Bernoulli's Theorem Or Principle
Bernoulli's theorem states that total energy of a small amount of a liquid from one point to another along a streamline flow remains constant throughout the displacement.
OR
Total energy of a fluid flow is a constant.
ie Kinetic Energy+Potential energy+Pressure Energy = a constant
If unit mass is considered ( 1 kg), we get the above equations as
If divided throughout the above equation by g (acceleration due to gravity), we get
Bernoulli's theorem may also be stated as the sum of pressure head, velocity head and gravitational head is a constant.
Proof for Bernoulli's Theorem
Consider the streamline flow of liquid of density 
, through a tube of non-uniform cross section.Consider two positions A and B of tube.A is at the height of h1 from the ground level and B is at the height of h2 from the ground level.
is the area of cross section of A and 
is the area of cross section at B. Let 
is the pressure at A and 
is the pressure at B.Let 
is the velocity at A and 
is the velocity with which the liquid leaving through the tube through B.
Here we applying the Work-Energy theorem.
ie. net work done = change in energy
Then
Then 
= 
Net work done on the system = 
= 
-
According to equation of continuity
Multiplying with 
This is the volume of fluid flowing through the section A and B in time
As the fluid is flowing from section A to B, there is change in Kinetic Energy and Potential Energy.
Change in Potential Energy = 
Change in Kinetic Energy = 
Total change in energy = 
+
According to work energy theorem
(
= 
- 
) + 
Using volume = 
, substituting v = 
(
m is cancelled and the equation become
Rearranging the above equation gives
ie 
+ 
= a constant
This is Bernoulli's equation
Special Case
When a fluid flowing through horizontal pipe then
Then above equation became
OR
This is the relation between pressure and velocity.
ie as pressure increases velocity decreases and velocity increases pressure decreases .