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What is bernoulli s principal with proof

Answer» Thanks<br>Bernoulli’s Principle\tFor 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.\tMathematically:- P+ ρv2/2 + ρgh = constant\twhere P= pressure ,E./ Volume=1/2mv2/V = 1/2v2(m/V) = 1/2ρv2E./Volume = mgh/V = (m/V)gh = ρgh\t\tDerive: Bernoulli’s equationAssumptions:\tFluid flow through a pipe of varying width.\tPipe is located at changing heights.\tFluid is incompressible.\tFlow is laminar.\tNo energy is lost due to friction:applicable only to non-viscous fluids.\tMathematically: -\tConsider the fluid initially lying between B and D. In an infinitesimal timeinterval Δt, this fluid would have moved.\tSuppose v1= speed at B and v2= speedat D, initial distance moved by fluid from to C=v1Δt.In the same interval Δtfluid distance moved by D to E = v2Δt.P1= Pressureat A1, 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)\t\t\tNet work done on the fluid is W1 – W2 = (P1A1v1Δt− P2A2v2Δt)\tBy the Equation of continuity Av=constant.\tP1A1 v1Δt - P2A2v2Δt where A1v1Δt =P1ΔV and A2v2Δt = P2ΔV.\t\t\tTherefore Work done = (P1− P2) ΔVequation (a)\tPart of this work goes in changing Kinetic energy, ΔK = (½)m (v22 – v12) and part in gravitational potential energy,ΔU =mg (h2 − h1).\t\t\tThe total change in energy ΔE= ΔK +ΔU = (½) m (v22 – v12) + mg (h2 − h1). (i)\tDensity of the fluid ρ =m/V or m=ρV\tTherefore in small interval of time Δt, small change in mass Δm\tΔm=ρΔV (ii)\t\t\tPutting the value from equation (ii) to (i)\tΔE = 1/2 ρΔV (v22 – v12) + ρgΔV (h2 − h1) equation(b)\tBy using work-energy theorem: W = ΔE\tFrom (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).\t\t\tAfter rearranging we get,P1 + (1/2) ρ v12 + ρg h1 = (1/2) ρ v22 + ρg h2\tP+(1/2) ρv2+ρg h = constant.\tThis is the Bernoulli’s equation.


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