Viscous Flow in Pipes.pdf

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• Since neither Δpnor are functions of the radialcoordinate, r , it follows that 2τr must also beindependent of r .i.e.,τ = Crlwhere C is a constant2τwC = whereτwis the wall shear stressD2τwr→ τ =D• Thus the pressure drop and wall shear stress arerelated by,4 τ wΔ p = l Dl >> 1 D• A small shear stress can produce a large pressuredifference if the pipe is relatively long
• The above analysis is valid for both laminar andturbulent flow.• To proceed further we must determine therelationship between shear stress and velocity.• For laminar Newtonian flow,uτ = μ ∂∂ydudrΔp 2 τ du ⎛ Δp ⎞= , =− ⎜ ⎟rl r dr ⎝2μl⎠Δp∫du=− rdr2μl∫For pipe flow τ = − μ ( τ > 0)Then⎛ Δp⎞⎜ ⎟⎝2μl⎠2u =− r + C1 C1, where is a constant2DΔpDB.C.: u = 0., r = , so that C1=2 16μl
Page 1 and 2: Chapter 8Viscous Flow in Pipes
Page 3 and 4: 8.1.1 Laminar or Turbulent Flow•
Page 5 and 6: 8.1.2 Entrance Region and FullyDeve
Page 7 and 8: 8.1.3 Pressure and Shear Stress•
Page 9 and 10: 2. Increased Wall Shear Stress:⎡
Page 11 and 12: Nature of flow• The nature of the
Page 13: 8.2.1 From F=ma Applied Directly to
Page 17 and 18: • Average velocityVQ2 2π RVcVcΔ
Page 19 and 20: 8.2.2 From the Navier-Stokes Equati
Page 21 and 22: • It is advantageous to describe
Page 23 and 24: 8.3 Fully developed turbulent flow
Page 25 and 26: 8.3.2 Turbulent shear stress• Tur
Page 27 and 28: Time-averaged Navier-Stokes Equatio
Page 29 and 30: Total stress• Total stressduτ =
Page 31 and 32: 8.3.3 Turbulent Velocity Profile•
Page 33 and 34: Turbulent Velocity Profile• Empir
Page 35 and 36: 8.3.5 Chaos and TurbulenceChaos the
Page 37 and 38: About the difficulty to understand
Page 39 and 40: Pressure drop• For turbulent flow
Page 41 and 42: Friction coefficient• For laminar
Page 43 and 44: • Energy Equation2 2p1 V1 p2 V2+
Page 45 and 46: 8.4.2 Minor Losses• Major losses
Page 47 and 48: Loss coefficients for minor lossFor
Page 49 and 50: Pressure loss• Flow patter and pr
Page 51 and 52: Loss coefficient• Loss coefficien
Page 53 and 54: Loss coefficient• Loss coefficien
Page 55 and 56: Loss coefficient
Page 57 and 58: 8.4.3 Noncircular Conduits• For n

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