PHYSICS

INVESTIGATORY
PROJECT
2018 – 2019

BERNOULLI’S THEOREM

MADE BY: AARYA RAJESH

 INDEX

PRESSURE
EQUATION OF

CONTINUITY
DANIEL

BERNOULLI
INTRODUCTION TO

BERNOULLI’S
THEOREM
BERNOULLI’S

EQUATION
EXPERIMENT

APPLICATIONS
CONCLUSION
BIBLIOGRAPHY

PRESSURE
 Pressure, in the physical sciences, is the
perpendicular force per unit area, or the stress at a point
within a confined fluid.

 In SI units, pressure is measured in pascals; one pascal

equals one newton per square metre.

 Absolute pressure of a gas or liquid is the total pressure it

exerts, including the effect of atmospheric pressure. An
absolute pressure of zero corresponds to empty space or
a complete vacuum.

 Pressure is a scalar quantity.

 EQUATION OF
CONTINUITY
 According to the equation of continuity Av =
constant. Where A =cross-sectional area and v=velocity
with which the fluid flows.
 It means that if any liquid is flowing in streamline flow in a
pipe of non-uniform cross-section area, then rate of flow
of liquid across any cross-section remains constant.
Consider a fluid flowing through a tube of varying thickness.
Let the cross-sectional area at one end (I) = A1 and cross-
sectional area of other end (II) = A2.
 The velocity and density of the fluid at one end (I)=v1, ρ1
respectively, velocity and density of fluid at other end (II)=
v2, ρ2
Volume covered by the fluid in a small interval of time ∆t,
across left cross-sectional is Area (I) =A1xv1x∆t
Volume covered by the fluid in a small interval of time ∆t,
across right cross-sectional Area (II) = A2x v2x∆t
Fluid inside is incompressible (volume of fluid does not
change by applying pressure) that is density remains same.
{ρ1=ρ2 .... (equation 1)}
Along (I) mass=ρ1 A1 v1∆t and along second point (II) ,
mass = ρ2A2 v2∆t
By using equation (1), we can conclude that
A1 v1 = A2 v2.This is the equation of continuity.
From Equation of continuity we can say that Av=constant.
This equation is also termed as “Conservation of mass of
incompressible fluids”.
 DANIEL BERNOULLI

Daniel Bernoulli (8 February 1700 – 17 March 1782) was a

Swiss mathematician and physicist and was one of the many
prominent mathematicians in the Bernoulli family. He is
particularly remembered for his applications of mathematics to
mechanics, especially fluid mechanics, and for his pioneering
work in probability and statistics. His name is commemorated
in the Bernoulli's principle, a particular example of
the conservation of energy, which describes the mathematics
of the mechanism underlying the operation of two important
technologies of the 20th century: the carburetor and the
airplane wing.
Together Bernoulli and Euler tried to discover more about the
flow of fluids. It was known that a moving body exchanges its
kinetic energy for potential energy when it gains height. Daniel
realised that in a similar way, a moving fluid exchanges its
specific kinetic energy for pressure, the former being the
kinetic energy per unit volume.

INTRODUCTION TO
BERNOULLI’S THEOREM
In fluid dynamics, Bernoulli's principle states that an increase in
the speed of a fluid occurs simultaneously with a decrease
in pressure or a decrease in the fluid's potential energy. The
principle is named after Daniel Bernoulli, as mentioned
before, who published it in his book Hydrodynamica in
1738. Although Bernoulli deduced that pressure decreases
when the flow speed increases, it was Leonhard Euler who
derived Bernoulli's equation in its usual form in 1752. The
principle is only applicable for isentropic flows: when the
effects of irreversible processes (like turbulence) and non-
adiabatic processes (e.g. heat radiation) are small and can be
neglected.
Bernoulli's principle can be applied to various types of fluid
flow, resulting in various forms of Bernoulli's equation; there
are different forms of Bernoulli's equation for different types
of flow. The simple form of Bernoulli's equation is valid
 for incompressible flows (e.g. most liquid flows
and gases moving at low Mach number). More advanced forms
may be applied to compressible flows at higher Mach numbers.
Bernoulli's principle can be derived from the principle
of conservation of energy. This states that, in a steady flow, the
sum of all forms of energy in a fluid along a streamline is the
same at all points on that streamline. This requires that the sum
of kinetic energy, potential energy and internal energy remains
constant. Thus an increase in the speed of the fluid – implying
an increase in its kinetic energy (dynamic pressure) – occurs
with a simultaneous decrease in (the sum of) its potential
energy (including the static pressure) and internal energy. If
the fluid is flowing out of a reservoir, the sum of all forms of
energy is the same on all streamlines because in a reservoir the
energy per unit volume (the sum of pressure and gravitational
potential ρ g h) is the same everywhere.

