2. Bernoulli’s Equation
Theory - Introduction
Bernoulli's principle states that an increase in the speed of a fluid ,
decrease in pressure.
The principle is named after Daniel Bernoulli who
published it in his book Hydrodynamica in 1738.
Leonhard Euler who derived Bernoulli's
equation in its usual form in 1752.
Leonhard Euler
(1707 - 1783)
Daniel Bernoulli
(1700 – 1782)
3. Bernoulli’s Principle
Theory - Statement
The total mechanical energy of the moving fluid comprising the
gravitational potential energy of elevation, the energy associated with
the fluid pressure and the kinetic energy of the fluid motion, remains
constant.
Mathematical form:
P+
1
2
𝑚𝑣2 +mgh = constant
pressure velocity height
Applicable :
• Incompressible
• Steady
• Non viscous 3
4. Bernoulli’s Equation
Explanation
4
Consider the following diagram where water flows from left to right in
a pipe that changes both area and height.
When fluid move upward, the water will be gaining gravitational
potential energy Ug as well as kinetic energy K.
5. Derivation
Work done on the fluid:
W1 = F1 ∆x1
As
P =
𝐹
𝐴
F = PA
Then
W1 = P1 A1 ∆x1
V1 =
∆x1
𝑡
In terms of velocity
∆x1 = V1 t
W1 = P1 A1V1 t
W2 = -F2 ∆x2
W2 = -P2 A2 ∆x2
W2 = -P2 A2V2 t
• The water at P2 will do negative work
on our system since it pushes in the
opposite direction as the motion of the
fluid.
6. Derivation
Net Work done on the fluid:
Wnet = W1 + W2
Wnet = P1 A1V1 t - P2 A2V2 t
• The Volume of both sections are equal
A1V1 t = A2V2 t = V
So
W = P1 V - P2 V
W = (P1 - P2 )V
As we know
(Density)
𝜌 =
𝑚
𝑉
V =
𝑚
𝜌
W = (P1 - P2 )V
W = (P1 - P2 )
𝑚
𝜌
7. Derivation
Work Energy Principle:
Work done = change in energy
W = ∆ (K.E) + ∆ (P.E)
Changing in ∆ (K.E) :
Changing in ∆ (P.E) :
∆ (K.E) =
1
2
mv2
∆ (K.E) =
1
2
mv2
2 -
1
2
mv1
2
∆ (P.E) = mgh
∆ (P.E) = mgh2 – mgh1
(1)
Put the values in equ (1)
(P1 - P2 )
𝑚
𝜌
=
1
2
mv2
2 -
1
2
mv1
2 + mgh2 – mgh1
(P1 - P2 )
𝑚
𝜌
= m (
1
2
v2
2 -
1
2
v1
2 + gh2 – gh1)
(P1 - P2 ) =
1
2
𝜌v2
2 -
1
2
𝜌v1
2 + 𝜌gh2 – 𝜌gh1)
This is Bernoulli's equation!
Generalize:
P+
1
2
𝑚𝑣2 +mgh = constant
This constant will be different for different fluid
systems, but the value of P+
1
2
𝑚𝑣2 +mgh will be the
same at any point along the flowing fluid.
9. Application - LIFT
9
The wings of plane have what is called an aerofoil shape.
The aerofoil shape helps us overcome weight which is the effect of gravity pulling
down on the mass of the aircraft.
The aerofoil shape gives us something called lift. This is the upward force required
to overcome gravity.
Something that slows us down is drag, which is the resistance to airflow through
the air . The drag force is opposite to the flight path.
Thrust is the forward force required to move an aircraft through the air. This must
be provided by an engine.
11. 11
Bernoulli’s principle helps to explain that an aircraft can achieve lift because of
the shape of its wings.
They are shaped so that that air flows faster over the top of the wing and slower
underneath.
Fast moving air = low air pressure while slow moving air = high air pressure.
The high air pressure underneath the wings will therefore push the aircraft up
through the lower air pressure.
12. Application - Baseball
12
One side will experience more pressure than the other thus having more air
turbulence and a slower air speed over the ball.
The other side would accelerate and move faster, because of lesser pressure.
This example explains the path of a baseball that’s thrown with
a clockwise spin.
If a ball is thrown with a counter-
clockwise spin, it will curve towards the
left.
13. Application–Atomizer
13
Atomizer is a device that is used to emit liquid droplets as fine spray. 'Atomize'
here means splitting up a large body into small particles.
It works on Bernoulli's principle.
When high speed horizontal air passes over a vertical tube, it creates a low
pressure and draws the air and liquid inside the vertical tube upward. Atomizer
has a nozzle at the end of the horizontal tube which causes the liquid to break
up into small drops and mixes it with the air.
