2. x
v1
v2
A1
A2
Y
Principle of continuity
When an incompressible and non – viscous liquid flows in
stream lined motion through a tube of non uniform cross-
section, then the product of the area of cross-section and the
velocity of flow is same at every point in the tube.
3. The velocity of liquid is smaller in the wider
parts of the tube and larger in the narrower parts.
A1 x v1 = A2 x v2
A x v = constant
4. Different energies of a flowing liquid
There are three types of energies in a following
liquid
1. Pressure energy
2. Kinetic energy
3. Potential energy
5. Pressure energy
If P is the pressure on an area A of a liquid, and the
liquid moves through a distance “l” due to this
pressure, then
pressure energy of liquid = work done = force x
distance
pressure x area x distance =
The volume of the liquid is A x l (area x distance).
pressure energy per unit volume of the liquid –
6. Kinetic Energy
If a liquid of mass in and volume V is t
owing with velocity t then its kinetic
energy is ½ mv2
kinetic energy per unit volume of the liquid =
½ ( m / v ) v2 = ½ v2
Where is the density of the liquid.
7. Potential Energy
If a liquid of mass ‘m’ is at a height ‘h’ from the
surface of the earth, then its potential energy is
mgh.
potential energy per unit volume of the liquid
= (m/V)g h = g h.
8. Bernoulli's Theorem
When an incompressible and non-viscous liquid (or
gas) flows in stream-lined motion from one
place to another, then at every point of its path the
total energy per unit volume (pressure energy +
kinetic energy + potential energy) is constant.
That is ,
P + ½ v2 + gh = constant.
9. Thus, Bernoulli's theorem is in one way the principle of
conservation of energy for a flowing liquid (or gas).
Proof : Suppose an incompressible and non-viscous liquid is
flowing in steam-lined motion through a tube XY of non-
uniform cross-section . Let Al and A2 be the areas
of cross-section of the tube at the ends X and Y respectively
which are at heights h1 and h2 from the surface of the earth.
Let P1 be the pressure and Lit the velocity of flow of the liquid
at X ; and P2 the respective quantities at Y. Since, the area A2
is smaller than A1 the velocity v2 is greater than v1(principle
of continuity).
11. work done per second on the liquid entering the tube at
X is
= P1 x A1 x v1
Similarly, work done per second against the force
P2 x A2 by the liquid leaving the tube at Y is ,
P2 x A2 x v2
: net work done on the liquid = (P1 A1 v1 -P2 A2 v2).
But A1 v1 and A2 v2 are respectively the volumes of the
liquid entering at X and leaving at Y per second.
These volumes must be equal and so
A1 v1 = A2 v2 =
12. where m is the mass of the liquid entering at X or
leaving at Y in 1 second, and is the density of liquid .
Substituting this in eq. , we have
net work done on the liquid = (P1 - P2) . …(ii)
The kinetic energy of the liquid entering at X in 1
second is ½ m v1
2 and that of the liquid leaving at Y
in 1 second is ½ mv2
2 .
increase in kinetic energy of the liquid = ½ m (v2
2-
v1
2).
13. The potential energy of the liquid at X is mgh1 and that
at Y is m g h2.
decrease in potential energy of the liquid = mg(h1 –
h2).
Thus, net increase in the energy of the liquid
= ½ m (v2
2 - v1
2) -m g (h1 - h2) ………..(iii)
This increase in energy is due to the net work done on
the liquid :
net work done = net increase in energy .
14. Hence by eq. (ii) and eq. (iii) , we can write
(P1 - P2) = ½ m (v2
2 –v1
2 ) - mg(h1 - h2)
P1 - P2 = ½ (v2
2-v1
2) – g (h1- h2)
P1 + ½ v1
2 + g h1 = P2 + ½ v2
2 + g h2
P+ ½ v2 + g h= constant
This is the Bernoulli's equation.
15. Pressure Head, Velocity Head and
Gravitational Head of a Flowing Liquid :
Dividing the above equation
by g, we have
P/( g) + v2/(2g)+ h = constant.
g is called the 'pressure-head', v2/2g is 'velocity head' and h is
'gravitational head'. The dimension of each of these three is the
dimension of height. The sum of these is called 'total head'.
Therefore,
Bernoulli's theorem may also be stated as :
“In stream-lined motion of an ideal liquid, the sum of pressure
head, velocity head and gravitational head at any point is
always constant.”
16. When the liquid flows in a horizontal plane, then h1=h2,
P1 + ½ v1
2 = P2 + ½ v2
2
P + ½ v2 = constant
Hence, in the horizontal stream-Intel motion of a liquid,
the sum of pressure and kinetic energy per
unit volume of the liquid at any point is constant ,
17. Applications or Examples Based on
Bernoulli's Theorem
Venturlmeter
Shape of the wing of an aeroplane
Bunsen’s burner
Atomizer
Filter pump
Blowing-off of Tin Roof in wind storm
Ping-pong ball placed on a water fountain
Magnus effect
Deep water runs calm
Blood flow and heart attack
18. Limitations of Bernoulli’s theorem
Bernoulli’s theorem is applicable under the
following restrictions:
The fluid is non-viscous.
The fluid is incompressible.
The fluid flow is streamlined, not turbulent.
The fluid flow is irrotational.
