Bernoulli Theorem

• For a streamline fluid flow, the sum of the pressure (P), the kinetic energy per unit volume (ρv²/2) and the potential energy per unit volume (ρgh) remain constant.
• Mathematically:- P+ ρv²/2 + ρgh = constant
  • where P= pressure ,
  • / Volume=1/2mv²/V = 1/2v²(m/V) = 1/2ρv²
  • /Volume = mgh/V = (m/V)gh = ρgh

Derive: Bernoulli’s equation

Assumptions:

1. Fluid flow through a pipe of varying width.
2. Pipe is located at changing heights.
3. Fluid is incompressible.
4. Flow is laminar.
5. No energy is lost due to friction:applicable only to non-viscous fluids.

Mathematically: -

• Consider the fluid initially lying between B and D. In an infinitesimal time interval Δt, this fluid would have moved.

• Suppose v₁= speed at B and v₂= speed at D, initial distance moved by fluid from to C=v₁Δt.
• In the same interval Δtfluid distance moved by D to E = v₂Δt.
• P₁= Pressureat A_1, P₂=Pressure at A₂.
• Work done on the fluid atleft end (BC) W₁ = P₁A₁(v₁Δt).
• Work done by the fluid at the other end (DE)W₂ = P₂A₂(v₂Δt)

• Net work done on the fluid is W₁– W₂ = (P₁A₁v₁Δt− P₂A₂v₂Δt)
• By the Equation of continuity Av=constant.
• P₁A₁ v₁Δt - P₂A₂v₂Δt where A₁v₁Δt =P₁ΔV and A₂v₂Δt = P₂ΔV.
• Therefore Work done = (P₁− P₂) ΔVequation (a)
  • Part of this work goes in changing Kinetic energy, ΔK = (½)m (v₂²– v₁²) and part in gravitational potential energy,ΔU =mg (h₂ − h₁).
• The total change in energy ΔE= ΔK +ΔU = (½) m (v₂²– v₁²) + mg (h₂ − h₁). (i)
• Density of the fluid ρ =m/V or m=ρV
• Therefore in small interval of time Δt, small change in mass Δm
  • Δm=ρΔV (ii)
• Putting the value from equation (ii) to (i)
• ΔE = 1/2 ρΔV (v₂²– v₁²) + ρgΔV (h₂ − h₁)  equation(b)
• By using work-energy theorem: W = ΔE
  • From (a) and (b)
  • (P₁-P₂) ΔV =(1/2) ρΔV (v₂²– v₁²) + ρgΔV (h₂ − h₁)
  • P₁-P₂= 1/2ρv₂² - 1/2ρv₁²+ρgh₂ -ρgh₁(By cancelling ΔV from both the sides).
• After rearranging we get,P₁+ (1/2) ρ v₁² + ρg h₁ = (1/2) ρ v₂² + ρg h₂
• P+(1/2) ρv²+ρg h = constant.
• This is the Bernoulli’s equation.

• The flow of an ideal fluid in a pipe ofvarying cross section. The fluid in asection of length v1Δt moves to the section of length v2Δt in time Δt.