Fluid Mechanics

1. BERNOULLI’S THEOREM &IT’S
APPLICATIONS
A lecture by…
S.Raghubir Singh
(Lecturer Civil Engg.)
Pt. Jagat Ram Govt. Polytechnic
College,Hoshiarpur.
Subject:-Fluid mechanics
3rd semester
(Civil Engg.)

2. CONTENTS
 Types of fluid flow
 Discharge, Continuity Equation
 Types of Energies/heads of Ideal fluid
• Potential Energy/head
• Kinetic Energy/ head
• Pressure Energy/ head
 Bernoulli’s theorem:-Assumptions, Limitations
 Applications of Bernoulli’s theorem :
• Venturimeter
• Orificemeter
• Pitot Tube

3. Types of Fluid Flow
The fluid in motion depends on various fluid
properties like temperature, density, velocity and
pressure. Based on these properties, the fluid can
be classified as :
1. Steady and unsteady flow
2. Uniform and non- uniform flow
3. Laminar and turbulent flow

4. STEADY FLOW
• Steady flow is defined as the type of flow in which
the fluid properties like pressure,density and velocity
at a point do not change with time. Mathematically,
( v/ t ) = 0
( p/ t ) = 0
( / t ) = 0
Example : Flow through a pipeline under
constant head.

5. STEADY FLOW

6. UNSTEADY FLOW
• UnSteady flow is defined as the type of flow in
which the fluid properties like pressure,density and
velocity at a point change with time.
Mathematically,
( v/ t )  0
( p/ t )  0
( / t )  0
Example : Flow through a pipeline when a valve is
opened or closed gradually.

7. STEADY FLOW CHANGING INTO
UNSTEADY ONE

8. UNIFORM FLOW
Uniform flow is that type of flow in which the
velocity of flow of a fluid is constant at any
section in the path of flow of fluid.
Mathematically,
( v/ s ) = 0 , t is constant
where v = change of velocity
s = change in displacement
Example :- Flow of a water through a pipeline of
uniform diameter.

9. UNIFORM FLOW

10. NON-UNIFORM FLOW
Non-Uniform flow is that type of flow in which
the velocity of flow of a fluid is not constant at
any section in the path of flow of fluid.
Mathematically,
( v/ s )  0 , t is constant
where v = change of velocity
s = change in displacement
Example :- Flow of a water through a pipeline of
variable diameter.

11. LAMINAR FLOW
• A flow is said to be laminar if
each particle fluid has a
definite path and the path of
one particle does not cross the
path of any other particle.
• It is also called streamline or
viscous flow.
• Examples :- Flow of blood in
veins and arteries,Ground water
flow.

12. TURBULENT FLOW
• A flow is said to be turbulent
if the fluid particles do not
have a definite path and path
of one particle crosses the path
of other particles during flow.
• It is also called non-laminar
flow. The fluid particles move
in zigzag way.
• Examples :- Flow of a river
water, flow of petrol through a
pipeline.

13. CHANGE OF LAMINAR FLOW INTO
TURBULENT FLOW

14. DISCHARGE
 Discharge can be defined as the quantity of a fluid
flowing per second through a section of a pipe or
channel.
 It is generally denoted by Q.
 Let A = Cross-sectional area of the pipe.
V = Average velocity of the liquid.
Discharge, Q = Area x Average velocity
Q = A x V
 Units of Discharge : m3/sec

15. CONTINUITY EQUATION
 Continuity equation states that if no fluid is added or
removed from the pipe in any length then the mass
passing across different sections shall be same.This
equation is based on the principle of Conservation of
mass.
 Mathematically, Continuity equation can be written as :
A1V1 = A2V2

16. CONTINUITY EQUATION
 Considering two cross sections
of a tapering pipe.
Let V1 =Average velocity at
Section 1-1
A1= Area of pipe at Section
1-1
1 = Density at section 1-1
and V2,A2, 2 are the values at
Section 2-2.
Section 1-1
Section 2-2

17. CONTINUITY EQUATION
Total Quantity of fluid mass passing through sec 1-1= 1 A1 V1
Total Quantity of fluid mass passing through sec 2-2= 2 A2 V2
According to law of conservation of mass,
1 A1 V1= 2 A2 V2 ------------(1)
For incompressible fluids, 1 = 2
From continuity equation (1)
A1 V1= A2 V2

18. ENERGIES OF AN IDEAL FLUID
There are three types of energies of the flowing fluids:
1.Kinetic Energy (E)
2.Potential Energy (U)
3.Pressure Energy (P)
Based on above three types of energies, we have
three different types of heads of flowing fluids
namely Kinetic head,potential head and pressure
head.
 A head is the specific measurement of water
pressure above datum(a reference from which
measurements are made)

19. 1. KINETIC ENERGY
• Kinetic energy of a fluid
particle is the energy
which it possesses due to
its motion or velocity.e.g
Waves striking at the
seashore.
• Kinetic head or velocity
head is the kinetic energy
of a fluid per unit of its
weight.

20. KINETIC HEAD
Let Mass of a fluid particle = m kg
Velocity of flow of the fluid =V m/sec
Kinetic Energy of the fluid particle E = 1/2 mV2 joules
 1 Joule = 1J = 1 kg(m2/s2) or 1 N-m
Weight of the fluid particle ,W = mg
Kinetic Head or Velocity head = E/W
= 1/2 mV2
mg
Kinetic Head = V2 metre of the fluid
2g

21. 2. POTENTIAL ENERGY
• Potential energy of a
fluid particle is the
energy which it
possesses due to its
position.
• Potential Head is the
potential energy of a
fluid per unit of its
weight.

