Bernoulli's theorem: Bernoullis's theorem states that " when a non viscous liquid flows between two points then the sum of pressure energy, kinetic energy and potential energy per unit mass is always constant at any point in the path of that liquid"
i.e., `P/rho+v^2/2+gh=constant`
`impliesP+rhogh+1/2rhov^2=constant`
Bernoulli's theorem is applicable to non viscous, incompressible and irrotational liquids in streamline flow only.
Proof: Let us consider that a liquid of density `'rho'` is flowing through a pipe of different area of cross sections `A_1` and `A_2` as shown.
Let the liquid enters at `A_1` with a velocity `V_1` and with a pressure `P_1` , density of liquid at `A_1` is say` rho`. Let the liquid leaves the pipe through `A_2` with a velocity `V_2` and pressure `P_2`. density of liquid at `A_2` is say `rho`. Since liquid is incompressible, `rho` is constant.
At region 1 the liquid will move at distance of `V_1 Deltat` where `Deltat` is very small time interval. Similarly at region 2 the liquid will move through a distance `V_2 Deltat`.
Work done on fluid at region `1=W_1=P_1A_1V_1 Deltat=P_1DeltaV`
Work done on fluid at region 2= `1=W_2=P_2A_2V_2 Deltat=P_2DeltaV`
Since same volume of liquid pass through the pipe `Delta` is constant.
`therefore` Work done by fluid `=W_1-W_2`
`=(P_1-P_2)DeltaV to 1`
`because` Liquid is uncompressible `'rho'` is constant . so mass of liquid entering the pipe and leaving the pipe `Deltam` is given by
`Deltam=rho A_1V_1Delta t=rho Delta V`
change in potential energy of liquid
`Delta U= rho g Delta v(h_2-h_1) to 2`
Change in kinetic energy of liquid
`Delta=1/2rhoDeltav(V_2^2-V_1^2)`
From work energy theorem work done = change in energy
` therefore Delta W= Delta U +Delta K`
`therefore(P_1-P_2)Deltav`
`=1/2rhoDeltaV(V_2^2-V_1^2)+rhog DeltaV(h_2-h_1)to 3`
`P_1-P_2=1//2rho(V_2^2-V_1^2)+rhog(h_2-h_1)or`
`P_1+1/2rhoV_1^2+rhogh_1=P_2+1/2rhoV_2^2+rhogh_2`
i.e., sum of pressure energy , potential energy and kinetic energy of the fluid is always constant.
Limitations: Bernoulli's theorem is applicable to non-viscous and uncompressible liquids only.
Applications of Bernoulli's theorem:
1) Dynamic lift on the wings of an aeroplane is due to Bernoulli's theorem.
2) Swinging of a spinning cricket ball is a consequences of Bernoulli's theorem.
3) During cyclones, the roof of thatched houses will fly away. This is a consequence of Bernoulli's theorem.
