Proof of Bernoulli's Theorem or Bernoulli's Principle ; Mechanical Properties Of Liquid - Physics

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

m+ mgh += a constant

If unit mass is considered ( 1 kg), we get the above equations as

   +gh+ = a constant    ; this is Bernoulli's equation

If divided throughout the above equation by g (acceleration due to gravity), we get

++ h = a constant

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

is the work done at section A in time  and the fluid has displacement  through the section A.

               
Then  = 

 is the work done at section B .

                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

+  = a constant

This is the relation between pressure and velocity.

  ie as pressure increases velocity decreases and velocity increases pressure decreases .