According to Bernoulli's theorem, for the streamline flow of a non-viscous and incompressible liquid, the sum of the pressure energy, kinetic energy and potential energy per unit mass is a constant.

**Bernoulli's theorem**

In 1738, Daniel Bernoulli proposed a theorem for the streamline
flow of a liquid based on the law of conservation of energy. According to Bernoulli's
theorem, for the streamline flow of a non-viscous and incompressible liquid,
the sum of the pressure energy, kinetic energy and potential energy per unit
mass is a constant.

P/p + v^{2} +gh = constant.

This equation is known as *Bernoulli's equation.*

Consider streamline flow of a liquid of density
ñ through a pipe AB of varying cross section.
Let *P _{1}* and

The mass *m* of the liquid
crossing per second through any section of the tube in accordance with the
equation of continuity is

a_{1}v_{1}ρ = a_{2}v_{2}ρ = m

or

a_{1}v_{1} = a_{2}v_{2} = m/p =V

As a_{1} > a_{2} , v_{1} < v_{2}

The force acting on the liquid at A = P_{1}a_{1}

The force acting on the liquid at B = P_{2} a_{2}

Work done per second on the liquid at A = P_{1}a_{1}
× v_{1} = P_{1}V

Work done by the liquid at B = P_{2}a_{2} × v_{2}
= P_{2}V

∴ Net work done per second on the
liquid by the pressure energy

in moving the liquid from A to B is = P_{1}V - P_{2}V
...(2)

If the mass of the liquid flowing in one second
from A to B is *m*, then increase in
potential energy per second of liquid from A to B is *mgh*_{2}* *-* mgh*_{1}

Increase in kinetic energy per second of the
liquid

= ½ mv_{2}^{2 }- ½ mv_{1}^{2}

According to work-energy principle, work done
per second by the pressure energy = Increase in potential energy per second +
Increase in kinetic energy per second

P/p + gh + ½ v^{2} = constant

This is Bernoulli's equation. Thus the total
energy of unit mass of liquid remains constant.

Dividing equation (3) by g, p/pg + v^{2}/2g
+ h = constant

Each term in this equation has the dimension of
length and hence is called head. p/pg is called pressure head, v^{2}/2g
is velocity head and h is the gravitational head.

**Special case :**

If the liquid flows through a horizontal tube, h_{1} = h_{2}.
Therefore there is no increase in potential energy of the liquid i.e. the
gravitational head becomes zero.

equation (3) becomes

P/p + 1/2v^{2} = a constant

This is another form of Bernoulli's equation.

**Application of Bernoulli's
theorem**

**(i) Lift of an aircraft wing**

A section of an aircraft wing and the flow lines are shown in Fig.
The orientation of the wing relative to the flow direction causes the flow
lines to crowd together above the wing. This corresponds to increased velocity
in this region and hence the pressure is
reduced. But below the wing, the pressure is nearly equal to the atmospheric
pressure. As a result of this, the upward force on the underside of the wing is
greater than the downward force on the topside. Thus there is a net upward
force or lift.

**(ii) Blowing of roofs**

During a storm, the roofs of huts or tinned roofs are blown off
without any damage to other parts of the hut. The blowing wind creates a low
pressure P_{1} on top of the roof. The pressure P_{2} under the
roof is however greater than P_{1}. Due to this pressure difference,
the roof is lifted and blown off with
the wind.

**(iii) Bunsen burner**

In a Bunsen burner, the gas comes out of the nozzle with high
velocity. Due to this the pressure in the stem of the burner decreases. So, air
from the atmosphere rushes into the burner.

**(iv) Motion of two parallel
boats**

When two boats separated by a small distance row parallel to each
other along the same direction, the velocity of water between the boats becomes
very large compared to that on the outer sides. Because of this, the pressure
in between the two boats gets reduced. The high pressure on the outer side
pushes the boats inwards. As a result of this, the boats come closer and may
even collide.

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