Bernoulli's principle can also be derived directly from Isaac

Newton's Second Law of Motion. If a small volume of fluid is
flowing horizontally from a region of high pressure to a region
of low pressure, then there is more pressure behind than in
front. This gives a net force on the volume, accelerating it
along the streamline.
Fluid particles are subject only to pressure and their own
weight. If a fluid is flowing horizontally and along a section of a
streamline, where the speed increases it can only be because
the fluid on that section has moved from a region of higher
pressure to a region of lower pressure; and if its speed
decreases, it can only be because it has moved from a region of
lower pressure to a region of higher pressure. Consequently,
within a fluid flowing horizontally, the highest speed occurs
where the pressure is lowest, and the lowest speed occurs
where the pressure is highest.

BERNOULLI’S EQUATION
The equation is given as,
P + 1/2(ρ v2) + ρgh = 0
Where P is pressure, ρ is the density of the fluid, v is its velocity,
g is the acceleration due to gravity and h is the height of the
fluid from the ground.

DERIVATION
 Finding the Work Done
First, we will calculate the work done (W1) on the fluid in the region BC.
Work done is

W1 = P1A1 (v1∆t) = P1∆V

Moreover, if we consider the equation of continuity, the same volume of

fluid will pass through BC and DE. Therefore, work done by the fluid on the
right-hand side of the pipe or DE region is

W2 = P2A2 (v2∆t) = P2∆V

Thus, we can consider the work done on the fluid as – P2∆V. Therefore, the
total work done on the fluid is

W1 – W2 = (P1 − P2) ∆V

The total work done helps to convert the gravitational potential energy and
kinetic energy of the fluid. Now, consider the fluid density as ρ and the mass
passing through the pipe as ∆m in the ∆t interval of time.

Hence, ∆m = ρA1 v1∆t = ρ∆V

Change in Gravitational Potential and Kinetic Energy

Now, we have to calculate the change in gravitational potential energy ∆U.

Similarly, the change in ∆K or kinetic energy can be written as

Calculation of Bernoulli’s Equation

Applying work-energy theorem in the volume of the fluid, the equation will
be
 Dividing each term by ∆V, we will obtain the equation

Rearranging the equation will yield

The above equation is the Bernoulli’s equation. However, the 1 and 2 of

both the sides of the equation denotes two different points along the pipe.
Thus, the general equation can be written as

EXPERIMENT
This experiment is aimed at investigating the validity of
Bernoulli’s equation when applied to a steady flow of water in
tapered duct and to measure the flow rate of steady flow
rates. Based on Bernoulli’s theorem relates the pressure,
velocity, and elevation in a moving fluid the compressibility and
viscosity (internal friction) of which are negligible and the flow
of which is steady, or laminar.
For this experiment, by using the FM 24 Bernoulli’s Apparatus
Test Equipment is to demonstrate the Bernoulli’s theorem. The
experiment was conducted in order to find the time taken to
collect 3L of water, the volumetric flow rates of the water, the
pressure difference at all manometer tube at different cross
section. The time to collect 0.003 m3 water is recorded based
on the different flow rate for each experiment.
The combination of venture meter complete with manometer
tube and hydraulic bench were used. During the experiment,
water is fed through a hose connector and the flow rate can be
adjusted at the flow regulator valve at the outlet of the test
section. The venture can be demonstrated as a means of flow
measurement and the discharge coefficient can be determined
the results show the reading of each manometer tubes
increase when the pressure difference increases. From the
reading of height can be calculated the data by applied the
Bernoulli equation to fin the velocity of the fluid moving.

