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 = P_{1}deltaV

W_{1} = P_{1}deltaV->Work done on the liquid

W_{2} = P2deltaV->Work done by the liquid

Net work done on the liquid = (P_{1} - P_{2})deltaV

Change in potential energy when liquid goes from X to Y = mg(h_{2} - h_{1})

Change in Kinetic Energy = mg(V_{2}^{2} - V_{1}^{2})

According to law of conservation of Energy,

(P_{1} - P_{2})delta V = mg(h_{2} - h_{1}) + 1/2m(V_{2}^{2} - V_{1}^{2})

(P_{1} - P_{2})deltaV = deltaV*rho*g(h_{2} - h_{1}) + 1/2deltaV*rho(V_{2}^{2} - V_{1}^{2})

P_{1} + rho*gh_{1} + 1/2*rho_{1}*V_{1}^{2} = P_{2} + rho*gh_{2} + 1/2*rho_{2}*V_{2}^{2})

Therefore, P + rho*gh + 1/2*rho*V^{2} = 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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