State and prove Bernoulli's theorem.

Explanation

According to Bernoulli's theorem, the sum of pressure energy, kinetic energy, and potential energy per unit mass of an incompressible, non-viscous fluid in a streamlined flow remains a constant.

Proof: Let us consider a flow of liquid through a pipe AB as shown in Figure. Let V be the volume of the liquid when it enters A in a time t which is equal to the volume of the liquid leaving B in the same time. Let `a_(A),v_(A)andP_(A)` be the area of cross section of the tube, velocity of the liquid and pressure exerted by the liquid at A respectively.
Let the force exerted by the liquid at A is
`F_(A)=P_(A)a_(A)`
Distance travelled by the liquid in time t is
`d=v_(A)t`
Therefore, the work done is
`W=F_(A)d=P_(A)a_(A)v_(A)t`
But `a_(A)v_(A)t=a_(A)d=V`, volume of the liquid entering at A.
Thus, the work done is the pressure energy (at A),
`W=F_(A)d=P_(A)V`
Pressure energy per unit volume at
`A=("pressure energy")/("volume")=(P_(A)V)/(V)=P_(A)`
Pressure energy per unit mass at
`A=("pressure energy")/("mass")=(P_(A)V)/(m)=(P_A)/(m//V)=(P_A)/(rho)`
Since m is the mass of the liquid entering at A in a given time, therefore, pressure energy of the liquid at A is
`E_(PA)=P_(A)V=P_(A)Vxx((m)/(m))=m(P_A)/(rho)`
Potential energy of the liquid at A,
`PE_(A)=mgh_(A)`,
Due to the flow of liquid, the kinetic energy of the liquid at A,
`KE_(A)=(1)/(2)mv_(A)^(2)`
Therefore, the total energy due to the flow of liquid at A, `E_(A)=EP_(A)+KE_(A)+PE_(A)`
`E_(A)=m(P_A)/(rho)+(1)/(2)mv_(A)^(2)+mgh_(A)`
Similarly, let `a_(B),v_(B)andP_(B)` be the area of cross section of the tube, velocity of the liquid, and pressure exerted by the liquid at B. Calculating the total energy at EB, we get
`E_(B)=m(P_B)/(rho)+(1)/(2)mv_(B)^(2)+mgh_(B)`
From the law of conservation of energy,
`E_(A)=E_(B)`
`m(P_A)/(rho)+(1)/(2)mv_(A)^(2)+mgh_(A)=m(P_B)/(rho)+(1)/(2)mv_(B)^(2)+mgh_(B)`
`(P_A)/(rho)+(1)/(2)mv_(A)^(2)+gh_(A)=(P_B)/(rho)+(1)/(2)mv_(B)^(2)_gh_(B)="constant"`
Thus, the above equation can be written as
`(P)/(rhog)+(1)/(2)(v^2)/(g)+h="constant"`