The pressure level and velocity reading for part A to E of the

tube is recorded. From Bernoulli theory, the relation between
the increase and decrease in the pressure value is
inversely proportional to its velocity. Bernoulli's Principle tells
that as the fluid flows more quickly through the narrow
sections, the pressure actually decreases rather than increases.
Thus, it proves the validity of Bernoulli’s theorem.
 APPLICATIONS
In modern everyday life there are many observations that can be
successfully explained by application of Bernoulli's principle, even
though no real fluid is entirely inviscid and a small viscosity often has a
large effect on the flow.
 An injector on a steam locomotive (or static boiler).

 A De Laval nozzle utilizes Bernoulli's principle to create a force by

turning pressure energy generated by the combustion
of propellants into velocity. This then generates thrust by way
of Newton's third law of motion.

 The pilot tube and static port on an aircraft are used to determine
the airspeed of the aircraft. These two devices are connected to
the airspeed indicator, which determines the dynamic pressure of
the airflow past the aircraft. Dynamic pressure is the difference
between stagnation pressure and static pressure. Bernoulli's
principle is used to calibrate the airspeed indicator so that it displays
the indicated airspeed appropriate to the dynamic pressure.

 Bernoulli's principle can be used to calculate the lift force on an

airfoil, if the behaviour of the fluid flow in the vicinity of the foil is
known. For example, if the air flowing past the top surface of an
aircraft wing is moving faster than the air flowing past the bottom
surface, then Bernoulli's principle implies that the pressure on the
surfaces of the wing will be lower above than below. This pressure
difference results in an upwards lifting force. Whenever the
distribution of speed past the top and bottom surfaces of a wing is
known, the lift forces can be calculated (to a good approximation)
using Bernoulli's equations – established by Bernoulli over a century
before the first man-made wings were used for the purpose of
flight. Bernoulli's principle does not explain why the air flows faster
past the top of the wing and slower past the underside.
 The Bernoulli grip relies on this principle to create a non-contact
adhesive force between a surface and the gripper.

 The carburettor used in many reciprocating engines contains

a venturi to create a region of low pressure to draw fuel into the
carburettor and mix it thoroughly with the incoming air. The low
pressure in the throat of a venturi can be explained by Bernoulli's
principle; in the narrow throat, the air is moving at its fastest speed
and therefore it is at its lowest pressure.

 The flow speed of a fluid can be measured using a device such as

a Venturi meter or an orifice plate, which can be placed into a
pipeline to reduce the diameter of the flow. For a horizontal device,
the continuity equation shows that for an incompressible fluid, the
reduction in diameter will cause an increase in the fluid flow speed.
 Subsequently, Bernoulli's principle then shows that there must be a
decrease in the pressure in the reduced diameter region. This
phenomenon is known as the Venturi effect.

 The maximum possible drain rate for a tank with a hole or tap at the
base can be calculated directly from Bernoulli's equation, and is
found to be proportional to the square root of the height of the
fluid in the tank. This is Torricelli's law, showing that Torricelli's law
is compatible with Bernoulli's principle. Viscosity lowers this drain
rate. This is reflected in the discharge coefficient, which is a function
of the Reynolds number and the shape of the orifice.

CONCLUSION
Bernoulli's law states that if a non-viscous fluid is flowing along
a pipe of varying cross section, then the pressure is lower at
constrictions where the velocity is higher, and the pressure is
higher where the pipe opens out and the fluid stagnate. Many
people find this situation paradoxical when they first encounter
it (higher velocity, lower pressure). 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.

BIBLIOGRAPHY
 www.sciencefare.com

 www.mycbseguide.com
 PHYSICS NCERT CLASS XI