Ses
Session 2016- 17
Su
u bj e ct - Ph ysics
cs
I n ve st iga
ga t or y pr oj e ct on Be r n ou lli’s
i’s Pr inciple
Su bm it t e d t o: M s. Ra m a Elisa la M a ’a m
Su bm it t e d by
by: M ohd. Am e e n I r fa n
Cla ss: XI ‘A’’
Roll n o:: 1 1 0 1 8
CERTIFICATE
This is to certify that Mohd. Ameen Irfan of class
11th ‘A’ has completed the project work in physics
in the year 2016-17 on “Bernoulli’s Theorem”
under the guidance of Ms. Rama Elisala Ma’am as
prescribed by CBSE course.
It is further certified that this project is the
individual work of the student.
Internal Examiner
Principal’s Signature
School Stamp
Acknowledgement
I would like to express my special thanks of
gratitude to my subject teacher Ms. Rama
Elisala Ma’am to give her guidance to make
the successful completion of this project.
I also want to give special thanks to our
principal ma’am Ms. Priya Chouhan
Ma’am who gave me this golden opportunity
to do this wonderful project on the topic
“Bernoulli’s Theorem”, so that I will get to
know about detailed information for the
same.
Secondly I would like to thank my parents
and classmates who helped me to complete
this project within the given time frame.
Content
• Pressure
• Pascal’s Law
• Hydraulics
• Continuity Equation
• Bernoulli’s Equation
• Derivation of Bernoulli’s Equation
• Venturi Tube
• Atomizer
• Torricelli and his Orifice
• Derivation of Torricelli’s Equation
• Streamlines
• Aerodynamic Lift
• Misconceptions of Lift
• Conclusion
• Bibliography
PRESSURE
1. Pressure is defined as force per unit
area.
2. Standard unit is Pascal, which is N/m2
3. For liquid pressure, the medium is
considered as a continuous
distribution of matter.
4. For gas pressure, it is calculated as
the average pressure of molecular
collisions on the container.
5. Pressure acts perpendicular on the
surface.
6. Pressure is a scalar quantity –
pressure has no particular direction (i.e.
acts in every direction).
Pascal’s Law
Pf = P0 + ρgh
1. “When there is an increase in
pressure at any point in a confined
fluid, there is an equal increase at
every point in the container.”
2. In a fluid, all points at the same
depth must be at the same pressure.
3. Consider a fluid in equilibrium.
Hydraulics
You have to push down the piston on the left
far down to achieve some change in the height
of the piston on the right.
1. Pressure is equal at the bottom of
both containers (because it’s the
same depth!)
2. P = F2/A2 = F1/A1 and since A1 < A2, F2 >
F1
3. There is a magnification of force,
just like a lever, but work stays the
same! (Conservation of energy). W =
F1* D1 = F2 * D2
∴ D1 > D2
Continuity Equation
1.
A1v1 = A2v2
2.
“What comes in comes out.”
3.
Av= V/s (volume flow rate) =
constant
Bernoulli’s Equation
P+1/2ρv*v+ρgh=constant
Where p is the pressure, ρ is the density, v
is the velocity, h is elevation, and g is
gravitational acceleration
Derivation of Bernoulli’s
Equation


Restrictions
Incompressible

Non-viscous fluid (i.e. no
friction)

Following a streamline motion (no
turbulence)