22. POTENTIAL HEAD
Let Mass of the fluid particle = ‘m’ kg
Height of the fluid particle above datum line = ‘Z’ m
Potential energy of the fluid particle U = m.g.Z N-m
Weight of the fluid particle ,W = mg
Potential Head of the fluid particle = U = m g Z
W m g
Potential Head = Z

23. 3. PRESSURE ENERGY
• The energy possessed
by the fluid particles by
virtue of its existing
pressure is called
pressure energy.
• Pressure head is the
pressure energy of a
fluid per unit of its
weight.

24. PRESSURE HEAD
Pressure Energy P in N-m = press. intensity x Volume
= p x V
= p x W/w since ( w = W/V)
Pressure Head of the liquid= P = p x W/w
W W
Pressure Head P = p/w

25. TOTAL ENERGY/ HEAD OF A LIQUID
PARTICLE IN MOTION
The Total energy of a fluid particle in motion is the
sum of its potential energy, kinetic energy and
pressure energy
Total Energy (E) = Z + V2/2g + p/w
Total head of a fluid a fluid particle in motion is the
sum of its potential head, kinetic head and pressure
head .
Total Head (H) = Z + V2/2g + p/w

26. BERNOULLI’S THEOREM
Dr. Daniell Bernoulli (1783)

27. BERNOULLI’S THEOREM
• This theorem states that the total energy of the
fluid particle remains the same for an ideal
incompressible fluid flowing from one section to
another section in a continuous stream.
Section 1-1
Section 2-2
Z1 Z2

28. BERNOULLI’S THEOREM
Bernoulli’s theorem can be derived from principle of
conservation of energy.
( Energy can neither be destroyed nor be created ;
it can only be transformed from one state to another)
Mathematically,
p/w + V2/2g + Z = Constant
where p/w = Pressure head
V2/2g = Velocity head
Z = Potential head

29. ASSUMPTIONS OF BERNOULLI’S
THEOREM
 The flow is steady and continuous.
 The flow is along a stream line
 The flow is ideal and incompressible.
 The velocity is uniform over the section .
 No external force acts on the fluid except force of
gravity.

30. LIMITATIONS OF BERNOULLI’S
THEOREM
 The Loss of energy due to pipe friction is
neglected in the Bernoulli’s equation.
 The loss of energy due to turbulency is not taken
into account by the Bernoulli’s equation.
 Bernoulli’s equation does not take into
consideration loss of energy due to change of
direction.

31. LIMITATIONS OF BERNOULLI’S
THEOREM
 In assumptions, the velocity of a liquid particle at
any point across a cross-section of a pipe is
assumed to be uniform but the velocity of the
liquid particle is maximum at centre of a pipe
and gradually decreases towards the walls of the
pipe. Hence, the mean velocity of flow should
have been taken into consideration.

32. APPLICATIONS OF BERNOULLI’S
THEOREM

33. APPLICATIONS OF BERNOULLI’S
THEOREM
The three important applications of
Bernoulli’s theorem are :-
1. Venturimeter
2. Orificemeter
3. Pitot Tube

34. VENTURIMETER
 A Venturimeter is a device used for measuring
discharge of a fluid flowing through a pipe.
 It is named after the Italian Engineer
Venturimeter in 18th century.
 The working Principle of Venturimeter is based
on Bernoulli’s Equation.
 The total energy of the fluid particle remains the same
for an ideal incompressible fluid flowing from one
section to another section in a continuous stream.

35. TYPES OF VENTURIMETER
There are three types of venturimeters:
1. Horizontal venturimeter
2. Vertical venturimeter
3. Inclined venturimeter
Construction of venturimeter
A venturimeter basically consists of three parts:-
a. Converging part
b. Throat
c. Diverging part (3 to 4 times longer than
converging part)

36. HORIZONTAL VENTURIMETER
Let D1 = Diameter at section 1-1
p1 = Intensity of pressure at section 1-1
V1 = Velocity of fluid at section 1-1
Z1 = Datum head at section 1-1
A1 = Area at section 1-1
and D2 , p2, V2, Z2, A2 are the corresponding values at
section 2-2.

37. HORIZONTAL VENTURIMETER
Expression of Actual discharge through venturimeter
Q actual = Cd A1 A2  2gh
 A1
2 - A2
2
where h = difference of pressure heads at section 1-1
and section 2-2
Cd = Co-efficient of venturimeter(discharge)
having value less than 1 (0.96 to 0.98 )
 Due to variation of Cd venturimeters are not suitable
for very low velocities.

38. ORIFICEMETER
 Orificemeter is the simple device used for
measuring the discharge of a fluid through a pipe.
 This device is comparitively cheaper than
venturimeter.
 The working Principle of orificemeter is also
based on Bernoulli’s Equation.
 Orificemeter consists of a flat circular plate
having a central circular hole in it called an
ORIFICE.

39. ORIFICEMETER
 The diameter of the orifice
varies from 0.4 to 0.8 times
the diameter of
pipe.(preferably 0.5 times).
 Expression of actual discharge
through orificemeter is given
by:-
Q actual = Cd A0  2gh
 1- (A0/A1)2
where A0 is the area of the orifice.

40. PITOT TUBE
 Pitot tube is used to measure the
velocity of flow at any point in
a pipe. It is a tube bent at right
angles.
 Pitot tube works on the principle
that if the velocity of flow at any
point becomes zero,the pressure
increases at that point.(V 1/P)
 Velocity at any point:
V = Cv  2gh
where Cv =Coefficient of Pitot tube

41. CONCLUSION
 Types of flow
 Continuity Equation
 Types of energies/ heads
 Bernoulli’s theorem :
Assumptions,Limitations
 Applications of Bernoulli’s
theorem : Venturimeter,
Orificemeter, Pitot tube
** ** ** ** **
Dr. Daniell Bernoulli (1783)