liquid at any point throughout its flow is constant when the flow is streamlined.

This implies that:

Pressure Energy + Kinetic Energy + Potential Energy = Constant.

For a unit mass of liquid:

If the pipe is horizontal, then h also is constant so:

The above equation makes it clear that when the velocity of the fluid increases, the pressure of the fluid decreases and vice versa. This principle can be illustrated by numerous demonstrations.

Everyday applications of Bernoulli’s Theorem

Venturimeter, atomiser and filter pump

Bernoulli’s principle is used in venturimeter to find the rate of flow of a liquid.

It is used in a carburettor to mix air and petrol vapour in an internal combustion engine. Bernoulli’s principle is used in an atomiser and filter pump.

Wings of Aeroplane

Wings of an aeroplane are made tapering. The upper surface is made convex and the lower surface is made concave. Due to this shape of the wing, the air currents at the top have a large velocity than at the bottom. Consequently the pressure above the surface of the wing is less as compared to the lower surface of the wing. This difference of pressure is helpful in giving a vertical lift to the plane.

How storms blow off the roofs?

Due to strong wind, storm or cyclone, the roofs are blown off. When a strong wind blows over the roof, there is lowering of pressure on the roof. As the pressure on the bottom side of the roof is higher, roofs are easily blown off without damaging the walls of the building.

How a moving train attracts a person standing nearby on a platform?

A suction effect is experienced by a person standing close to the platform at railway station when a fast train passes the person. This is because the fast moving air between the person and train produces a decrease in pressure and the excess air pressure on the other side pushes the person towards the train.