Constant density
*There exists an extended form of
equation that takes friction and
compressibility into account, but that is too
complicated for our level of study.
Et ot al = 1 / 2 m v 2 +
m gh
W = F/ A* A* d =
PV
• Consider the change in total energy of
the fluid as it moves from the inlet to
the outlet.
Δ Etotal = Wdone on fluid - Wdone by fluid
Δ Etotal = (1/2mv22 + mgh1) – (1/2mv12 + mgh2)
Wdone on fluid - Wdone by fluid = (1/2mv22 + mgh1)
– (1/2mv12 + mgh2)
P2V2 - P1V1 = (1/2mv22 + mgh1) – (1/2mv12 +
mgh2)
P2 – P1 = (1/2ρ v12 + ρ gh1) – (1/2ρ v12 + ρ gh1)
Venturi Tube
1. A2 < A1 ; V2 > V1
2. According to Bernoulli’s Law,
pressure at A2 is lower.
3. Choked flow: Because pressure
cannot be negative, total flow
rate will be limited. This is
useful in controlling fluid
velocity.
P2 + 1 / 2 ρ v 1 2 = P1 +
Δ P = ρ/ 2 * ( v 2 2 – v 1 2 )
1
/ 2ρ v 12
;
Atomizer
• This is an atomizer, which uses
the Venturi effect to spray liquid.
• When the air stream from the
hose flows over the straw, the
resulting low pressure on the top
lifts up the fluid.
Torricelli and his
Orifice
• In 1843, Evangelista Torricelli proved
that the flow of liquid through an
opening is proportional to the square
root of the height of the opening.
• Q = A*√(2g(h1-h2))
where Q is flow
rate, A is area, h is height
Depending on the contour and shape of
the opening, different discharge
coefficients can be applied to the
equation (of course we assume simpler
situation here-
Derivation of Torricelli’s
Equation
1. We use the Bernoulli Equation:
2. In the original diagram A1 [top] is
much larger than A2 [the opening].
Since A1V1 = A2V2 and A1 >> A2, V1 ≈
0
3. Since both the top and the opening
are open to atmospheric pressure,
P1 = P2 = 0 (in gauge pressure).
The equation simplifies down to:
ρgh1 = 1/2 ρv22 + ρgh2
/2 ρv22 = pg(h1-h2)
1
V22 = 2g(h1-h2)
∴ V2 = √(2g(h1-h2))
Q = Av2 = A √(2g(h1-h2))
Streamlines
1
A streamline is a path traced out by
a mass less particle as it moves with
the flow.
2. Velocity is zero at the surface.
1. As you move away from the surface,
the velocity uniformly approaches the
free stream value (fluid molecules
nearby the surface are dragged due to
viscosity).
2. The layer at which the velocity reaches
the free stream value is called
boundary layer. It does not necessarily
match the shape of the object –
boundary layer can be detached,
creating turbulence (wing stall in
aerodynamic terms).
Aerodynamic Lift
1. Lift is the fort that keeps
an aircraft in the air.
2. In Bernoulli-an view, lift is
produced by the different
of pressure (faster
velocity on the top, slower
velocity in the bottom)
3. In Newtonian view, lift is
the reaction force that
results from the
downward deflection of
the air.
3.
4.
Both views are correct, but the current
argument arises from the misapplication of
either view.
The most accurate explanation would take into
account the simultaneous conservation of mass,
momentum, and energy of a fluid, but that
involves multivariable calculus.
Misconceptions of Lift
1. In many popular literature,
encyclopedia, and even textbooks,
Bernoulli’s Law is used incorrectly to
explain the aerodynamic lift.
#1: Equal transit time
- The air on the upper side of the
wing travels faster because it has to
travel a longer path and must “catch up”
with the air on the lower side.
The error lies in the specification of
velocity. Air is not forced to “catch up”
with the downside air. Also, this theory
predicts slower velocity than in reality.
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).
Venturi-meter, atomizer and filter pump Bernoulli’s
principle is used in venturi-meter to find the rate of
flow of a liquid. It is used in a carburetor to mix air
and petrol vapor in an internal combustion engine.
Bernoulli’s principle is used in an atomizer and filter
pump. Wings of Aero plane Wings of an aero plane
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
1. Help from Internet
• www.sceincefare.com
• www.mycbsegide.com
• www.slideshare.com
2. Help from books
• Referenced from H.C. Verma
• Referenced from physics
NCERT
3. Help from teachers
• Rama Elisala Ma’am