State and prove Bernoulli's theorem. Give two applications and limitations of it.

Bernoulli's Equation is an expression that relates the pressure difference between 2 points to the velocity change and elevation change.
Work has to be done on the liquid to make it move from X to Y.
Work done on the liquid to displace it from X to Y.
dw = P₁deltaV
W₁ = P₁deltaV->Work done on the liquid 
W₂ = P2deltaV->Work done by the liquid
Net work done on the liquid = (P₁ - P₂)deltaV
​Change in potential energy when liquid goes from X to Y = mg(h₂ - h₁)
Change in Kinetic Energy = mg(V₂² - V₁²)
According to law of conservation of Energy,
                                                                      (P₁ - P₂)delta V = mg(h₂ - h₁) + 1/2m(V₂² - V₁²)
                                                                      (P₁ - P₂)deltaV = deltaV*rho*g(h₂ - h₁) + 1/2​deltaV*rho(V₂² - V₁²)​
                                                                      P₁ + rho*gh₁ + 1/2*rho₁*V₁²​ = P₂ + rho*gh₂ + 1/2*rho₂*V₂²)
                                                 Therefore,    P + rho*gh + 1/2*rho*V² = constant
Sum total of potential energy, pressure energy and kinetic energy remains constant.
Two applications of Bernoulli's Theorm:-
1.Atomiser or Sprayer
2.Blowing off the roofs during storm
Two limitations of Bernouilli;s Theorm:-
1.There are always some external Forces acting on the Liquid, which affects the Flow of Liquid. So neglect all such external forces.  
2.If the Liquid is Flowing through curved path, the energy due to Centrifugal Forces should also be taken into account. ​
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